Game Simulation

SNAKES_AND_LADDERS, C++ programs which investigate 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 GNU LGPL license.


SNAKES_AND_LADDERS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

DUEL_SIMULATION, a C++ program which simulates N repetitions of a duel between two players, each of whom has a known firing accuracy.

FAIR_DICE_SIMULATION, a C++ program which simulates N tosses of 2 dice, making a histogram of the results.

HIGH_CARD_SIMULATION, a C++ program which simulates a situation in which you see the cards in a deck one by one, and must select the one you think is the highest and stop.

ISING_2D_SIMULATION, a C++ program which carries out a Monte Carlo simulation of an Ising model, a 2D array of positive and negative charges, each of which is likely to flip to be in agreement with neighbors.

POISSON_SIMULATION, a C++ library which simulates a Poisson process in which events randomly occur with an average waiting time of Lambda.

REACTOR_SIMULATION, a C++ program which is a simple Monte Carlo simulation of the shielding effect of a slab of a certain thickness in front of a neutron source. This program was provided as an example with the book "Numerical Methods and Software."


  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:

Examples and Tests:

Some graphics files are created.

List of Routines:

You can go up one level to the C++ source codes.

Last modified on 19 September 2014