Rolling the Doubling Cube

by Douglas Zare
1 February 2014

Douglas Zare

Backgammon stands out from other games of skill and chance because of the doubling cube. In fact, the doubling cube could be added to many other activities, even spectator sports. However, many casual players don't know how the doubling cube is used. While it is good to learn the rules of checker play first, many experienced casual players have never learned to use the doubling cube. One said, "I think after you play the game, you roll the doubling cube to see how many points you win." Well, that's creative. The doubling cube does resemble a large die. What can you learn from rolling it? In this column we will see two applications of the idea of rolling the doubling cube.

A game of dice

Suppose you get to choose between rolling the doubling cube (labeled 2, 4, 8, 16, 32, 64) and an ordinary die (labeled 1, 2, 3, 4, 5, 6). I'll roll the other. The game will be to roll the larger number. Given the choice, you would pick the doubling cube, of course. Because the nth number on the doubling cube is larger than the nth number on the regular die, it is clearly better to choose to roll the cube even though it is possible to lose by rolling a 2 or 4 against a 5, for example, or we could tie.

To make it more fair, let's say that when you roll the ordinary die, the numbers are multiplied by 5. So, a roll showing 3 pips would be 15. This has the side benefit of eliminating ties. Which die should you choose? Now the game is much closer, and perhaps it isn't obvious which has an advantage.

      2     4     8     16     32     64  
  5             X     X     X     X  
  10                 X     X     X  
  15                 X     X     X  
  20                     X     X  
  25                     X     X  
  30                     X     X  

The Xs mark where the doubling cube wins. There are 16 Xs out of 36, so you should choose to roll the regular die.

Next, suppose I have a special die. On 2 faces, it has 0, and on 4 faces, it has 31. It commemorates an historic football game between American Samoa and Australia. To be perfectly fair, I offer you the choice of which to roll, the doubling cube, the ordinary die (multiplied by 5), or the special 0-31 die. After you choose what you think is best, I'll choose the better of the remaining dice, and we will roll to try to make the higher number again. Which die should you choose?

In fact, every choice loses! You might think that going first and getting to choose the best die is an advantage, but it isn't. Suppose you choose the regular die. I will choose the 0-31 die. Regardless of what you roll, I win if I roll a 31 (2/3 of the time) and I lose if I roll a 0 (1/3).

      0     0     31     31     31     31  
  5             X     X     X     X  
  10             X     X     X     X  
  15             X     X     X     X  
  20             X     X     X     X  
  25             X     X     X     X  
  30             X     X     X     X  

Ok, so the special die is better. But if you choose the 0-31 die, I will roll the doubling cube.
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