Snakes and Ladders has no decisions, so you can compute it exactly. Here is how I got the number I use in other posts: one counter alone needs 39.23 throws on average to finish the classic board.
The model
The state is the square you stand on, 0 to 100. A throw moves you 1 to 6 squares with equal chance. If the new square is the head of a snake or the foot of a ladder, you move to its other end. If you would pass 100, you stay. That last rule is the exact-finish rule.
That makes a Markov chain with 101 states and one absorbing state, 100. The expected number of throws from each state satisfies a linear system. For a state s that is not 100, E[s] = 1 + (1/6) * sum of E[next(s, d)] over the six throws, with E[100] = 0.
The result
I ran the numbers on 17 October 2026, as a Markov chain and as a simulation of the rules. The mean is 39.23 throws. The median is 32, the 90th percentile 72, and the 99th 128. The fastest possible game is 7 throws.
Some other numbers from the same model:
- 89.4% of games touch at least one snake.
- A game averages 4.23 snake landings and 3.32 ladder landings.
- With no snakes or ladders at all, the mean would be 33.33 throws.
- Ladders only: 21.84. Snakes only: 118.11.
- The exact-finish rule alone costs about 3.4 throws.
Why the median is below the mean
A few long games drag the mean up. A counter that lands on the long snake near the end can lose most of a game, so the distribution has a long tail.
A sanity check you can do
Line up the variants and see whether they come out in the order you would expect. With ladders only, the mean is 21.84 throws. With no snakes or ladders at all, 33.33. On the classic board, 39.23. With snakes only, 118.11. Ladders help, snakes hurt, and the classic board sits between them, closer to the plain track than to the snakes-only board because the ladders cancel part of the damage.
If your own version of the model does not give the same order, look for a bug in how it treats a throw that would pass 100.
Play it
Boardit's Snakes and Ladders uses this board in its Classic setting. It is the board with nine ladders and ten snakes, and it supports two to eight players. The other boards, Ladder Garden and Snake Pit, change the numbers, so the figures above apply to Classic only.
The source is in the game's code if you want to rerun it.
Ahmed Moaz builds Boardit, a free browser game site.
Top comments (0)