The Verification Venue · pointed at a game with no decisions in it

The Board That Loiters at Sixty

You cannot play snakes and ladders badly. There are no choices, so the whole game is a machine, and a machine can be solved rather than played. This page prints its board, solves the machine in your browser, and shows you where a long game actually lives: not crawling toward the end, but turning back, over and over, to a preferred stretch of the board.

Everything below is computed the moment the page loads, from the layout printed under the board. Press Play one game to watch a single token ride the chutes, or Solve the chain already done: the exact distribution of game lengths is the gold histogram, and pressing Spin 5,000 games lays a seeded simulation over it so you can watch the two agree.

The layout this page solves, printed in full

This is one specific arrangement of chutes and ladders. Commercial boards differ from it and from each other, so every number on this page is a theorem about this object and about nothing else. No layout here is attributed to any manufacturer. ↓

The rules this page implements, stated so there is no ambiguity. One token starts on square 1. Each turn a fair six-sided spinner (or die, the maths does not care) gives 1 to 6. Move forward that many squares. If the landing square is the foot of a ladder, climb it; if it is the head of a snake, slide down; chains do not cascade, a single application per turn. You must land exactly on square 100 to finish. If a spin would carry you past 100, you do not move at all and the turn is spent. That last rule matters enormously: change it to "reach or pass 100 wins" and the answer below drops by several whole spins. Every published analysis you have ever skimmed made some choice here, and they do not all make the same one.

With no decisions anywhere, the game is an absorbing Markov chain. Number the 99 non-final squares as transient states and build Q, the 99 by 99 matrix where Q[i][j] is the chance that a turn spent on square i ends with the token on square j. Each row of Q plus the chance of finishing that very turn sums to exactly 1, which is a conservation law the check panel tests. The expected number of turns from every square at once is the vector h satisfying:

No simulation is needed for that number. It is a theorem about the printed layout, and the page solves it directly, then hands you the simulation anyway so you can watch the histogram converge onto the exact curve. The distribution is not bell-shaped. Its mode sits well below its mean, and the mean is dragged upward by a long thin tail: the snakes on 93, 95 and 98 sit close enough to the finish that a game can be one bad spin from done and then lose fifty more turns. That tension between "usually quick" and "sometimes enormous" is the whole character of the game.

Where the long games live. Here is the thing no amount of playing will show you. Take the chain restricted to the 99 transient squares and ask: conditioned on not having finished yet, where is the token? As the condition "still going" persists, that conditional distribution converges to the quasi-stationary distribution, the dominant left eigenvector of Q. Press Paint the residence above and the board repaints itself with it. A long game is not a slow walk to the finish. It is a residence: the token keeps coming back to the far stretch of the board while the game refuses to end. How tightly it bunches is a computed quantity, not a flourish, and words make it easy to overstate. The busiest ten squares hold the share printed in the check panel below, weighed there against a plain yardstick: twice the share that a perfectly even spread over all 99 squares would give any ten squares. The residence clears that bar, so the weighting is real, and it is spread wider than the word "cluster" suggests. The peak square is named in the caption under the board.