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.[1][2]

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]

  • 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 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 1 1 4
Die 2 2 3

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 1 1 5 10 11 13 17
Die 2 3 4 7 12 15 16
Die 3 2 6 8 9 14 18

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 1 1 8 11 14 19 22 27 30 35 38 41 48
Die 2 2 7 10 15 18 23 26 31 34 39 42 47
Die 3 3 6 12 13 17 24 25 32 36 37 43 46
Die 4 4 5 9 16 20 21 28 29 33 40 44 45

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 1 1 10 19 20 21 22 39 40 41 42 51 60 61 62 71 80 81 90 99 100
109 118 119 120 121 122 123 132 133 150 151 168 169 178 179 180 181 182 183 192
201 202 211 220 221 230 239 240 241 250 259 260 261 262 279 280 281 282 291 300
Die 2 2 9 13 16 25 28 33 36 45 48 52 59 65 68 72 79 85 86 94 95
101 108 112 115 126 129 134 141 145 146 155 156 160 167 172 175 187 188 196 197
203 210 212 219 225 226 234 235 244 247 251 258 266 267 274 275 283 290 294 297
Die 3 3 8 12 17 24 29 32 37 44 49 53 58 64 69 73 78 83 88 92 97
102 107 111 116 125 130 135 140 143 148 153 158 161 166 171 176 185 190 194 199
204 209 213 218 223 228 232 237 243 248 252 257 264 269 272 277 284 289 293 298
Die 4 4 7 11 18 26 27 34 35 43 50 54 57 63 70 74 77 84 87 93 96
103 106 110 117 127 128 137 138 142 149 152 159 163 164 173 174 184 191 195 198
205 208 214 217 224 227 231 238 245 246 254 255 263 270 271 278 286 287 295 296
Die 5 5 6 14 15 23 30 31 38 46 47 55 56 66 67 75 76 82 89 91 98
104 105 113 114 124 131 136 139 144 147 154 157 162 165 170 177 186 189 193 200
206 207 215 216 222 229 233 236 242 249 253 256 265 268 273 276 285 288 292 299

See also

References

  1. ^ a b c d e 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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