Building Sets of Permutation-Fair Dice

Figures & data

Table 1 A set of four 12-sided dice that are permutation fair.

*Die	Faces
	i	ii	iii	iv	v	vi	vii	viii	ix	x	xi	xii
a	1	8	11	14	19	22	27	30	35	38	41	48
b	2	7	10	15	18	23	26	31	34	39	42	47
c	3	6	12	13	17	24	25	32	36	37	43	46
d	4	5	9	16	20	21	28	29	33	40	44	45

Table 2 A set $\mathcal{F}$ of three 6-sided dice that are permutation fair.

Die	Faces
	i	ii	iii	iv	v	vi
a	1	6	9	10	14	17
b	2	5	7	12	15	16
c	3	4	8	11	13	18

Table 3 Counts for each three-long pattern in the blocks of f.

Pattern	Counts
	${\mathbf{f}}_{1\le j\le 6}$	${\mathbf{f}}_{7\le j\le 12}$	${\mathbf{f}}_{13\le j\le 18}$
abc	2	2	0
acb	2	0	2
bac	0	2	2
bca	2	0	2
cab	0	2	2
cba	2	2	0

Fig. 1 A set of five 120-sided dice that is 5/5 permutation fair. This set of dice was produced by David Blanco and this image is used here with his permission.