The Math Behind Backgammon

-Sanskruti Mehta

We have all heard of the game Backgammon, right? Not many of us play it.  For me, Backgammon was that confusing and old age game that was on the other side of my Chess board like a “Buy one, Get one free”. It is only now I know that Backgammon is quite interesting and was and is still a popular game. But did you know that you can raise your chances of winning by using a little Math? Turns out- there is Math behind backgammon.

How is it Played?

The image below shows how the backgammon board is setup in the beginning of the game.

The triangles seen are known as pips or points and the playing pieces are known as checkers or chips.

The points are numbered from 1 to 24. In the commonly used setup, each player begins with fifteen chips, two are placed on their 24-point, three on their 8-point, and five each on their 13-point and their 6-point. The two players move their chips in opposing directions, from the 24-point towards the 1-point. Points 1 through 6 are called the home board or inner board, and points 7 through 12 are called the outer board.

The objective is for players to remove (bear off) all their checkers from the board before their opponent can do the same.

To start the game, each player rolls one die, and the player with the higher number moves first using the numbers shown on both dice. In the course of a move, a checker may land on any point that is unoccupied or is occupied by one or more of the player’s own checkers. It may also land on a point occupied by exactly one opposing checker, or “blot”. In this case, the blot has been “hit”, and is placed in the middle of the board on the bar that divides the two sides of the playing surface. A checker may never land on a point occupied by two or more opposing checkers; thus, no point is ever occupied by checkers from both players simultaneously. There is no limit to the number of checkers that can occupy a point at any given time. Checkers placed on the bar must re-enter the game through the opponent’s home board before any other move can be made. When all of a player’s checkers are in that player’s home board, that player may start removing them; this is called “bearing off”.

The Math:

Now that we have understood how to play Backgammon, let us get to the main topic. As I revealed earlier, Backgammon is played often to win money. A doubling cube is used for this purpose. It is like poker in that sense. There is a big role of luck in the game. However, there are certain calculations that you can do to estimate how fast you can win and whether you have a chance to win: –

1) Pip Counting

Determining your position in this race is achieved by calculating the difference between the number of pips that you need to get all your checkers home and off the board, and the number our opponent needs. The technique is known as pip counting.

The pip count= the no of the point/pip x the no of checkers on that point/pip

The pip count is 167 for each player at the start of the game. This can be better understood with the help of an example.

In this example at the closing stage of a game, Red has two checkers on the 6 point and White has one checker on his 6 point and two checkers on his 2 point. Red requires 12 pips to completely bear off (6 × 2 = 12), while White requires 10 pips (6 × 1) + (2 × 2) = 10. Therefore, the pip count informs us that Red is 2 pips behind in the race.

The pip count is an essential factor in many cube decisions and can help in determining the appropriate choice of strategy while playing.

2) Probability

While you cannot predict what will be the 2 numbers that come face up on your dice, you at least can know the probability of a certain number or numbers turning up face up. Since each die has 6 numbers the number of possibilities or the summation of the sample space is 36. This can draw many probabilities.

For example, the probability of a double coming up (same number on both dies) is 6/36 or 1/6. Now how can you use your probability skills to improve your strategy?

Let’s say, that this is how your board looks like. This reveals that one of the most dangerous points on which to leave a blot is 7. Twenty-four dice permutations can hit a blot exposed on that point. To a beginner it looks almost suicidal to move onto 7 with an opening move, 66% chance that your opponent will hit you. Yet many of the most expert players use this as an opening move. If you do this there are:

• 19 chances in 36 that he can hit you but is forced to leave a blot.
• 12 chances in 36 that he will miss your blot.
• 5 chances in 36 that he can hit you and cover his blot. (i.e. 6:6, 3:3, 6:1, 1:6 and 1:1)

This is just an example of where probability is used. Even knowing the probability of getting each number on one die or a certain sum of number on both dice can be helpful to you.

Nowadays Backgammon is played online by many. So, all these calculations are done by the computer for you. However, if you ever want to play it in physical form, you may want to brush up on your math skills and learn to calculate fast. You can only get so far by luck. Math can help you to win. Happy playing!

References

https://en.wikipedia.org/wiki/Backgammon

https://bkgm.com/articles/Koca/bgtalkpaper2.html

https://bkgm.com/articles/Driver/GuideToCountingPips/

https://www.backgammononlineguide.com/odds.html