snakes_probability, a MATLAB code which computes the game length probabilities for Snakes and Ladders, by Desmond Higham and Nicholas Higham.
Snakes and Ladders is a children's game played on a 10x10 numbered board. A player's turn consists of rolling a single die, and moving the indicated number of squares. If the final square is the foot of a ladder, the player moves up to a higher numbered square. If the final square is the mouth of a snake, the player moves downward.
For the one-player version of the game, it is interesting to pose the question of the probability that a particular game will take a certain number of moves.
By adding a square 0, where the player begins, the game board can be modeled as a vector of length 101, and the transitions from one square to another can be modeled by a transition matrix. Most commonly, the entries in row I will be zero except that columns I+1 through I+6 will have the value 1/6. However, rows which correspond to a snake or ladder, and rows for which I+6 is greater than 100, must be handled specially.
Given the transition matrix A, the one player game can be modeled as a Markov Chain Monte Carlo system. In particular, given an initial starting vector v, the probability distribution after one move is the vector A' * v, and repeated multiplication by A' will display the exact probability distribution at every step.
The computer code and data files made available on this web page are distributed under the MIT license
snakes_probability is available in a MATLAB version.
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Desmond Higham, Nicholas Higham