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.
Properties There are three properties of fairness, with increasing strength: Go-first-fair - Each player has an equal chance of rolling the highest number (going first). Place-fair - When all the rolls are ranked in order, each player has an equal chance of receiving each rank. Permutation-fair - Every possible ordering of players has an equal probability, which also ensures it is "place-fair".
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 dice 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.
Configurations Two players
The two player case is somewhat trivial. Two coins (2-sided dice) can be used:
Die 114 Die 223
Three players
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.
+ Numbers on each 6-sided die Die 11510111317 Die 2347121516 Die 326891418
Four players
An optimal and permutation-fair solution for four 12-sided dice was found by Robert Ford in 2010. Alternative optimal configurations for mismatched dice were found by Eric Harshbarger.
+ Numbers on each 12-sided die Die 11811141922273035384148 Die 22710151823263134394247 Die 33612131724253236374346 Die 4459162021282933404445
Five players
Several candidates exist for a set of five dice, but none is known to be optimal.
A not-permutation-fair solution for five 60-sided dice was found by James Grime and Brian Pollock. A permutation-fair solution for a mixed set of one 36-sided die, two 48-sided dice, one 54-sided die, and one 20-sided die was found by Eric Harshbarger in 2023.https://intapi.sciendo.com/pdf/10.2478/rmm-2023-0004 Note that some of these dice would not be able to have more than one rotation axis of order more than 2 (as regular polyhedra have)see Point groups in three dimensions.
A permutation-fair solution for five 60-sided dice was found by Paul Meyer in 2023.
+ Numbers on each 60-sided die Die 11101920212239404142516061627180819099100109118119120121122123132133150151168169178179180181182183192201202211220221230239240241250259260261262279280281282291300 Die 229131625283336454852596568727985869495101108112115126129134141145146155156160167172175187188196197203210212219225226234235244247251258266267274275283290294297 Die 338121724293237444953586469737883889297102107111116125130135140143148153158161166171176185190194199204209213218223228232237243248252257264269272277284289293298 Die 447111826273435435054576370747784879396103106110117127128137138142149152159163164173174184191195198205208214217224227231238245246254255263270271278286287295296 Die 556141523303138464755566667757682899198104105113114124131136139144147154157162165170177186189193200206207215216222229233236242249253256265268273276285288292299
See also
Intransitive dice
References
External links Go First Dice - Numberphile Go First Dice Wiki