# 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 (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.

## 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 3 six-sided dice was found by Robert Ford in 2010. There are several optimal alternatives using mismatched dice.
Numbers on each 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 4 twelve-sided dice was found by Robert Ford in 2010. Alternative optimal configurations for mismatched dice were found by Eric Harshbarger.
Numbers on each 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 5 dice, but none is known to be optimal.
A not-permutation-fair solution for 5 sixty-sided dice was found by James Grime and Brian Pollock. A permutation-fair solution for a mixed set of 1 thirty-six-sided die, 2 forty-eight-sided dice, 1 fifty-four-sided die, and 1 twenty-sided die was found by Eric Harshbarger in 2023.
A permutation-fair solution for 5 sixty-sided dice was found by Paul Meyer in 2023.
Numbers on each die :
| Die 1 | 1 | 10 | 19 | 20 | 21 | 22 | 39 | 40 | 41 | 42 | 51 | 60 | 61 | 62 | 71 | 80 | 81 | 90 | 99 | 100 |
| Die 1 | 109 | 118 | 119 | 120 | 121 | 122 | 123 | 132 | 133 | 150 | 151 | 168 | 169 | 178 | 179 | 180 | 181 | 182 | 183 | 192 |
| Die 1 | 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 |
| Die 2 | 101 | 108 | 112 | 115 | 126 | 129 | 134 | 141 | 145 | 146 | 155 | 156 | 160 | 167 | 172 | 175 | 187 | 188 | 196 | 197 |
| Die 2 | 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 |
| Die 3 | 102 | 107 | 111 | 116 | 125 | 130 | 135 | 140 | 143 | 148 | 153 | 158 | 161 | 166 | 171 | 176 | 185 | 190 | 194 | 199 |
| Die 3 | 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 |
| Die 4 | 103 | 106 | 110 | 117 | 127 | 128 | 137 | 138 | 142 | 149 | 152 | 159 | 163 | 164 | 173 | 174 | 184 | 191 | 195 | 198 |
| Die 4 | 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 |
| Die 5 | 104 | 105 | 113 | 114 | 124 | 131 | 136 | 139 | 144 | 147 | 154 | 157 | 162 | 165 | 170 | 177 | 186 | 189 | 193 | 200 |
| Die 5 | 206 | 207 | 215 | 216 | 222 | 229 | 233 | 236 | 242 | 249 | 253 | 256 | 265 | 268 | 273 | 276 | 285 | 288 | 292 | 299 |