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Go First Dice
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Go First Dice
Go First Dice are a set of dice in which, when rolled together, each die has an equal chance of showing the highest number, the second highest number, and so on.
The dice are intended for fairly deciding the order of play in, for example, a board game. The number on each side is unique among the set, so that no ties can be formed.
There are three properties of fairness, with increasing strength:
It is also desired that any subset of dice taken from the set and rolled together should also have the same properties, so they can be used for fewer players as well.
Configurations where all die have the same number of sides are presented here, but alternative configurations might instead choose mismatched dice to minimize the number of sides, or minimize the largest number of sides on a single die.
Sets may be optimized for smallest least common multiple, fewest total sides, or fewest sides on the largest die. Optimal results in each of these categories have been proven by exhaustion for up to 4 dice.
The two player case is somewhat trivial. Two coins (2-sided die) can be used:
An optimal and permutation-fair solution for three 6-sided dice was found by Robert Ford in 2010. There are several optimal alternatives using mismatched dice.
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Go First Dice AI simulator
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Go First Dice
Go First Dice are a set of dice in which, when rolled together, each die has an equal chance of showing the highest number, the second highest number, and so on.
The dice are intended for fairly deciding the order of play in, for example, a board game. The number on each side is unique among the set, so that no ties can be formed.
There are three properties of fairness, with increasing strength:
It is also desired that any subset of dice taken from the set and rolled together should also have the same properties, so they can be used for fewer players as well.
Configurations where all die have the same number of sides are presented here, but alternative configurations might instead choose mismatched dice to minimize the number of sides, or minimize the largest number of sides on a single die.
Sets may be optimized for smallest least common multiple, fewest total sides, or fewest sides on the largest die. Optimal results in each of these categories have been proven by exhaustion for up to 4 dice.
The two player case is somewhat trivial. Two coins (2-sided die) can be used:
An optimal and permutation-fair solution for three 6-sided dice was found by Robert Ford in 2010. There are several optimal alternatives using mismatched dice.