Intransitive dice

A set of dice is intransitive (or nontransitive) if it contains X>2 dice, X1, X2, and X3... with the property that X1 rolls higher than X2 more than half the time, and X2 rolls higher than X3 etc... more than half the time, but where it is not true that X1 rolls higher than Xn more than half the time. In other words, a set of dice is intransitive if the binary relation – X rolls a higher number than Y more than half the time – on its elements is not transitive. More simply, X1 normally beats X2, X2 normally beats X3, but X1 does not normally beat Xn.

It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. This is different in that instead of only "A does not normally beat C" it is now "C normally beats A". Using such a set of dice, one can invent games which are biased in ways that people unused to intransitive dice might not expect (see Example).

Example

An example of intransitive dice (opposite sides have the same value as those shown).

Consider the following set of dice.

• Die A has sides 2, 2, 4, 4, 9, 9.
• Die B has sides 1, 1, 6, 6, 8, 8.
• Die C has sides 3, 3, 5, 5, 7, 7.

The probability that A rolls a higher number than B, the probability that B rolls higher than C, and the probability that C rolls higher than A are all ⁠5/9⁠, so this set of dice is intransitive. In fact, it has the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time.

Now, consider the following game, which is played with a set of dice.

1. The first player chooses a die from the set.
2. The second player chooses one die from the remaining dice.
3. Both players roll their die; the player who rolls the higher number wins.

If this game is played with a transitive set of dice, it is either fair or biased in favor of the first player, because the first player can always find a die that will not be beaten by any other dice more than half the time. If it is played with the set of dice described above, however, the game is biased in favor of the second player, because the second player can always find a die that will beat the first player's die with probability ⁠5/9⁠. The following tables show all possible outcomes for all three pairs of dice.

Player 1 chooses die A Player 2 chooses die C					Player 1 chooses die B Player 2 chooses die A					Player 1 chooses die C Player 2 chooses die B
A C	2	4	9		B A	1	6	8		C B	3	5	7
3	C	A	A		2	A	B	B		1	C	C	C
5	C	C	A		4	A	B	B		6	B	B	C
7	C	C	A		9	A	A	A		8	B	B	B

If one allows weighted dice, i.e., with unequal probability weights for each side, then alternative sets of three dice can achieve even larger probabilities than ${\frac {5}{9}}\approx 0.56$ that each die beats the next one in the cycle. The largest possible probability is one over the golden ratio, ${\frac {1}{\varphi }}\approx 0.62$.

Variations

Efron's dice

Efron's dice are a set of four intransitive dice invented by Bradley Efron.

Representation of Efron's dice. The back side of each die has the same faces as the front except for the 5, 5, 1 die (where the back side of 5 is 1, and the back side of 1 is 5).

The four dice A, B, C, D have the following numbers on their six faces:

• A: 4, 4, 4, 4, 0, 0
• B: 3, 3, 3, 3, 3, 3
• C: 6, 6, 2, 2, 2, 2
• D: 5, 5, 5, 1, 1, 1

Each die is beaten by the previous die in the list with wraparound, with probability ⁠2/3⁠. C beats A with probability ⁠5/9⁠, and B and D have equal chances of beating the other. If each player has one set of Efron's dice, there is a continuum of optimal strategies for one player, in which they choose their die with the following probabilities, where 0 ≤ x ≤ ⁠3/7⁠:

P(choose A) = x

P(choose B) = ⁠1/2⁠ - ⁠5/6⁠x

P(choose C) = x

P(choose D) = ⁠1/2⁠ - ⁠7/6⁠x

Miwin's dice

Miwin's dice

Main article: Miwin's dice

Miwin's Dice were invented in 1975 by the physicist Michael Winkelmann.

Consider a set of three dice, III, IV and V such that

• die III has sides 1, 2, 5, 6, 7, 9
• die IV has sides 1, 3, 4, 5, 8, 9
• die V has sides 2, 3, 4, 6, 7, 8

Then:

• the probability that III rolls a higher number than IV is ⁠17/36⁠
• the probability that IV rolls a higher number than V is ⁠17/36⁠
• the probability that V rolls a higher number than III is ⁠17/36⁠

Intransitive dice set for more than two players

A number of people have introduced variations of intransitive dice where one can compete against more than one opponent.

Three players

Oskar dice

Oskar van Deventer introduced a set of seven dice (all faces with probability ⁠1/6⁠) as follows:

• A: 2, 02, 14, 14, 17, 17
• B: 7, 07, 10, 10, 16, 16
• C: 5, 05, 13, 13, 15, 15
• D: 3, 03, 09, 09, 21, 21
• E: 1, 01, 12, 12, 20, 20
• F: 6, 06, 08, 08, 19, 19
• G: 4, 04, 11, 11, 18, 18

One can verify that A beats {B,C,E}; B beats {C,D,F}; C beats {D,E,G}; D beats {A,E,F}; E beats {B,F,G}; F beats {A,C,G}; G beats {A,B,D}. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. Namely,

• G beats {A,B}; F beats {A,C}; G beats {A,D}; D beats {A,E}; D beats {A,F}; F beats {A,G};
• A beats {B,C}; G beats {B,D}; A beats {B,E}; E beats {B,F}; E beats {B,G};
• B beats {C,D}; A beats {C,E}; B beats {C,F}; F beats {C,G};
• C beats {D,E}; B beats {D,F}; C beats {D,G};
• D beats {E,F}; C beats {E,G};
• E beats {F,G}.

Whatever the two opponents choose, the third player will find one of the remaining dice that beats both opponents' dice.

Grime dice

Dr. James Grime discovered a set of five dice as follows:

• A: 2, 2, 2, 7, 7, 7
• B: 1, 1, 6, 6, 6, 6
• C: 0, 5, 5, 5, 5, 5
• D: 4, 4, 4, 4, 4, 9
• E: 3, 3, 3, 3, 8, 8

One can verify that, when the game is played with one set of Grime dice:

• A beats B beats C beats D beats E beats A (first chain);
• A beats C beats E beats B beats D beats A (second chain).

However, when the game is played with two such sets, then the first chain remains the same, except that D beats C, but the second chain is reversed (i.e. A beats D beats B beats E beats C beats A). Consequently, whatever dice the two opponents choose, the third player can always find one of the remaining dice that beats them both (as long as the player is then allowed to choose between the one-die option and the two-die option):

Sets chosen by opponents		Winning set of dice
		Type	Number
A	B	E	1
A	C	E	2
A	D	C	2
A	E	D	1
B	C	A	1
B	D	A	2
B	E	D	2
C	D	B	1
C	E	B	2
D	E	C	1

Four players

It has been proved that a four player set would require at least 19 dice. In July 2024 GitHub user NGeorgescu published a set of 23 eleven sided dice which satisfy the constraints of the four player intransitive dice problem. The set has not been published in an academic journal or been peer-reviewed.

Four players

A four-player set is proved to require at least 19 dice.

Georgescu dice

In 2024, American scientist Nicholas S. Georgescu discovered a set of 23 dice which solve the four-player intransitive dice problem.

Four-player intransitive dice win graph

0	40	61	83	105	116	158	173	203	213	234
1	29	46	89	109	119	153	175	196	226	243
2	41	54	72	113	122	148	177	189	216	252
3	30	62	78	94	125	143	179	205	229	238
4	42	47	84	98	128	138	181	198	219	247
5	31	55	90	102	131	156	183	191	209	233
6	43	63	73	106	134	151	162	184	222	242
7	32	48	79	110	137	146	164	200	212	251
8	44	56	85	114	117	141	166	193	225	237
9	33	64	91	95	120	159	168	186	215	246
10	45	49	74	99	123	154	170	202	228	232
11	34	57	80	103	126	149	172	195	218	241
12	23	65	86	107	129	144	174	188	208	250
13	35	50	69	111	132	139	176	204	221	236
14	24	58	75	92	135	157	178	197	211	245
15	36	66	81	96	115	152	180	190	224	231
16	25	51	87	100	118	147	182	206	214	240
17	37	59	70	104	121	142	161	199	227	249
18	26	67	76	108	124	160	163	192	217	235
19	38	52	82	112	127	155	165	185	207	244
20	27	60	88	93	130	150	167	201	220	230
21	39	68	71	97	133	145	169	194	210	239
22	28	53	77	101	136	140	171	187	223	248

Li dice

Youhua Li subsequently developed a set of 19 dice with 171 faces each that solves the four-player problem. This has been shown to be extensible for any number of dice given a domination graph with n nodes, producing dice with n(n−1)/2 faces.

Intransitive 12-sided dice

In analogy to the intransitive six-sided dice, there are also dodecahedra which serve as intransitive twelve-sided dice. The points on each of the dice result in the sum of 114. There are no repetitive numbers on each of the dodecahedra.

Miwin's dodecahedra (set 1) win cyclically against each other in a ratio of 35:34.

The miwin's dodecahedra (set 2) win cyclically against each other in a ratio of 71:67.

Set 1:

D III	purple	1	2			5	6	7		9	10	11			14	15	16		18
D IV	red	1		3	4	5			8	9	10		12	13	14			17	18
D V	dark grey		2	3	4		6	7	8			11	12	13		15	16	17

• 
  D III
• 
  D IV
• 
  D V

Set 2:

D VI	cyan	1	2	3	4					9	10	11	12	13	14			17	18
D VII	pear green	1	2			5	6	7	8	9	10					15	16	17	18
D VIII	light grey			3	4	5	6	7	8			11	12	13	14	15	16

• 
  D VI
• 
  D VII
• 
  D VIII

Intransitive prime-numbered 12-sided dice

It is also possible to construct sets of intransitive dodecahedra such that there are no repeated numbers and all numbers are primes. Miwin's intransitive prime-numbered dodecahedra win cyclically against each other in a ratio of 35:34.

Set 1: The numbers add up to 564.

PD 11	grey to blue	13	17	29	31	37	43	47	53	67	71	73	83
PD 12	grey to red	13	19	23	29	41	43	47	59	61	67	79	83
PD 13	grey to green	17	19	23	31	37	41	53	59	61	71	73	79

• 
  PD 11
• 
  PD 12
• 
  PD 13

Set 2: The numbers add up to 468.

PD 1	olive to blue	7	11	19	23	29	37	43	47	53	61	67	71
PD 2	teal to red	7	13	17	19	31	37	41	43	59	61	67	73
PD 3	purple to green	11	13	17	23	29	31	41	47	53	59	71	73

• 
  PD 1
• 
  PD 2
• 
  PD 3

Generalized Muñoz-Perera's intransitive dice

A generalization of sets of intransitive dice with $N$ faces is possible. Given $N\geq 3$, we define the set of dice $\{D_{n}\}_{n=1}^{N}$ as the random variables taking values each in the set $\{v_{n,j}\}_{j=1}^{J}$ with

$\mathbb {P} \left[D_{n}=v_{n,j}\right]={\frac {1}{J}}$,

so we have $N$ fair dice of $J$ faces.

To obtain a set of intransitive dice is enough to set the values $v_{n,j}$ for $n,j=1,2,\ldots ,N$ with the expression

$v_{n,j}=(j-1)N+(n-j){\text{mod}}(N)+1$,

obtaining a set of $N$ fair dice of $N$ faces

Using this expression, it can be verified that

$\mathbb {P} \left[D_{m}<D_{n}\right]={\frac {1}{2}}+{\frac {1}{2N}}-{\frac {(n-m){\text{mod}}(N)}{N^{2}}}$,

So each die beats $\lfloor N/2-1\rfloor$ dice in the set.

Examples

3 faces

$D_{1}$	1	6	8
$D_{2}$	2	4	9
$D_{3}$	3	5	7

The set of dice obtained in this case is equivalent to the first example on this page, but removing repeated faces. It can be verified that $D_{3}>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{3}$.

4 faces

$D_{1}$	1	8	11	14
$D_{2}$	2	5	12	15
$D_{3}$	3	6	9	16
$D_{4}$	4	7	10	13

Again it can be verified that $D_{4}>D_{3},D_{3}>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{4}$.

6 faces

$D_{1}$	1	12	17	22	27	32
$D_{2}$	2	7	18	23	28	33
$D_{3}$	3	8	13	24	29	34
$D_{4}$	4	9	14	19	30	35
$D_{5}$	5	10	15	20	25	36
$D_{6}$	6	11	16	21	26	31

Again $D_{6}>D_{5},D_{5}>D_{4},D_{4}>D_{3},D_{3}>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{6}$. Moreover $D_{6}>\{D_{5},D_{4}\},D_{5}>\{D_{4},D_{3}\},D_{4}>\{D_{3},D_{2}\},D_{3}>\{D_{2},D_{1}\},D_{2}>\{D_{1},D_{6}\}\ {\text{and}}\ D_{1}>\{D_{6},D_{5}\}$.

See also

• Blotto games
• Freivalds' algorithm
• Go First Dice
• Nontransitive game
• Rock paper scissors
• Condorcet's voting paradox

Sources

• Gardner, Martin (2001). The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems: Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics (1st ed.). New York: W. W. Norton & Company. p. 286–311.
• Spielerische Mathematik mit Miwin'schen Würfeln (in German). Bildungsverlag Lemberger. ISBN 978-3-85221-531-0.

External links

• MathWorld page
• Ivars Peterson's MathTrek - Tricky Dice Revisited (April 15, 2002)
• Jim Loy's Puzzle Page
• Miwin official site (in German)
• Open Source nontransitive dice finder
• Non-transitive Dice by James Grime
• Maths Gear
• Conrey, B., Gabbard, J., Grant, K., Liu, A., & Morrison, K. (2016). Intransitive dice. Mathematics Magazine, 89(2), 133-143. Awarded by Mathematical Association of America
• Timothy Gowers' project on intransitive dice
• Klarreich, Erica (2023-01-19). "Mathematicians Roll Dice and Get Rock-Paper-Scissors". Quanta Magazine.
• Adrián Muñoz Perera's site'
• Introduction to Non-Transitive Gambling Bets for Magicians by Bruce Carlley. This is the ONLY book on this topic.