Kids and Teens

Analysis of Chutes and Ladders

Analysis of Chutes and Ladders Game

The internet allows you to trace the game’s origins back to the 2nd century B.C.

This system was created by Hindu spiritual leaders to show children the positive consequences of bad deeds as well as the rewards. The ladders are for virtues and good morality, while the snakes signify vices and poor choices. 

It was first introduced to England in 1892. In the United States, it was sold by Milton Bradley in 1943 under the name Chutes and Ladders. Square 100 is believed to represent the Hindu idea of Nirvana. You can easily calculate the probability of an event using the probability calculator. And you can easily find the probability calculator.


The 10×10 gridded board is used for the game. It is numbered sequentially in a zigzag pattern, starting at 1 (the beginning), and ending at 100 (the end), in the top left corner. There are snakes and ladders at various places on the board, which connect a pair of squares. The right is a representation of the board our family uses. (Snakes are in red with a dot representing the head, while ladders are in light green with an arrow indicating the direction of travel.

All players begin by placing their hands on the board. They then roll one die or, in the case of Milton Bradley, flip a spinner that is labeled 1 through 6. Each player moves the appropriate number of squares. Your token will land at the foot or the top of a ladder if your move is successful. 

In the same way, if you land on the head or tail of a snake with your move, you will be forced to slide to the previous square. Landing on the ladder’s top or tail of a snake has no consequences. These snakes/ladders can be viewed as one-way events. The first player to win 100 is the winner.

What is the average time it takes to play a game?

Due to the cyclic nature of the game, landing on a snake could send you backward. If you’re extremely unlucky, another …), can be landed on you. There is no limit to how many moves a game can take.

However, the likelihood of long-duration games diminishes rapidly. The analysis below will show that a billion Chutes and Ladders were played and that the longest game ever recorded took 394 moves.

It takes 97.6% of all games, thankfully, 100 moves or less.

A Bayesian and an Objectivist walk into a bar.

We have two main mechanisms that can be used to calculate the probabilities of an event:

  • Experimentation (Objectivizing)
  • Formal modeling (Subjectivizing) – Also known as a Bayesian approach.

These words are complex, but the concepts are simple.

  • Objective Approach

An objective approach is to repeat the experiment multiple times and note the frequency of the results. Less likely events occur more often than more. The results will tell you the relative probabilities. Your confidence in your results will increase the more experiments or samples you do.

You can flip a coin once, twice, a hundred times, or a million times, and the ratio of heads to tails will tell you the probability of it landing heads or tails. (Hopefully, close to 50/50 for non-gimmicked coins).

While a coin flip may seem trivial, how about the probability of winning a solitaire game? Or winning a Blackjack hand against a dealer who has an exposed seven. A gut-shot flush draw in Texas Hold’em Poker?

  • Monte-Carlo Simulation

A Monte-Carlo simulation, named after the Casino, is a model of a system that is then executed multiple times using random inputs. These algorithms are used when a deterministic algorithm, which is one that predictably produces the same output regardless of input, is not possible.

Monte-Carlo simulations can be written quickly and are very useful. These simulations allow you to get results without having to understand the inner workings of your engine.

  • Monte-Carlo Chutes & Ladders

It is easy to create a program for playing Chutes and Ladders. A simple data structure can store the graph. To keep track of the position of a player, all you need is a single pointer.

To calculate the expected game length, there is no player-player interaction. We only need to take into account the movement of one player.

  • Defensive Coding

While the chances of a game running and running are very low (constantly landing in snakes and going round and round in circles), there is the possibility that the main game loop could continue to execute for hours. This will lock your code. A smart developer would implement an algorithm such as this to program a counter that increments on every roll of the dice. The counter should be checked and, if it exceeds a threshold, the loop should be closed.

Any coded implementation without a failsafe such as this should alarm your ears. Even if you win the lottery more often than being struck by lightning the third time the day, it is still possible.

a) You are correct in estimating that the odds of success are small. Are you good at estimating Black Swan events?

b) Someone alters the parameters of a game at a later time (adding new dice, rules, or additional features dramatically changing the dynamics of a system).

c) Your random number generator might not be as random as you think. You may get into a harmonic resonance, which oscillates between states so you don’t progress.

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