Watch an LLM Think

How Monte Carlo Tree Search turns single-pass reasoning into a branching exploration of ideas.

/* =========================================================
CSS CUSTOM PROPERTIES (scoped to container)
========================================================= */

mcts-explorable {

--bg: #1a1a2e;
--bg-dark: #0d0d1a;
--accent: #e94560;
--accent2: #4a9eff;
--text: #e0e0e0;
--text-dim: #888;

/* Base styles (were on body) */
background: var(--bg);
color: var(--text);
font-family: 'Courier New', Courier, monospace;

/* Break out of Hugo's article container to go full-width */
width: 100vw;
position: relative;
left: 50%;
right: 50%;
margin-left: -50vw;
margin-right: -50vw;
}

mcts-explorable *, #mcts-explorable *::before, #mcts-explorable *::after {

box-sizing: border-box;
}

/* =========================================================
HERO SECTION
========================================================= */

mcts-explorable #hero {

background: linear-gradient(180deg, var(--bg-dark) 0%, var(--bg) 100%);
text-align: center;
padding: 80px 24px 60px;
}

mcts-explorable #hero .subtitle {

font-size: 11px;
letter-spacing: 4px;
color: var(--accent);
text-transform: uppercase;
margin: 0 0 16px 0;
font-variant: small-caps;
}

mcts-explorable .scroll-indicator {

margin-top: 30px;
color: var(--text-dim);
font-size: 1.5rem;
animation: bounce 2s ease infinite;
}

@keyframes bounce {
0%, 100% { transform: translateY(0); }
50% { transform: translateY(8px); }
}

mcts-explorable #hero h1 {

font-size: 2.5rem;
color: #ffffff;
font-weight: bold;
margin: 0 0 20px 0;
line-height: 1.2;
}

mcts-explorable #hero .description {

color: var(--text-dim);
font-size: 1rem;
max-width: 600px;
margin: 0 auto;
line-height: 1.6;
}

/* =========================================================
TWO-PANEL SCROLLYTELLING LAYOUT
========================================================= */

mcts-explorable #scrollytelling {

display: flex;
align-items: flex-start;
}

/* Left: narrative column */

mcts-explorable #narrative {

width: 40%;
padding: 24px;
overflow-y: auto;
line-height: 1.7;
font-size: 0.95rem;
}

mcts-explorable #narrative p {

max-width: 45ch;
margin-bottom: 16px;
}

mcts-explorable #narrative h2 {

margin-top: 40px;
margin-bottom: 16px;
}

/* Right: sticky tree panel */

mcts-explorable #tree-panel {

width: 60%;
position: sticky;
top: 0;
height: 100vh;
background: var(--bg-dark);
display: flex;
flex-direction: column;
align-items: center;
justify-content: center;
}

mcts-explorable #tree-svg {

width: 100%;
height: 80%;
}

/* Node detail inspector */

mcts-explorable #node-detail {

width: 100%;
padding: 12px 20px;
font-family: 'Courier New', Courier, monospace;
font-size: 0.8rem;
color: var(--text-dim);
border-left: 3px solid var(--accent);
background: rgba(233, 69, 96, 0.05);
min-height: 60px;
margin: 0 20px;
max-width: calc(100% - 40px);
white-space: pre-wrap;
word-break: break-word;
}

/* =========================================================
NARRATIVE SECTIONS
========================================================= */

mcts-explorable #narrative section {

min-height: 80vh;
padding: 40px 0;
border-bottom: 1px solid rgba(255, 255, 255, 0.1);
}

mcts-explorable #narrative section:last-child {

border-bottom: none;
}

mcts-explorable #narrative section h2 {

color: var(--accent);
letter-spacing: 1px;
text-transform: uppercase;
font-size: 0.9rem;
margin: 0 0 20px 0;
font-weight: bold;
}

mcts-explorable #narrative section p {

color: var(--text);
line-height: 1.7;
font-size: 0.9rem;
margin: 0 0 16px 0;
}

/* =========================================================
NODE & EDGE ANIMATIONS
========================================================= */

mcts-explorable .node-group {

transition: opacity 300ms ease, transform 300ms ease;
}

mcts-explorable .node-group.entering {

opacity: 0;
transform-origin: center;
}

@keyframes pulse {
0% { transform: scale(1); }
50% { transform: scale(1.15); }
100% { transform: scale(1); }
}

mcts-explorable .node-group.pulsing circle {

stroke: var(--accent);
transition: stroke 300ms ease;
}

mcts-explorable .edge {

transition: stroke 400ms ease, stroke-width 400ms ease;
}

mcts-explorable .edge.pulsing {

stroke: var(--accent) !important;
stroke-width: 3 !important;
}

@keyframes glow-pulse {
0%, 100% { opacity: 1; }
50% { opacity: 0.7; }
}

mcts-explorable .node-group.highlighted circle {

animation: glow-pulse 1.5s ease infinite;
}

/* =========================================================
NARRATIVE CONTENT STYLES
========================================================= */

mcts-explorable .puzzle-box {

border: 1px solid var(--accent);
padding: 16px 20px;
margin: 16px 0;
border-radius: 4px;
background: rgba(233, 69, 96, 0.05);
}

mcts-explorable .puzzle-box p { margin: 4px 0; }

mcts-explorable .puzzle-hint {

color: var(--text-dim);
font-size: 0.85rem;
font-style: italic;
margin-top: 8px !important;
}

mcts-explorable .llm-output {

background: var(--bg-dark);
border-left: 3px solid var(--accent);
padding: 16px;
margin: 16px 0;
font-size: 0.85rem;
line-height: 1.6;
white-space: pre-wrap;
}

mcts-explorable .flaw-annotation {

color: #ff6b6b;
font-size: 0.85rem;
font-style: italic;
}

mcts-explorable .transition-text {

color: var(--accent);
font-style: italic;
font-size: 1.1rem;
margin-top: 24px;
line-height: 1.6;
}

mcts-explorable .formula-box {

background: var(--bg-dark);
padding: 12px 20px;
margin: 16px 0;
border-radius: 4px;
font-size: 1.1rem;
text-align: center;
letter-spacing: 1px;
}

mcts-explorable .accent { color: var(--accent); }

mcts-explorable .accent2 { color: var(--accent2); }

mcts-explorable .branch-comparison {

margin: 16px 0;
}

mcts-explorable .branch-example {

background: var(--bg-dark);
padding: 10px 16px;
margin: 8px 0;
border-radius: 4px;
font-size: 0.9rem;
}

mcts-explorable .branch-label {

color: var(--accent);
font-weight: bold;
margin-right: 8px;
}

mcts-explorable .hint-text {

color: var(--text-dim);
font-size: 0.85rem;
font-style: italic;
}

mcts-explorable .score-high { color: #4ade80; font-weight: bold; }

mcts-explorable .score-low { color: #f87171; font-weight: bold; }

/* =========================================================
INTERACTIVE CONTROLS: UCB1, ROLLOUT, BACKPROP
========================================================= */

mcts-explorable .ucb1-slider { margin: 16px 0; }

mcts-explorable .ucb1-slider label { display: block; color: var(--text-dim); font-size: 0.8rem; margin-bottom: 6px; }

mcts-explorable .ucb1-slider input[type="range"] { width: 100%; accent-color: var(--accent); }

mcts-explorable .slider-labels { display: flex; justify-content: space-between; font-size: 0.75rem; color: var(--text-dim); margin-top: 4px; }

mcts-explorable .slider-labels .current-value { color: var(--accent); font-weight: bold; }

mcts-explorable .rollout-controls,

mcts-explorable .backprop-controls { margin: 16px 0; display: flex; align-items: center; gap: 12px; }

mcts-explorable .control-btn { background: var(--bg-dark); border: 1px solid var(--accent); color: var(--accent); padding: 6px 16px; cursor: pointer; font-family: inherit; font-size: 0.85rem; border-radius: 3px; }

mcts-explorable .control-btn:hover { background: rgba(233, 69, 96, 0.1); }

mcts-explorable .speed-buttons { display: flex; gap: 4px; }

mcts-explorable .speed-btn { background: var(--bg-dark); border: 1px solid #ffffff20; color: var(--text-dim); padding: 4px 10px; cursor: pointer; font-family: inherit; font-size: 0.8rem; border-radius: 3px; }

mcts-explorable .speed-btn.active { border-color: var(--accent); color: var(--accent); }

mcts-explorable .math-display {

background: var(--bg-dark);
padding: 12px 16px;
margin: 12px 0;
border-radius: 4px;
font-size: 0.85rem;
line-height: 1.6;
min-height: 40px;
border-left: 2px solid var(--accent2);
}

/* =========================================================
ASIDE / COLLAPSIBLE
========================================================= */

mcts-explorable .aside { margin: 20px 0; border: 1px solid #ffffff15; border-radius: 4px; }

mcts-explorable .aside summary { padding: 10px 16px; cursor: pointer; color: var(--accent2); font-size: 0.9rem; }

mcts-explorable .aside summary:hover { background: rgba(74, 158, 255, 0.05); }

mcts-explorable .aside-content { padding: 0 16px 16px; }

mcts-explorable .split-comparison { display: flex; gap: 16px; margin: 12px 0; }

mcts-explorable .comparison-panel { flex: 1; background: var(--bg-dark); padding: 12px; border-radius: 4px; text-align: center; }

mcts-explorable .comparison-panel h4 { margin: 0 0 8px; font-size: 0.8rem; color: var(--text-dim); text-transform: uppercase; letter-spacing: 1px; }

/* =========================================================
SAMPLING CONTROLS
========================================================= */

mcts-explorable .sampling-controls,

mcts-explorable .voting-controls { margin: 12px 0; }

mcts-explorable .strategy-toggles,

mcts-explorable .voting-controls { display: flex; gap: 6px; flex-wrap: wrap; }

mcts-explorable .strategy-btn { background: var(--bg-dark); border: 1px solid #ffffff20; color: var(--text-dim); padding: 5px 12px; cursor: pointer; font-family: inherit; font-size: 0.8rem; border-radius: 3px; }

mcts-explorable .strategy-btn.active { border-color: var(--accent); color: var(--accent); }

mcts-explorable .strategy-btn:hover { background: rgba(233, 69, 96, 0.05); }

mcts-explorable #answer-histogram { margin: 16px 0; min-height: 80px; }

/* =========================================================
UCB1 TOOLTIP
========================================================= */

mcts-explorable #ucb1-tooltip {

position: fixed;
display: none;
background: var(--bg-dark);
border: 1px solid var(--accent);
padding: 8px 12px;
font-family: 'Courier New', Courier, monospace;
font-size: 11px;
color: var(--text);
line-height: 1.5;
z-index: 1000;
pointer-events: none;
white-space: pre;
border-radius: 4px;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.5);
}

/* =========================================================
RESPONSIVE BREAKPOINT — STACKED LAYOUT
========================================================= */
@media (max-width: 1000px) {
#mcts-explorable #scrollytelling {
flex-direction: column;
}

#mcts-explorable #narrative {
width: 100%;
order: 2;
}

#mcts-explorable #tree-panel {
width: 100%;
height: 50vh;
position: relative;
top: auto;
order: 1;
}
}

/* =========================================================
KEYBOARD ACCESSIBILITY — FOCUS INDICATORS
========================================================= */

mcts-explorable .control-btn:focus,

mcts-explorable .strategy-btn:focus,

mcts-explorable .speed-btn:focus,

mcts-explorable input[type="range"]:focus {

outline: 2px solid var(--accent);
outline-offset: 2px;
}

An Explorable Explanation

Watch an LLM Think

How Monte Carlo Tree Search turns single-pass reasoning into a branching exploration of ideas

▼

One Shot, One Chance

Here is a classic logic puzzle:

A says: "B is a knave."

B says: "We are the same type."

Is A a knight or a knave?

(Knights always tell the truth. Knaves always lie.)

Let's ask an LLM to solve it:

The LLM assumed A is a knave and never tested the alternative. It reached a confident answer without checking all the constraints.

What if the LLM could explore both assumptions simultaneously, and backtrack from the one that leads to contradiction?

The Tree Appears

Instead of committing to a single reasoning path, MCTS explores multiple paths simultaneously. It builds a tree where each branch represents a different line of reasoning.

But how does it decide which branch to explore next? It uses a formula called UCB1:

UCB1 = value + c · √(ln(parent_visits) / visits)

The first term (value) measures how promising a path has been so far. The second term (exploration) favors paths that haven't been tried much.

The constant c controls the balance. Try adjusting it:

exploration constant (c)

0 (greedy)
√2
4 (explore)

Branching Thoughts

Once UCB1 selects a node, the LLM generates a new reasoning step from that point. The tree grows.

Branch A:
"Let's assume A is a knight..."

Branch B:
"Let's assume A is a knave..."

Each branch represents a different assumption the LLM is testing. This is structured exploration, not random guessing.

Click any node in the tree to see its full reasoning.

Following the Thread

From the expanded node, the LLM keeps reasoning until it reaches an answer or hits a maximum depth. This is called a rollout.

▶ Play

1x
2x
4x

Watch the nodes appear one by one as the LLM follows its reasoning to a conclusion.

How is this different from regular MCTS?

In classical MCTS, rollout nodes are simulated but discarded. Only the evaluation score is kept. Here, every reasoning step is preserved in the tree.

This means we keep a complete record of every reasoning trace the LLM explored.

What's a Good Answer?

The rollout reaches an answer, but how do we score it? The evaluator checks whether the reasoning is logically consistent with the puzzle's constraints.

A correct derivation with no contradictions scores 1.0. A derivation that hits a contradiction scores 0.0. Weak reasoning that reaches the right answer by luck scores somewhere in between.

Scores Flow Upward

When a reasoning path reaches a conclusion and gets scored, the result propagates back up the tree. Each ancestor node updates its average value and visit count.

Step ↑
Reset

Good answers make nearby paths more attractive for future exploration. The tree learns where to search next.

Many Paths, One Answer

Which paths should we look at?

After 20 simulations, the tree has explored many reasoning paths. But which ones matter? Different sampling strategies highlight different aspects:

value
visits
diverse
top-k

Showing paths with the highest average values.

How do we pick the final answer?

With multiple paths reaching conclusions, we can use voting to decide:

majority vote
weighted vote

When multiple independent reasoning paths agree, we can be more confident in the answer.

Try It Yourself

This section requires a local Ollama instance to run live MCTS searches.

Connect Ollama at localhost:11434 to try this.

Click a node to inspect it.

"use strict";(()=>{var f="http://www.w3.org/2000/svg",B=class{constructor(e,s={}){this.svg=e,this.nodeRadius=s.nodeRadius||18,this.levelHeight=s.levelHeight||90,this.siblingSpacing=s.siblingSpacing||50,this.nodes=new Map,this.currentTree=null,this._nextLeafX=0,this._tooltip=document.getElementById("ucb1-tooltip"),this._nodeDetail=document.getElementById("node-detail")}layout(e){this._nextLeafX=0;let s=new Map;this._assignPositions(e,0,s),this._separateSubtrees(e,s);let t=this.nodeRadius+10,i=1/0;for(let n of s.values())n.x<i&&(i=n.x);let a=t-i;if(a!==0)for(let n of s.values())n.x+=a;return s}_assignPositions(e,s,t){let i=s*this.levelHeight+this.nodeRadius+10;if(!e.children||e.children.length===0){let l=this._nextLeafX;this._nextLeafX+=this.siblingSpacing,t.set(e.id,{x:l,y:i,data:e});return}for(let l of e.children)this._assignPositions(l,s+1,t);let a=0;for(let l of e.children)a+=t.get(l.id).x;let n=a/e.children.length;t.set(e.id,{x:n,y:i,data:e})}_separateSubtrees(e,s){if(!e.children||e.children.length<=1){e.children&&e.children.length===1&&this._separateSubtrees(e.children[0],s);return}for(let i of e.children)this._separateSubtrees(i,s);for(let i=1;i<e.children.length;i++){let a=e.children[i-1],n=e.children[i],l=this._getRightmostX(a,s),r=this._getLeftmostX(n,s)-l;if(r<this.siblingSpacing){let u=this.siblingSpacing-r;this._shiftSubtree(n,u,s)}}let t=0;for(let i of e.children)t+=s.get(i.id).x;s.get(e.id).x=t/e.children.length}_getRightmostX(e,s){let t=s.get(e.id).x;if(e.children)for(let i of e.children){let a=this._getRightmostX(i,s);a>t&&(t=a)}return t}_getLeftmostX(e,s){let t=s.get(e.id).x;if(e.children)for(let i of e.children){let a=this._getLeftmostX(i,s);a<t&&(t=a)}return t}_shiftSubtree(e,s,t){if(t.get(e.id).x+=s,e.children)for(let i of e.children)this._shiftSubtree(i,s,t)}render(e,s={}){for(this.currentTree=e;this.svg.firstChild;)this.svg.removeChild(this.svg.firstChild);this.nodes.clear();let t=this.layout(e),i=document.createElementNS(f,"g");i.setAttribute("data-layer","edges"),this.svg.appendChild(i);let a=document.createElementNS(f,"g");a.setAttribute("data-layer","nodes"),this.svg.appendChild(a),this._renderEdges(e,t,i),this._renderNodes(e,t,s,a),this.autoFit()}_renderEdges(e,s,t){if(!e.children)return;let i=s.get(e.id);for(let a of e.children){let n=s.get(a.id),l=document.createElementNS(f,"line");l.setAttribute("x1",String(i.x)),l.setAttribute("y1",String(i.y+this.nodeRadius)),l.setAttribute("x2",String(n.x)),l.setAttribute("y2",String(n.y-this.nodeRadius)),l.setAttribute("stroke","#ffffff15"),l.setAttribute("stroke-width","1.5"),l.setAttribute("data-from",e.id),l.setAttribute("data-to",a.id),l.classList.add("edge"),t.appendChild(l),this._renderEdges(a,s,t)}}_renderNodes(e,s,t,i){let a=s.get(e.id),n=t[e.id],l=document.createElementNS(f,"g");l.setAttribute("transform",translate(${a.x}, ${a.y})),l.setAttribute("data-node-id",e.id),l.classList.add("node-group"),l.style.cursor="pointer",l.style.transformOrigin="center",l.setAttribute("role","button");let v=(e.state||"").slice(0,50);l.setAttribute("aria-label","Node: "+v+". Value "+(e.value||0).toFixed(2)+", "+(e.visits||0)+" visits"),l.setAttribute("tabindex","0");let r=document.createElementNS(f,"circle");r.setAttribute("cx","0"),r.setAttribute("cy","0"),r.setAttribute("r",String(this.nodeRadius)),r.setAttribute("stroke-width","2"),e.is_terminal?(r.setAttribute("stroke","var(--accent)"),r.setAttribute("fill","rgba(233,69,96,0.15)")):(r.setAttribute("stroke",n||"#ffffff30"),r.setAttribute("fill","transparent")),n&&r.setAttribute("stroke",n),l.appendChild(r);let u=document.createElementNS(f,"text");u.setAttribute("x","0"),u.setAttribute("y",String(this.nodeRadius+14)),u.setAttribute("text-anchor","middle"),u.setAttribute("font-family","monospace"),u.setAttribute("font-size","10"),u.setAttribute("fill","var(--text-dim)"),u.textContent=v=${(e.value||0).toFixed(2)},l.appendChild(u);let h=document.createElementNS(f,"text");if(h.setAttribute("x","0"),h.setAttribute("y",String(this.nodeRadius+26)),h.setAttribute("text-anchor","middle"),h.setAttribute("font-family","monospace"),h.setAttribute("font-size","10"),h.setAttribute("fill","var(--text-dim)"),h.textContent=n=${e.visits||0},l.appendChild(h),e.answer){let m=document.createElementNS(f,"text");m.setAttribute("x","0"),m.setAttribute("y",String(-this.nodeRadius-8)),m.setAttribute("text-anchor","middle"),m.setAttribute("font-family","monospace"),m.setAttribute("font-size","9"),m.setAttribute("fill","var(--accent)"),m.textContent=e.answer,l.appendChild(m)}if(l.addEventListener("click",()=>{this._nodeDetail&&(this._nodeDetail.textContent=e.state||"(no state)")}),l.addEventListener("keydown",m=>{(m.key==="Enter"||m.key===" ")&&(m.preventDefault(),this._nodeDetail&&(this._nodeDetail.textContent=e.state||"(no state)"))}),l.addEventListener("mouseenter",m=>{this._showTooltip(m,e)}),l.addEventListener("mousemove",m=>{this._tooltip&&this._tooltip.style.display==="block"&&this._positionTooltip(m)}),l.addEventListener("mouseleave",()=>{this._hideTooltip()}),i.appendChild(l),this.nodes.set(e.id,{x:a.x,y:a.y,data:e,element:l}),e.children)for(let m of e.children)this._renderNodes(m,s,t,i)}_showTooltip(e,s){if(!this._tooltip)return;let t=1.414,i;if(!s.visits||s.visits===0)i="not yet visited";else{let a=this._findParentVisits(s.id);if(a===null)i=exploitation: ${(s.value||0).toFixed(3)}
(root node);else{let n=s.value||0,l=t*Math.sqrt(Math.log(a)/s.visits),v=n+l;i=exploitation: ${n.toFixed(3)}
exploration: ${l.toFixed(3)}
UCB1: ${v.toFixed(3)}}}this._tooltip.textContent=i,this._tooltip.style.display="block",this._positionTooltip(e)}_positionTooltip(e){if(!this._tooltip)return;let s=12;this._tooltip.style.left=e.clientX+s+"px",this._tooltip.style.top=e.clientY+s+"px"}_hideTooltip(){this._tooltip&&(this._tooltip.style.display="none")}_findParentVisits(e){for(let[,s]of this.nodes)if(s.data.children){for(let t of s.data.children)if(t.id===e)return s.data.visits||0}return null}autoFit(){let s=1/0,t=1/0,i=-1/0,a=-1/0;for(let u of this.nodes.values()){let h=u.x-this.nodeRadius,m=u.x+this.nodeRadius,c=u.y-this.nodeRadius-20,o=u.y+this.nodeRadius+30;h<s&&(s=h),m>i&&(i=m),c<t&&(t=c),o>a&&(a=o)}if(s===1/0)return;let n=s-40,l=t-40,v=i-s+2*40,r=a-t+2*40;this.svg.setAttribute("viewBox",${n} ${l} ${v} ${r})}updateNode(e,s){let t=this.nodes.get(e);if(!t)return;let i=t.element,a=i.querySelector("circle"),n=i.querySelectorAll("text");s.stroke&&a&&a.setAttribute("stroke",s.stroke),s.fill&&a&&a.setAttribute("fill",s.fill),s.value!==void 0&&n.length>0&&(n[0].textContent=v=${s.value.toFixed(2)}),s.visits!==void 0&&n.length>1&&(n[1].textContent=n=${s.visits})}highlightPath(e,s){let t=new Set(e);for(let a of this.nodes.values())a.element.style.opacity="0.3";let i=this.svg.querySelectorAll("line");for(let a of i)a.style.opacity="0.3";for(let a of e){let n=this.nodes.get(a);if(n){n.element.style.opacity="1",n.element.classList.add("highlighted");let l=n.element.querySelector("circle");l&&l.setAttribute("stroke",s)}}for(let a of i){let n=a.getAttribute("data-from"),l=a.getAttribute("data-to");n&&l&&t.has(n)&&t.has(l)&&(a.style.opacity="1",a.setAttribute("stroke",s))}}addNodeAnimated(e){let s=this.nodes.get(e);if(!s)return;let t=s.element,{x:i,y:a}=s;t.style.opacity="0",t.style.transform=translate(${i}px, ${a}px) scale(0),requestAnimationFrame(()=>{t.style.opacity="1",t.style.transform=translate(${i}px, ${a}px) scale(1)})}pulseNode(e){let s=this.nodes.get(e);if(!s)return;let t=s.element;t.classList.remove("pulsing"),t.offsetWidth,t.style.animation="pulse 400ms ease",t.classList.add("pulsing");let i=()=>{t.style.animation="",t.classList.remove("pulsing"),t.removeEventListener("animationend",i)};t.addEventListener("animationend",i)}pulseEdge(e,s){let t=this.svg.querySelector(line[data-from="${e}"][data-to="${s}"]);t&&(t.classList.add("pulsing"),setTimeout(()=>{t.classList.remove("pulsing")},400))}clearHighlights(){for(let s of this.nodes.values()){s.element.style.opacity="1",s.element.classList.remove("highlighted");let t=s.element.querySelector("circle");t&&(s.data.is_terminal?t.setAttribute("stroke","var(--accent)"):t.setAttribute("stroke","#ffffff30"))}let e=this.svg.querySelectorAll("line");for(let s of e)s.style.opacity="1",s.setAttribute("stroke","#ffffff15")}};var w=class{constructor(e,s){this.renderer=e,this.data=s,this.currentSection=null,this.pendingAnimation=null}init(){let e=new IntersectionObserver(s=>{s.forEach(t=>{if(t.isIntersecting){let i=t.target.dataset.section;i&&i!==this.currentSection&&this.onSectionEnter(i)}})},{threshold:.3});document.querySelectorAll("section[data-section]").forEach(s=>e.observe(s))}onSectionEnter(e){this.pendingAnimation&&(cancelAnimationFrame(this.pendingAnimation),this.pendingAnimation=null),this.currentSection=e,this.transitionTo(e)}_findPhase(e,s){return this.data.simulations.find(t=>t.step===e&&t.phase===s)}transitionTo(e){let s=document.getElementById("node-detail");switch(e){case"1":this.renderer.svg.style.opacity="0",s&&(s.textContent="");break;case"2":{this.renderer.svg.style.opacity="1";let t=this._findPhase(5,"backprop");t&&this.renderer.render(t.tree);break}case"3":{let t=this._findPhase(6,"expand");t&&(this.renderer.render(t.tree),t.new_node_id&&setTimeout(()=>this.renderer.pulseNode(t.new_node_id),300));break}case"4":{let t=this._findPhase(6,"rollout");t&&this.renderer.render(t.tree);break}case"4.5":{let t=this._findPhase(6,"backprop");if(t&&(this.renderer.render(t.tree),t.backprop_path&&t.backprop_path.length>0)){let i=t.backprop_path[0];this.renderer.updateNode(i,{stroke:"#e94560"})}break}case"5":{let t=this._findPhase(6,"backprop");t&&this.renderer.render(t.tree);break}case"6":{let t=this.data.simulations[this.data.simulations.length-1].tree;this.renderer.svg.style.opacity="1",this.renderer.render(t),document.querySelector(".strategy-btn[data-strategy].active")?.click();break}case"7":{for(;this.renderer.svg.firstChild;)this.renderer.svg.removeChild(this.renderer.svg.firstChild);break}}}};var E={puzzle:{question:"A says 'B is a knave.' B says 'We are the same type.' What is A?",correct_answer:"A is a knight",single_pass_wrong:"A says 'B is a knave,' which suggests A might be a knave trying to cast suspicion on B. If A is a knave, then A's statement is a lie, meaning B is actually a knight. B says 'We are the same type,' and since B would be a knight telling the truth, A and B must indeed be the same type. This confirms A and B are both knaves \u2014 wait, but B is a knight in this chain, so they'd have to both be knights. That seems fine: knaves lie and knights tell the truth, and both statements hold together neatly. ANSWER: A is a knave."},simulations:[{step:1,phase:"select",selected_path:["0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:0,visits:0,is_terminal:!1,answer:null,children:[]}},{step:1,phase:"expand",selected_path:["0"],new_node_id:"0.0",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]}},{step:1,phase:"rollout",rollout_path:["0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]}},{step:1,phase:"backprop",backprop_path:["0.0","0"],score:1,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:2,phase:"select",selected_path:["0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:2,phase:"expand",selected_path:["0"],new_node_id:"0.1",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]}},{step:2,phase:"rollout",rollout_path:["0.1"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]}},{step:2,phase:"backprop",backprop_path:["0.1","0"],score:0,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:3,phase:"select",selected_path:["0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:3,phase:"expand",selected_path:["0"],new_node_id:"0.2",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]}},{step:3,phase:"rollout",rollout_path:["0.2"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]}},{step:3,phase:"backprop",backprop_path:["0.2","0"],score:.6,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:4,phase:"select",selected_path:["0","0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:4,phase:"expand",selected_path:["0","0.0"],new_node_id:"0.0.0",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:4,phase:"rollout",rollout_path:["0.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:4,phase:"backprop",backprop_path:["0.0.0","0.0","0"],score:1,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.65,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:5,phase:"select",selected_path:["0","0.1"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.65,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:5,phase:"expand",selected_path:["0","0.1"],new_node_id:"0.1.0",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.65,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:5,phase:"rollout",rollout_path:["0.1.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.65,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:1,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:5,phase:"backprop",backprop_path:["0.1.0","0.1","0"],score:0,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.52,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:6,phase:"select",selected_path:["0","0.2"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.52,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}},{step:6,phase:"expand",selected_path:["0","0.2"],new_node_id:"0.2.0",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.52,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]}]}},{step:6,phase:"rollout",rollout_path:["0.2.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.52,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:1,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]}]}},{step:6,phase:"backprop",backprop_path:["0.2.0","0.2","0"],score:.6,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:7,phase:"select",selected_path:["0","0.0","0.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:7,phase:"expand",selected_path:["0","0.0","0.0.0"],new_node_id:"0.0.0.0",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:7,phase:"rollout",rollout_path:["0.0.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:7,phase:"backprop",backprop_path:["0.0.0.0","0.0.0","0.0","0"],score:1,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:8,phase:"select",selected_path:["0","0.1","0.1.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:8,phase:"expand",selected_path:["0","0.1","0.1.0"],new_node_id:"0.1.0.0",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:0,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:8,phase:"rollout",rollout_path:["0.1.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:1,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:0,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:8,phase:"backprop",backprop_path:["0.1.0.0","0.1.0","0.1","0"],score:0,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.525,visits:8,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:9,phase:"select",selected_path:["0","0.2","0.2.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.525,visits:8,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[]}]}]}},{step:9,phase:"expand",selected_path:["0","0.2","0.2.0"],new_node_id:"0.2.0.0",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.525,visits:8,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:9,phase:"rollout",rollout_path:["0.2.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.525,visits:8,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:1,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:9,phase:"backprop",backprop_path:["0.2.0.0","0.2.0","0.2","0"],score:.6,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:9,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:10,phase:"select",selected_path:["0","0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:9,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:10,phase:"expand",selected_path:["0","0.0"],new_node_id:"0.0.1",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:9,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:10,phase:"rollout",rollout_path:["0.0.1","0.0.1.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5333333333333333,visits:9,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:1,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:10,phase:"backprop",backprop_path:["0.0.1.0","0.0.1","0.0","0"],score:.9,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5700000000000001,visits:10,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.975,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:11,phase:"select",selected_path:["0","0.0","0.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5700000000000001,visits:10,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.975,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:11,phase:"expand",selected_path:["0","0.0","0.0.0"],new_node_id:"0.0.0.1",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5700000000000001,visits:10,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.975,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:11,phase:"rollout",rollout_path:["0.0.0.1","0.0.0.1.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5700000000000001,visits:10,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.975,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:1,visits:2,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:11,phase:"backprop",backprop_path:["0.0.0.1.0","0.0.0.1","0.0.0","0.0","0"],score:.85,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5954545454545456,visits:11,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.95,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9500000000000001,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:12,phase:"select",selected_path:["0","0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5954545454545456,visits:11,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.95,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9500000000000001,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:12,phase:"expand",selected_path:["0","0.0"],new_node_id:"0.0.2",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5954545454545456,visits:11,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.95,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9500000000000001,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:12,phase:"rollout",rollout_path:["0.0.2","0.0.2.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.5954545454545456,visits:11,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.95,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9500000000000001,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:12,phase:"backprop",backprop_path:["0.0.2.0","0.0.2","0.0","0"],score:.95,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6250000000000001,visits:12,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9500000000000001,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9500000000000001,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:13,phase:"select",selected_path:["0","0.0","0.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6250000000000001,visits:12,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9500000000000001,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9500000000000001,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:13,phase:"expand",selected_path:["0","0.0","0.0.0"],new_node_id:"0.0.0.2",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6250000000000001,visits:12,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9500000000000001,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9500000000000001,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:13,phase:"rollout",rollout_path:["0.0.0.2","0.0.0.2.0","0.0.0.2.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6250000000000001,visits:12,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9500000000000001,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9500000000000001,visits:3,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:13,phase:"backprop",backprop_path:["0.0.0.2.0.0","0.0.0.2.0","0.0.0.2","0.0.0","0.0","0"],score:.8,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6384615384615386,visits:13,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9285714285714286,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9125000000000001,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:14,phase:"select",selected_path:["0","0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6384615384615386,visits:13,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9285714285714286,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9125000000000001,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:14,phase:"expand",selected_path:["0","0.0"],new_node_id:"0.0.3",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6384615384615386,visits:13,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9285714285714286,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9125000000000001,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:14,phase:"rollout",rollout_path:["0.0.3","0.0.3.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6384615384615386,visits:13,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9285714285714286,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9125000000000001,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:14,phase:"backprop",backprop_path:["0.0.3.0","0.0.3","0.0","0"],score:.9,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6571428571428574,visits:14,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.925,visits:8,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9125000000000001,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:15,phase:"select",selected_path:["0","0.0","0.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6571428571428574,visits:14,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.925,visits:8,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9125000000000001,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:15,phase:"expand",selected_path:["0","0.0","0.0.0"],new_node_id:"0.0.0.3",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6571428571428574,visits:14,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.925,visits:8,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9125000000000001,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:15,phase:"rollout",rollout_path:["0.0.0.3","0.0.0.3.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6571428571428574,visits:14,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.925,visits:8,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9125000000000001,visits:4,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:15,phase:"backprop",backprop_path:["0.0.0.3.0","0.0.0.3","0.0.0","0.0","0"],score:1,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6800000000000002,visits:15,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9333333333333333,visits:9,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.93,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:16,phase:"select",selected_path:["0","0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6800000000000002,visits:15,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9333333333333333,visits:9,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.93,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:16,phase:"expand",selected_path:["0","0.0"],new_node_id:"0.0.4",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6800000000000002,visits:15,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9333333333333333,visits:9,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.93,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:16,phase:"rollout",rollout_path:["0.0.4","0.0.4.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6800000000000002,visits:15,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9333333333333333,visits:9,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.93,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:16,phase:"backprop",backprop_path:["0.0.4.0","0.0.4","0.0","0"],score:.7,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6812500000000001,visits:16,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9099999999999999,visits:10,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.93,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:17,phase:"select",selected_path:["0","0.0","0.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6812500000000001,visits:16,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9099999999999999,visits:10,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.93,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:17,phase:"expand",selected_path:["0","0.0","0.0.0"],new_node_id:"0.0.0.4",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6812500000000001,visits:16,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9099999999999999,visits:10,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.93,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:17,phase:"rollout",rollout_path:["0.0.0.4","0.0.0.4.0","0.0.0.4.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6812500000000001,visits:16,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9099999999999999,visits:10,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.93,visits:5,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:17,phase:"backprop",backprop_path:["0.0.0.4.0.0","0.0.0.4.0","0.0.0.4","0.0.0","0.0","0"],score:.85,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6911764705882354,visits:17,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9045454545454544,visits:11,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9166666666666666,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:18,phase:"select",selected_path:["0","0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6911764705882354,visits:17,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9045454545454544,visits:11,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9166666666666666,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:18,phase:"expand",selected_path:["0","0.0"],new_node_id:"0.0.5",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6911764705882354,visits:17,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9045454545454544,visits:11,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9166666666666666,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:18,phase:"rollout",rollout_path:["0.0.5","0.0.5.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.6911764705882354,visits:17,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9045454545454544,visits:11,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9166666666666666,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:18,phase:"backprop",backprop_path:["0.0.5.0","0.0.5","0.0","0"],score:1,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7083333333333335,visits:18,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9125,visits:12,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9166666666666666,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:19,phase:"select",selected_path:["0","0.0","0.0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7083333333333335,visits:18,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9125,visits:12,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9166666666666666,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:19,phase:"expand",selected_path:["0","0.0","0.0.0"],new_node_id:"0.0.0.5",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7083333333333335,visits:18,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9125,visits:12,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9166666666666666,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.5",state:"Sanity check: no other assignment of types works without contradiction.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:19,phase:"rollout",rollout_path:["0.0.0.5","0.0.0.5.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7083333333333335,visits:18,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9125,visits:12,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.9166666666666666,visits:6,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.5",state:"Sanity check: no other assignment of types works without contradiction.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.0.5.0",state:"Uniqueness established. ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:19,phase:"backprop",backprop_path:["0.0.0.5.0","0.0.0.5","0.0.0","0.0","0"],score:.75,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7105263157894739,visits:19,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.8999999999999999,visits:13,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.8928571428571429,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.5",state:"Sanity check: no other assignment of types works without contradiction.",value:.75,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.5.0",state:"Uniqueness established. ANSWER: A is a knight.",value:.75,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:20,phase:"select",selected_path:["0","0.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7105263157894739,visits:19,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.8999999999999999,visits:13,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.8928571428571429,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.5",state:"Sanity check: no other assignment of types works without contradiction.",value:.75,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.5.0",state:"Uniqueness established. ANSWER: A is a knight.",value:.75,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:20,phase:"expand",selected_path:["0","0.0"],new_node_id:"0.0.6",tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7105263157894739,visits:19,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.8999999999999999,visits:13,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.8928571428571429,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.5",state:"Sanity check: no other assignment of types works without contradiction.",value:.75,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.5.0",state:"Uniqueness established. ANSWER: A is a knight.",value:.75,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.6",state:"Final verification: the unique solution is A is a knight, B is a knave.",value:0,visits:0,is_terminal:!1,answer:null,children:[]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:20,phase:"rollout",rollout_path:["0.0.6","0.0.6.0"],tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7105263157894739,visits:19,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.8999999999999999,visits:13,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.8928571428571429,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.5",state:"Sanity check: no other assignment of types works without contradiction.",value:.75,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.5.0",state:"Uniqueness established. ANSWER: A is a knight.",value:.75,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.6",state:"Final verification: the unique solution is A is a knight, B is a knave.",value:0,visits:0,is_terminal:!1,answer:null,children:[{id:"0.0.6.0",state:"Verified: ANSWER: A is a knight.",value:0,visits:0,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}},{step:20,phase:"backprop",backprop_path:["0.0.6.0","0.0.6","0.0","0"],score:.9,tree:{id:"0",state:"Question: A says 'B is a knave.' B says 'We are the same type.' What is A?",value:.7200000000000002,visits:20,is_terminal:!1,answer:null,children:[{id:"0.0",state:"Let's assume A is a knight. Knights always tell the truth.",value:.9,visits:14,is_terminal:!1,answer:null,children:[{id:"0.0.0",state:"A says 'B is a knave.' Since A is truthful, B must be a knave.",value:.8928571428571429,visits:7,is_terminal:!1,answer:null,children:[{id:"0.0.0.0",state:"B says 'We are the same type.' Since B is a knave, B lies. A is a knight, B is a knave: they are different types. B's lie is consistent. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]},{id:"0.0.0.1",state:"Double-checking: B is a knave, so B's statement 'We are the same type' is a lie.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.1.0",state:"B's lie is confirmed. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.2",state:"Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0",state:"Continuing: Alternative angle: What if both statements are evaluated simultaneously?",value:.8,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.2.0.0",state:"Simultaneous evaluation confirms. ANSWER: A is a knight.",value:.8,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.3",state:"Verifying consistency: knight A, knave B satisfies all constraints.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.3.0",state:"No contradictions found. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.0.4",state:"Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0",state:"Continuing: Rechecking B's statement under the A-is-knight assumption.",value:.85,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.4.0.0",state:"B's statement is indeed a lie. ANSWER: A is a knight.",value:.85,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.0.5",state:"Sanity check: no other assignment of types works without contradiction.",value:.75,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.0.5.0",state:"Uniqueness established. ANSWER: A is a knight.",value:.75,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.0.1",state:"Let me verify: if A is a knight, A's claim about B being a knave must be true.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.1.0",state:"All constraints satisfied. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.2",state:"Confirming: A (knight) and B (knave) are different types, matching B's lie.",value:.95,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.2.0",state:"Types are consistent. ANSWER: A is a knight.",value:.95,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.3",state:"Cross-checking: A's truth-telling constrains B to be a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.3.0",state:"Constraint propagation yields. ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.4",state:"Exploring: Could there be another valid assignment? Let me check.",value:.7,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.4.0",state:"No alternative assignment works. ANSWER: A is a knight.",value:.7,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.5",state:"Attempting a formal proof: A=knight => B=knave => B lies => consistent.",value:1,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.5.0",state:"Formal proof complete. ANSWER: A is a knight.",value:1,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]},{id:"0.0.6",state:"Final verification: the unique solution is A is a knight, B is a knave.",value:.9,visits:1,is_terminal:!1,answer:null,children:[{id:"0.0.6.0",state:"Verified: ANSWER: A is a knight.",value:.9,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]},{id:"0.1",state:"Let's assume A is a knave. Knaves always lie.",value:0,visits:3,is_terminal:!1,answer:null,children:[{id:"0.1.0",state:"A says 'B is a knave.' Since A lies, B must be a knight.",value:0,visits:2,is_terminal:!1,answer:null,children:[{id:"0.1.0.0",state:"B says 'We are the same type.' Since B is a knight, B tells truth. So A and B are the same type. But A is a knave and B is a knight: they are different types. Contradiction. ANSWER: A is a knave.",value:0,visits:1,is_terminal:!0,answer:"A is a knave",children:[]}]}]},{id:"0.2",state:"Let me start with B's statement. B says 'We are the same type.' That's interesting.",value:.6,visits:3,is_terminal:!1,answer:null,children:[{id:"0.2.0",state:"If B is telling the truth, they're the same type. If B is lying, they're different types. Either way, B's statement constrains the relationship between A and B.",value:.6,visits:2,is_terminal:!1,answer:null,children:[{id:"0.2.0.0",state:"A's statement about B being a knave seems important. If A is right about B, then B is a knave, and B's statement is a lie. That seems to work. ANSWER: A is a knight.",value:.6,visits:1,is_terminal:!0,answer:"A is a knight",children:[]}]}]}]}}]};var y=E;function D(d,e){let s=[d.id],t={},i=d;for(;i.children&&i.children.length>0;){let a=null,n=-1/0,l=[];for(let v of i.children){let r;if(v.visits===0)r=1/0;else{let u=v.value,h=e*Math.sqrt(Math.log(i.visits)/v.visits);r=u+h}l.push({id:v.id,score:r===1/0?"\u221E":r.toFixed(2)}),r>n&&(n=r,a=v)}if(!a)break;t[a.id]={chosen:n===1/0?"\u221E":n.toFixed(2),siblings:l},s.push(a.id),i=a}return{path:s,scores:t}}function L(d){let e=document.getElementById("ucb1-c"),s=document.getElementById("ucb1-value");e&&e.addEventListener("input",()=>{let t=parseFloat(e.value);if(s&&(s.textContent=t.toFixed(1)),!d.currentTree)return;let{path:i}=D(d.currentTree,t);d.highlightPath(i,"#e94560")})}var _=class{constructor(e,s){this.renderer=e,this.path=s,this.baseDelay=1500,this.speed=1,this.currentStep=0,this.playing=!1,this.timer=null}play(){this.currentStep>=this.path.length&&(this.currentStep=0),this.playing=!0;let e=document.getElementById("rollout-play");e&&(e.textContent="\u23F8 Pause"),this._tick()}pause(){this.playing=!1,this.timer&&clearTimeout(this.timer);let e=document.getElementById("rollout-play");e&&(e.textContent="\u25B6 Play")}_tick(){if(!this.playing||this.currentStep>=this.path.length){this.pause();return}let e=this.path[this.currentStep];this.renderer.pulseNode(e),this.renderer.highlightPath(this.path.slice(0,this.currentStep+1),"#e94560"),this.currentStep++,this.timer=setTimeout(()=>this._tick(),this.baseDelay/this.speed)}setSpeed(e){this.speed=e,document.querySelectorAll(".speed-btn").forEach(s=>{let t=s;t.classList.toggle("active",parseInt(t.dataset.speed||"1",10)===e)})}};function C(d,e){let s=document.getElementById("rollout-play");if(!s)return;let i=e.simulations.find(n=>n.step===7&&n.phase==="rollout")?.rollout_path??[],a=null;s.addEventListener("click",()=>{a||(a=new _(d,i)),a.playing?a.pause():a.play()}),document.querySelectorAll(".speed-btn").forEach(n=>{n.addEventListener("click",()=>{a&&a.setSpeed(parseInt(n.dataset.speed||"1",10))})})}var b=class{constructor(e,s,t,i){this.renderer=e,this.path=s,this.score=t,this.tree=i,this.currentStep=-1,this.mathDisplay=document.getElementById("backprop-math")}step(){if(this.currentStep++,this.currentStep>=this.path.length){this.mathDisplay&&this._setMathComplete();return}let e=this.path[this.currentStep],s=this._findNode(this.tree,e);if(this.renderer.highlightPath(this.path.slice(0,this.currentStep+1),"#4a9eff"),this.renderer.pulseNode(e),this.currentStep>0){let t=this.path[this.currentStep-1];this.renderer.pulseEdge(t,e)}s&&this.mathDisplay&&this._setMathStep(e,s)}_setMathComplete(){if(!this.mathDisplay)return;this.mathDisplay.textContent="";let e=document.createElement("span");e.style.color="var(--accent)",e.textContent="Backpropagation complete.",this.mathDisplay.appendChild(e)}_setMathStep(e,s){if(!this.mathDisplay)return;this.mathDisplay.textContent="";let t=document.createElement("strong");t.textContent=Node ${e},this.mathDisplay.appendChild(t),this.mathDisplay.appendChild(document.createElement("br"));let i=document.createTextNode(visits: ${s.visits} | value: ${s.value.toFixed(2)});this.mathDisplay.appendChild(i),this.mathDisplay.appendChild(document.createElement("br"));let a=document.createElement("span");a.style.color="var(--text-dim)",a.textContent=Score ${this.score.toFixed(1)} propagated upward,this.mathDisplay.appendChild(a)}reset(){this.currentStep=-1,this.renderer.clearHighlights(),this.mathDisplay&&(this.mathDisplay.textContent="")}_findNode(e,s){if(e.id===s)return e;for(let t of e.children||[]){let i=this._findNode(t,s);if(i)return i}return null}};function T(d,e){let s=document.getElementById("backprop-step"),t=document.getElementById("backprop-reset");if(!s)return;let i=e.simulations.find(n=>n.step===7&&n.phase==="backprop");if(!i)return;let a=null;s.addEventListener("click",()=>{a||(a=new b(d,i.backprop_path,i.score,i.tree)),a.step()}),t&&t.addEventListener("click",()=>{a&&a.reset()})}function S(d){let e=[];d.is_terminal&&e.push(d);for(let s of d.children||[])e.push(...S(s));return e}function Q(d,e){function s(t,i){if(i.push(t.id),t.id===e)return[...i];for(let a of t.children||[]){let n=s(a,i);if(n)return n}return i.pop(),null}return s(d,[])||[]}function V(d,e,s=5){let t=S(d);if(t.length===0)return[];switch(e){case"value":return t.sort((i,a)=>a.value-i.value).slice(0,s);case"visits":return t.sort((i,a)=>a.visits-i.visits).slice(0,s);case"diverse":{let i={};t.forEach(v=>{let r=v.answer||"unknown";i[r]||(i[r]=[]),i[r].push(v)});let a=[];for(let v of Object.keys(i))i[v].sort((r,u)=>u.value-r.value),i[v].length>0&&a.push(i[v][0]);let n=new Set(a.map(v=>v.id)),l=t.sort((v,r)=>r.value-v.value).filter(v=>!n.has(v.id));return a.concat(l).slice(0,s)}case"topk":return t.sort((i,a)=>a.value-i.value).slice(0,s);default:return t.slice(0,s)}}var M={value:"Showing paths with the highest average values.",visits:"Showing the most-explored paths (highest visit counts).",diverse:"Showing one path per unique answer, prioritizing variety.",topk:"Showing the top-k terminal nodes by score."};function P(d,e){let s=S(d),t={},i={};s.forEach(h=>{let m=h.answer||"unknown";t[m]=(t[m]||0)+1,i[m]=(i[m]||0)+h.value});let a="http://www.w3.org/2000/svg",n=Math.max(...Object.values(t),1),l=Object.keys(t).sort((h,m)=&gt;t[m]-t[h]),v={"A is a knight":"#4ade80","A is a knave":"#f87171"};e.textContent="";let r=document.createElementNS(a,"svg");r.setAttribute("width","100%"),r.setAttribute("height","80"),r.setAttribute("viewBox","0 0 300 80");let u=Math.min(120,280/l.length);return l.forEach((h,m)=>{let c=10+m*(u+10),o=t[h]/n*50,p=v[h]||"#888",A=document.createElementNS(a,"rect");A.setAttribute("x",String(c)),A.setAttribute("y",String(60-o)),A.setAttribute("width",String(u-5)),A.setAttribute("height",String(o)),A.setAttribute("fill",p),A.setAttribute("opacity","0.7"),A.setAttribute("rx","2"),r.appendChild(A);let k=document.createElementNS(a,"text");k.setAttribute("x",String(c+(u-5)/2)),k.setAttribute("y","74"),k.setAttribute("text-anchor","middle"),k.setAttribute("fill",p),k.setAttribute("font-size","9"),k.setAttribute("font-family","monospace"),k.textContent=h.replace("A is a ",""),r.appendChild(k);let g=document.createElementNS(a,"text");g.setAttribute("x",String(c+(u-5)/2)),g.setAttribute("y",String(56-o)),g.setAttribute("text-anchor","middle"),g.setAttribute("fill",p),g.setAttribute("font-size","10"),g.setAttribute("font-family","monospace"),g.textContent=String(t[h]),r.appendChild(g)}),e.appendChild(r),{counts:t,weightedScores:i}}function N(d,e,s){let t=document.getElementById("confidence-display");if(!t)return;let i,a,n=Object.values(d).reduce((r,u)=>r+u,0);if(s==="majority"){let r=Object.entries(d).sort((u,h)=>h[1]-u[1]);i=r[0]?.[0]??"unknown",a=r[0]?(r[0][1]/n*100).toFixed(0):"0"}else{let r=Object.entries(e).sort((h,m)=>m[1]-h[1]),u=Object.values(e).reduce((h,m)=>h+m,0);i=r[0]?.[0]??"unknown",a=r[0]?(r[0][1]/u*100).toFixed(0):"0"}t.textContent="";let l=document.createElement("strong");l.textContent="Answer: ";let v=document.createElement("span");v.style.color="var(--accent)",v.textContent=i,l.appendChild(v),t.appendChild(l),t.appendChild(document.createTextNode((${a}% confidence, ${s} vote)))}function I(d,e){let s=document.querySelectorAll(".strategy-btn[data-strategy]"),t=document.querySelectorAll(".strategy-btn[data-vote]"),i=document.getElementById("answer-histogram"),a=document.getElementById("strategy-description");if(s.length===0)return;let n="value",l="majority",v=null;function r(){let u=d.currentTree;if(!u)return;let h=V(u,n,5);if(d.clearHighlights(),h.length>0){let m=new Set;h.forEach(c=>Q(u,c.id).forEach(o=>m.add(o))),d.highlightPath([...m],"#e94560")}a&&(a.textContent=M[n]||""),i&&(v=P(u,i)),v&&N(v.counts,v.weightedScores,l)}s.forEach(u=>{u.addEventListener("click",()=>{s.forEach(h=>h.classList.remove("active")),u.classList.add("active"),n=u.dataset.strategy||"value",r()})}),t.forEach(u=>{u.addEventListener("click",()=>{t.forEach(h=>h.classList.remove("active")),u.classList.add("active"),l=u.dataset.vote||"majority",v&&N(v.counts,v.weightedScores,l)})})}function R(d,e){let s="http://www.w3.org/2000/svg",t=[{x:100,y:15,r:10,permanent:!0},{x:60,y:55,r:8,permanent:!0},{x:140,y:55,r:8,permanent:!0}],i=[{x:140,y:95,r:7,permanent:!1}];for(;d.firstChild;)d.removeChild(d.firstChild);let a=(n,l,v,r,u)=>{let h=document.createElementNS(s,"line");h.setAttribute("x1",String(n)),h.setAttribute("y1",String(l+10)),h.setAttribute("x2",String(v)),h.setAttribute("y2",String(r-7)),h.setAttribute("stroke","#ffffff30"),h.setAttribute("stroke-width","1.5"),u&&h.setAttribute("class",u),d.appendChild(h)};a(100,15,60,55),a(100,15,140,55),a(140,55,140,95,"rollout-edge"),[...t,...i].forEach(n=>{let l=document.createElementNS(s,"circle");l.setAttribute("cx",String(n.x)),l.setAttribute("cy",String(n.y)),l.setAttribute("r",String(n.r)),l.setAttribute("fill",n.permanent?"transparent":"rgba(233,69,96,0.15)"),l.setAttribute("stroke",n.permanent?"#ffffff40":"#e94560"),l.setAttribute("stroke-width","1.5"),n.permanent||l.setAttribute("class","rollout-node"),d.appendChild(l)}),e&&setTimeout(()=>{d.querySelectorAll(".rollout-node").forEach(n=>{n.style.transition="opacity 1s ease",n.style.opacity="0.2",n.setAttribute("stroke-dasharray","3,3")}),d.querySelectorAll(".rollout-edge").forEach(n=>{n.style.transition="opacity 1s ease",n.style.opacity="0.2",n.setAttribute("stroke-dasharray","3,3")})},800)}function x(){let d=document.querySelector(".aside");d&&d.addEventListener("toggle",()=>{if(!d.open)return;let e=document.getElementById("rollout-comparison");if(!e||e.dataset.rendered)return;e.dataset.rendered="true";let s="http://www.w3.org/2000/svg",t=(n,l,v)=&gt;{let r=document.createElement("div");r.className="comparison-panel";let u=document.createElement("h4");u.textContent=n,r.appendChild(u);let h=document.createElementNS(s,"svg");h.setAttribute("id",l),h.setAttribute("width","100%"),h.setAttribute("height","120"),h.setAttribute("viewBox","0 0 200 120"),r.appendChild(h);let m=document.createElement("p");return m.style.fontSize="0.75rem",m.style.color="var(--text-dim)",m.style.marginTop="8px",m.textContent=v,r.appendChild(m),r};e.appendChild(t("Classical MCTS","classical-tree","Rollout nodes fade away")),e.appendChild(t("Tree-Building (this implementation)","treebuild-tree","Rollout nodes stay in the tree"));let i=document.getElementById("classical-tree"),a=document.getElementById("treebuild-tree");i&&R(i,!0),a&&R(a,!1)})}function W(){let d=document.getElementById("tree-svg");if(!d)return;let e=new B(d);new w(e,y).init(),L(e),C(e,y),T(e,y),I(e,y),x();let t=document.getElementById("single-pass-output");t&&y.puzzle.single_pass_wrong&&(t.textContent=y.puzzle.single_pass_wrong)}document.readyState==="loading"?document.addEventListener("DOMContentLoaded",W):W();})();