Snakes and Ladders and Intransitivity, or what mathematicians do in their time off

TL;DR

This paper explores the intransitivity in the game of Snakes and Ladders, illustrating complex mathematical concepts like Markov chains and intransitive dice through recreational analysis.

Contribution

It demonstrates the intransitivity property in Snakes and Ladders and connects it to intransitive dice, showcasing mathematical insights through recreational mathematics.

Findings

Snakes and Ladders exhibits intransitivity among certain squares.

Analysis uses Markov chains, simulations, and size-biased sampling.

Minimal intransitive dice examples are constructed with few faces.

Abstract

This recreational mathematics article shows that the game of Snakes and Ladders is intransitive: square 69 has a winning edge over 79, which in turn beats 73, which beats 69. Analysis of the game is a nice illustration of Markov chains, simulations of different sorts, and size-biased sampling. Connecting this to "intransitive dice" illustrates the power of a name, and the joy of working with colleagues. When draws do not count, we show a minimal example of intransitive dice, with one die having just a single "face" and two dice each having two faces.