The Diagonal 3s Pattern in Slitherlink: A Complete Guide

The diagonal 3-3 pattern is one of the most satisfying deductions in Slitherlink. When two 3s sit diagonally adjacent (sharing only a corner vertex), you can immediately determine two edges — without knowing anything else about the puzzle.

The Setup

Two 3s share a single vertex (diagonally adjacent). In this example, the 3s are at top-left and bottom-right of a 2×2 area:

·   ·   ·
  3      
·   ·   ·
      3  
·   ·   ·

The shared vertex is the center dot. Four edges meet at this vertex: up, down, left, right (relative to the center).

What Gets Determined

At the shared vertex, exactly 2 of the 4 edges must be lines (vertex degree = 2). These 2 lines form an L-shape, with one line pointing toward each 3:

Case A (up + right):

·   ·   ·
  3   │
·   ·   ·───·
          3  
·   ·   ·   ·

Case B (left + down):

·   ·   ·   ·
  3          
·───·   ·   ·
    │     3  
·   ·   ·   ·

Why It Works

Each 3 needs 3 of its 4 edges to be lines. The shared vertex connects to exactly one edge of each 3.

Consider the shared vertex. It must have degree 0 or 2. If degree 0, neither 3 gets a line from this vertex — each 3 would need all 3 of its other edges to be lines. But trace what happens: the vertices at the outer corners of each 3 would then need specific configurations that conflict with each other (they'd create isolated segments that can't form a single loop).

So the shared vertex must have degree 2. Of those 2 lines, exactly 1 points toward the top-left 3, and exactly 1 points toward the bottom-right 3. You don't know which specific pair yet, but you know the vertex has 2 lines.

Combining with Other Clues

The real power comes when neighboring cells resolve which case (A or B) applies:

With an adjacent 0

If a 0 sits next to the shared vertex, it eliminates edges, often forcing one specific case:

·   ·   ·   ·
  3     0    
·   ·   ·   ·
      3      
·   ·   ·   ·

The 0 in the top-right eliminates the edge going right from the shared vertex. This forces Case B (left + down).

With an adjacent 1

A 1 near the shared vertex limits possibilities similarly, often cascading into a full resolution of both 3s.

The Extended Chain: Multiple Diagonal 3s

When three or more 3s form a diagonal chain, each consecutive pair triggers the pattern independently, and the results compound:

·   ·   ·   ·
  3          
·   ·   ·   ·
      3      
·   ·   ·   ·
          3  
·   ·   ·   ·

The top pair determines 2 edges at vertex (row 1, col 1). The bottom pair determines 2 edges at vertex (row 2, col 2). Together, they constrain the middle 3 from both sides, often fully resolving it.

How to Spot It

Train yourself to scan for diagonal 3s as part of your opening routine:

1. Scan all 3s on the board
2. For each 3, check its four diagonal neighbors
3. If another 3 sits diagonally → mark the shared vertex as degree 2
4. Check if surrounding clues resolve which case applies

This pattern appears in approximately 30% of puzzles at difficulty Level 4 and above. Recognizing it saves significant time compared to deducing those edges through constraint propagation alone.

---

Practice diagonal patterns: slitherlinks.com — try 7×7 Level 4-5 puzzles where diagonal 3s frequently appear.