The Doubling Cube

The doubling cube is a way of increasing the wager in a game. For those readers who don’t understand how to use a doubling cube in a game, I explain the rules below.

1. 1. There is a doubling cube numbered with powers of 2, as follows – 2, 4, 8, 16, 32, 64.
2. 2. The cube starts “in the middle,” which means either player may elect to offer a double. It does not matter what side is up if in the middle. It is assumed to have a value of 1 if in the middle.
3. 3. When the cube is in the middle or owned by a particular player, he has the option to offer a double at the beginning of his turn.
4. 4. If a player chooses to double, he should verbalize the offer by saying “Shall we double?” or something to that effect. He should also place the cube before his opponent. If it is the initial double, he should place the 2 face up. Otherwise, he should double the value previously face up.
5. 5. The opponent has two options, accept the doubled stakes of the game, as shown on the cube, or forfeit for the value on the cube before the challenge was made. If the cube was in the middle and there is a forfeit, then the losing player must pay the original wager on the game.
6. 6. If a double is accepted, the player who was challenged now owns the cube and is the only one who may redouble.

 

As an example, two players wager $1 on a game. Player A offers to double, which player B accepts. The cube now shows a 2. Later, player B redoubles and player A declines. Player A must pay player B $2.

In my experience, I have only seen the doubling cube used in backgammon. However, it could be used in any two-player game where there are multiple turns. It could also be used in racing, for example in a horse race, as long as decisions were made instantly.

I recently did a lot of thinking and analysis of when to double. To simplify the math, I assumed the probability of winning moved on a continuous basis, as in a race. This is much unlike backgammon, where the probability of winning can change significantly with every roll of the dice.

After a lot of math, which I won’t get into, I find either player should make an initial double if his probability of winning is 65% or more. The other player should accept if his probability of winning is 35% or more.

After an initial double, the player who owns the cube should redouble if his probability of winning is 80% or more. Likewise, his opponent should accept if his probability of winning is 20% or more.

What could you expect to happen if two perfect players played? Assume both players try to maximize their expected win first and are averse to risk second. You would never see a double accepted. It would become a contest of who was the first to have a 65% chance of winning, at which time he would offer to double and the opponent would decline, being risk averse.

So why do you see so much doubling accepted in backgammon? It is become the probability of winning does not change continuously, but discretely in jumps, according to the role of the dice.

While my continuous assumption is not applicable in backgammon, I think my analysis is in the ballpark for backgammon purposes. In fact, it seems to agree fairly closely with respected experts as well as my own Backgammon NJ software.

August 15, 2024 Puzzle Question

Four people need to cross a bridge at night. The bridge is in poor repair. At most, two people can be on the bridge at a time. Also, there are boards missing, so a flashlight is required to cross. Throwing the flashlight is not allowed. If two people cross at the same time, they must go at the slower person’s speed. It takes the four people 1, 2, 5 and 10 minutes respectively to cross. How can they all get across in 17 minutes?

August 15, 2024 Puzzle Answer

1. 1. One-minute and two-minute people cross over (2 minutes consumed).
2. 2. One-minute person crosses back (3 minutes consumed).
3. 3. Five-minute and ten-minute people cross over (13 minutes consumed).
4. 4. Two-minute person crosses back (15 minutes consumed).
5. 5. One-minute and two-minute people cross over (17 minutes consumed).

 

It can also be done in the same time by sending the two-minute person over at step 2 and the one-minute person in step 4.

August 22, 2024 Puzzle Question

An evil warden gathers ten prisoners from his prison. He explains to them that in 24 hours he will line them up in height order, starting with the tallest on the left. Every prisoner will face the right (being able to see all shorter prisoners). He will then place either a black or white hat on each prisoner, being careful not to let the prisoners see the color of their own hat. After this step, each prisoner will be able to see the hats of all shorter prisoners only.

Then, starting from the left, with the tallest prisoner, he will ask each prisoner the color of his hat.Such responses, “black” or “white,” are the only allowed communication. Any coughing, tapping or other attempts to communicate will result in an immediate and painful death of all ten prisoners.If 9 or more are correct, they shall all be released immediately. Otherwise, if 8 or less are correct, they will all immediately be executed.

The prisoners are then given 24 hours to discuss strategy. There is a way to ensure everybody is set free. What should be their strategy?