snakes_and_ladders, a Python code which provides tools for studying the game of Snakes and Ladders.

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.

It is a simple exercise to create a simulation of the game for several players.

Since the game is essentially a race, with no other competition between the players, it can be studied in a simplified version in which there is only one player.

For the one-player version of the game, it is interesting to pose the question of the average length of a game, that is, how many rolls of the die it takes in order to reach the final square.

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_and_ladders is available in a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

python_simulation, a Python code which uses simulation to study card games, contests, and other processes which have a random element. Usually, the purpose is to try to predict the average behavior of the system over many trials.


  1. Steve Althoen, Larry King, Kenneth Schilling,
    How long is a game of Snakes and Ladders?,
    The Mathematical Gazette,
    Volume 77, Number 478, March 1993, pages 71-76.
  2. Nick Berry,
    A Mathematical Analysis of Snakes and Ladders,
  3. Desmond Higham, Nicholas Higham,
    MATLAB Guide,
    SIAM, 2005,
    ISBN13: 9780898717891.

Source Code:

Last modified on 29 November 2022.