Go First Dice

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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. [1] [2]

Contents

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: [1]

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. [1]

Configurations

Two players

The two player case is somewhat trivial. Two coins (2-sided die) 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. [1] 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. [1]

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. [3]

A permutation-fair solution for five 60-sided dice was found by Paul Meyer in 2023. [4]

Numbers on each 60-sided die
Die 11101920212239404142516061627180819099100
109118119120121122123132133150151168169178179180181182183192
201202211220221230239240241250259260261262279280281282291300
Die 229131625283336454852596568727985869495
101108112115126129134141145146155156160167172175187188196197
203210212219225226234235244247251258266267274275283290294297
Die 338121724293237444953586469737883889297
102107111116125130135140143148153158161166171176185190194199
204209213218223228232237243248252257264269272277284289293298
Die 447111826273435435054576370747784879396
103106110117127128137138142149152159163164173174184191195198
205208214217224227231238245246254255263270271278286287295296
Die 556141523303138464755566667757682899198
104105113114124131136139144147154157162165170177186189193200
206207215216222229233236242249253256265268273276285288292299

See also

References

  1. 1 2 3 4 5 Harshbarger, Eric (2015). "Go First Dice" . Retrieved 9 Oct 2019.
  2. ↑ Bellos, Alex (18 Sep 2012). "Puzzler develops game-changing Go First dice". The Guardian . Retrieved 9 Oct 2019.
  3. ↑ https://intapi.sciendo.com/pdf/10.2478/rmm-2023-0004
  4. ↑ "significant_solutions", Go First Dice Wiki, archived from the original on 2023-10-02
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