# SNAKES_MATRIX The Snakes and Ladders Transition Matrix

SNAKES_MATRIX, a MATLAB library which computes the transition matrix for 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.

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.

### Languages:

SNAKES_MATRIX is available in a MATLAB version.

### Related Data and Programs:

DICE_SIMULATION, a MATLAB program which simulates N tosses of M dice, making a histogram of the results.

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

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

GAMBLERS_RUIN_SIMULATION, a MATLAB program which simulates the game of gambler's ruin.

HIGH_CARD_SIMULATION, a MATLAB 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 MATLAB 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 MATLAB library which simulates a Poisson process in which events randomly occur with an average waiting time of Lambda.

RANDOM_WALK_1D_SIMULATION, a MATLAB program which simulates a random walk in a 1-dimensional region.

RANDOM_WALK_2D_AVOID_SIMULATION, a MATLAB program which simulates a self-avoiding random walk in a 2-dimensional region.

RANDOM_WALK_2D_AVOID_TASKS, a MATLAB program which computes many self avoiding random walks in a 2-dimensional region by creating a job which defines each walk as a task, and then computes these independently using MATLAB's Parallel Computing Toolbox task computing capability.

RANDOM_WALK_2D_SIMULATION, a MATLAB program which simulates a random walk in a 2-dimensional region.

RANDOM_WALK_3D_SIMULATION, a MATLAB program which simulates a random walk in a 3-dimensional region.

REACTOR_SIMULATION, a MATLAB 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."

ROULETTE_SIMULATION, a MATLAB program which simulates the spinning of a roulette wheel and the evaluation of certain common roulette bets.

SIR_SIMULATION, a MATLAB program which simulates the spread of a disease through a hospital room of M by N beds, using the Susceptible/Infected/Recovered (SIR) model.

SNAKES_AND_LADDERS, a MATLAB program which simulates the game of Snakes and Ladders.

SNAKES_BAR, a MATLAB library which produces bar charts of the count, PDF and CDF estimates for the length of a one-player game of Snakes and Ladders, produced by simulating N games.

SNAKES_HISTOGRAM, a MATLAB library which produces histograms of the count, PDF and CDF estimates for the length of a one-player game of Snakes and Ladders, produced by simulating N games.

SNAKES_PROBABILITY, a MATLAB program which computes the game length probabilities for Snakes and Ladders, by Desmond Higham and Nicholas Higham.

TRAFFIC_SIMULATION, a MATLAB program which simulates the cars waiting to get through a traffic light.

TRUEL_SIMULATION, a MATLAB program which simulates N repetitions of a duel between three players, each of whom has a known firing accuracy.

### Reference:

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,
http://www.datagenetics.com/blog/november12011/index.html

### Source Code:

You can go up one level to the MATLAB source codes.