The Doubling Cube

The doubling cube serves as a tool for increasing the stakes in a game. For those unfamiliar with its application, I will clarify the rules in the following sections.

1. 1. The doubling cube is marked with numbers that are powers of 2: 2, 4, 8, 16, 32, and 64.
2. 2. When the cube is positioned in the center, it means that either player has the option to propose a doubling of the stakes. The orientation of the cube doesn’t matter if it is in this neutral position, and it essentially holds a value of 1.
3. 3. A player can offer a double at the start of their turn if the cube is either in the neutral position or is owned by them.
4. 4. To initiate a double, the player must voice their proposal, saying something like, \"Shall we double?\" They should then place the cube in front of their opponent. If this is the first doubling, the face showing must be 2; in subsequent doubles, they should present the number that reflects the last value.
5. 5. The opposing player has two choices: they can either agree to the increased stakes as indicated by the cube or concede, paying the stake shown before the challenge was made. If the cube was in a neutral position during a concession, the losing player must pay the original betting amount.
6. 6. If the opponent accepts the double, they then take ownership of the cube and are now the only one who can propose a redouble.

For instance, if two players are betting $1 in a game, and player A suggests a double, player B accepts this. The cube now displays 2. Later on, player B might decide to redouble, but if player A refuses, they owe player B $2.

From what I have observed, the doubling cube is predominantly seen in backgammon, but its application could extend to any turn-based game with two players. It may even be applicable in racing scenarios, such as horse racing, allowing for quick decision-making.

I have spent considerable time reflecting on the optimal timing for doubling. To make things simpler, I approached the probabilities of winning as if they were fluid, akin to a race, which contrasts sharply with backgammon where probabilities fluctuate drastically with each dice roll.

After extensive calculations, which I won't delve into here, I determined that players should propose an initial double if they believe their chance of winning exceeds 65%. Conversely, the other player should consider accepting if they believe their chances are at least 35%.

Following an initial double, the player who holds the cube should suggest a redouble if their winning probability is 80% or higher, while the opponent ought to accept if they see their odds at 20% or greater.

What would transpire if two flawless players competed? Assuming both aim to maximize their expected wins and are hesitant to take risks, a doubling acceptance would be rare. Instead, it would turn into a race to be the first to achieve a winning probability of 65%, at which point one would propose a double that the other would decline, due to their risk aversion.

Why is it that we witness numerous doubling acceptances in backgammon? This phenomenon occurs because the winning probabilities shift in distinct jumps, dictated by the outcomes of the dice rather than continuously.

Though my assumption about continuous probabilities may not apply to backgammon, I believe the general insights gleaned from my analysis hold relevance for the game. This is consistent with the views of respected experts and aligns well with my own Backgammon NJ software.

August 15, 2024 Puzzle Question

A scenario unfolds where four individuals must cross a bridge during nighttime. Due to its poor condition, the bridge can support a maximum of two people at once. Additionally, due to missing planks, a flashlight is mandatory for crossing, and tossing it is prohibited. If two people cross simultaneously, they must proceed at the pace of the slower individual. Their respective crossing times are 1, 2, 5, and 10 minutes. How can they successfully traverse within 17 minutes?

August 15, 2024 Puzzle Answer

1. 1. The individuals who take 1 and 2 minutes cross together (totaling 2 minutes).
2. 2. The person taking 1 minute returns (making it 3 minutes total).
3. 3. The individuals taking 5 and 10 minutes cross together (bringing the total to 13 minutes).
4. 4. The 2-minute individual returns (now totaling 15 minutes).
5. 5. Finally, the individuals taking 1 and 2 minutes cross again (reaching 17 minutes).

An alternative method to accomplish the same feat involves sending the 2-minute individual during step 2 and the 1-minute individual during step 4.

August 22, 2024 Puzzle Question

A sinister warden gathers ten inmates and informs them that in 24 hours, he will position them in order of height, with the tallest on the left. Each prisoner will face the right and will be unable to see his own hat's color. He will place either a black or white hat on each prisoner without them knowing their own hat's color. After this, prisoners will only see the hats of those shorter than themselves.

Beginning with the tallest prisoner on the left, each will be asked to state the color of his hat. The only allowed responses are \"black\" or \"white,\" and any other form of communication, such as coughing or signaling, will ensure the immediate and severe punishment of all ten inmates. If 9 or more answers are correct, they will gain freedom; otherwise, if 8 or fewer are correct, they will face execution.

The prisoners are allotted 24 hours to devise a strategy that guarantees their release. What should their approach be?