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.

Properties

There are three properties of fairness, with increasing strength:
  • Go-first-fair - Each player has an equal chance of rolling the highest number.
  • 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 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.

Configurations

Two players

The two player case is somewhat trivial. Two coins 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.
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.
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.
A permutation-fair solution for five 60-sided dice was found by Paul Meyer in 2023.
Die 11101920212239404142516061627180819099100
Die 1109118119120121122123132133150151168169178179180181182183192
Die 1201202211220221230239240241250259260261262279280281282291300
Die 229131625283336454852596568727985869495
Die 2101108112115126129134141145146155156160167172175187188196197
Die 2203210212219225226234235244247251258266267274275283290294297
Die 338121724293237444953586469737883889297
Die 3102107111116125130135140143148153158161166171176185190194199
Die 3204209213218223228232237243248252257264269272277284289293298
Die 447111826273435435054576370747784879396
Die 4103106110117127128137138142149152159163164173174184191195198
Die 4205208214217224227231238245246254255263270271278286287295296
Die 556141523303138464755566667757682899198
Die 5104105113114124131136139144147154157162165170177186189193200
Die 5206207215216222229233236242249253256265268273276285288292299