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 3 six-sided dice was found by Robert Ford in 2010.^[1] 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.^[1]

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.^[3]

A permutation-fair solution for 5 sixty-sided dice was found by Paul Meyer in 2023.^[4]

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
	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