# Calculate Average Duration of Snakes & Ladders Game

• Crumbles
In summary: If each player has a different probability of winning (say, 50% for player 1 and 50% for player 2), then the average game length is:
Crumbles
I want to calculate the average duration of a snakes and ladders game in terms of the number of moves. Does anybody have any idea on how to tackle the problem?

How do you play snakes and ladders?

It's a board game that consists of an array of numbers from 1 to 100 that you play by rolling a die. You can find an online version http://www.bbc.co.uk/schools/numbertime/games/snakes.shtml.

You start the game off the board [if you get a 1 from your first throw, you move to square 1, if you get a 2, square 2, etc] and move according to the roll of the die. You win when you get to square 100. On your way there, you come across snakes and ladders. If the roll of your die leads you to the foot of a ladder, you get to move up the ladder. But if it leads you to the head of a snake, you follow the snake to the square that contains its tail.

You will need shockwave to load the link above. You can also find a jpeg of the board http://www.hants.gov.uk/museum/toys/history/images/snakes.jpg .

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It looks like it's too complicated to do manually and you should probably just write a program to simulate it.

Yes, writing a program is my plan. But I have been advised against doing it by literally taking an average with the computer playing the game many times. It is apparently very time consuming and yields very inaccurate answers.

It can be worked out by probabilities from what I got told... Any ideas?

I don't believe it is such a bad idea to do it with the computer playing the game many times--probably no more than a couple hundred moves a game so you could play a million games. But if you don't want to do it that way then you could do it like this: start out with 0 on each square except for the first square that has 1. Then put 1/6 on each of the 6 squares following square 1 and make the first square 0. If any of the 1/6's are on a ladder or a snake, then put them at the end of the ladder or snake. You should probably use linked nodes that can also be referenced by array to represent the board. On each iteration, take the value on each square and distribute it evenly over the following 6 squares, accounting for snakes or ladders. (You should use a second board as a buffer for the next step). When a value appears at the finish, you record the value and what iteration it finished at in another array (this second array should be huge, it must have as many elements as the number of game moves you simulate). When, say, 99.9% of the value is at the finish, you terminate. That other array then has the probability distribution of 1 player finishing in x moves if he is playing alone. Then from that array you can find the probability distribution of 2 players (assuming there are 2) finishing in n moves which is what I assume you want.

Actually, I think there is a way to calculate the average for the 2 player game as you go, so you don't need the huge array. Of course if you are only interested in the average for the 1 player game then it's easy, you just multiply each incoming value by the move it arrives at, and add that to a running total. But there is a way to calculate it for a 2 player game as well.

## 1. How do you calculate the average duration of a Snakes & Ladders game?

The average duration of a Snakes & Ladders game can be calculated by dividing the total number of turns taken by the total number of games played. This will provide the average number of turns per game.

## 2. What is considered a standard duration for a Snakes & Ladders game?

The standard duration for a Snakes & Ladders game can vary, but it is typically between 10-20 minutes. This can depend on the number of players and the luck of the dice rolls.

## 3. Is it possible to calculate the average duration for a specific number of players?

Yes, it is possible to calculate the average duration for a specific number of players. This can be done by tracking the number of turns taken for each game played with that number of players and then calculating the average as mentioned in the first question.

## 4. Does the size of the Snakes & Ladders board affect the average duration of the game?

Yes, the size of the board can have an impact on the average duration of the game. A larger board may result in longer games, while a smaller board may result in shorter games. This can also depend on the number of snakes and ladders on the board.

## 5. Can the average duration of a Snakes & Ladders game be affected by player strategy?

Yes, player strategy can have an impact on the average duration of a Snakes & Ladders game. Some players may choose to take longer routes to avoid snakes, while others may try to reach the end as quickly as possible. This can result in variations in the average duration of the game.

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