Intransitive dice


A set of dice is intransitive if it contains dice, with the property that rolls higher than more than half the time, rolls higher than more than half the time, and so on, but does roll higher than more than half the time. In other words, a set of dice is intransitive if the binary relation – rolls a higher number than more than half the time – on its elements is not transitive. More simply, normally beats, normally beats, but does normally beat.
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 " does not normally beat " it is now " normally beats ". Using such a set of dice, one can invent games which are biased in ways that people unused to intransitive dice might not expect.

Example

[Image:Intransitive dice 2.svg|thumb|240px|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, 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. The following tables show all possible outcomes for all three pairs of dice.
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 that each die beats the next one in the cycle. The largest possible probability is one over the golden ratio,.

Variations

Efron's dice

Efron's dice are a set of four intransitive dice invented by Bradley Efron.
[Image:Efron dice 2.svg|thumb|320px|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. C beats A with probability, 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 :

Miwin's dice

[Image:Miwin Wuerfel Titan.gif|thumb|Miwin's dice IX, X, XI]
Miwin's dice were invented in 1975 by the physicist Michael Winkelmann. Miwin's dice are a set of nontransitive dice invented in 1975 by the physicist Michael Winkelmann. They consist of three different dice with faces bearing numbers from one to nine; opposite faces sum to nine, ten or eleven.Miwin's dice facilitate generating numbers at random, within a given range, such that each included number is equally-likely to occur. In order to obtain a range that does not begin with 1 or 0, simply add a constant value to bring it into that range.1 – 9: 1 die is rolled : P = P =... = P = 1/90 – 80: 2 dice are rolled, always subtract 1: P = P =... = P = 1/9² = 1/81
The numbers on each die give the sum of 30 and have an arithmetic mean of five. Miwin's dice have six sides, each of which bear a number, depicted in a pattern of dots. The standard set is made of wood; special designs are made of titanium or other materials.
  • 1/3 of the die-face values can be divided by three without carry over.
  • 1/3 of the die-face values can be divided by three having a carry over of one.
  • 1/3 of the die-face values can be divided by three having a carry over of two.
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
  • the probability that IV rolls a higher number than V is
  • the probability that V rolls a higher number than III is
This is because the probability for a given number with all three dice is 11/36, for a given rolled double is 1/36, for any rolled double 1/4. The probability to obtain a rolled double is only 50% compared to normal dice.
The dice in the first and second Miwin sets have similar attributes: each die bears each of its numbers exactly once, the sum of the numbers is 30, and each number from one to nine is spread twice over the three dice. This attribute characterizes the implementation of intransitive dice, enabling the different game variants. All the games need only three dice, in comparison to other theoretical nontransitive dice, designed in view of mathematics, such as Efron's dice. In the first set, each die is named for the sum of its two lowest numbers. The dots on each die are colored blue, red or black. Each die has the following numbers:
Die IIIwith red dots125679
Die IVwith blue dots134589
Die Vwith black dots234678

Numbers 1 and 9, 2 and 7, and 3 and 8 are on opposite sides on all three dice. Additional numbers are 5 and 6 on die III, 4 and 5 on die IV, and 4 and 6 on die V. The dice are designed in such a way that, for every die, another will usually win against it. The probability that a given die in the sequence will roll a higher number than the next in the sequence is 17/36; a lower number, 16/36. Thus, die III tends to win against IV, IV against V, and V against III. Such dice are known as nontransitive.
In the second set, each die is named for the sum of its lowest and highest numbers. The dots on each die are colored yellow, white or green. Each die has the following numbers:
Die IXwith yellow dots135678
Die Xwith white dots124689
Die XIwith green dots234579

The probability that a given die in the sequence will roll a higher number than the next in the sequence is 17/36; a lower number, 16/36. Thus, die XI tends to win against X, X against IX, and IX against XI.
In the third set:
Die MW 5with blue numbers56781516
Die MW 3with red numbers3411121314
Die MW 1with black numbers129101718

In the fourth set:
Die MW 6with yellow numbers569101314
Die MW 4with white numbers34781718
Die MW 2with green numbers1211121516

The probability that a given die in the first sequence or the second sequence will roll a higher number than the next in the sequence is 5/9; a lower number, 4/9.

Other distributions

In the 0 – 90 distribution, the governing probability is P = P =... = P = 8/9³ = 8/729. To obtain an equal distribution with numbers from 0 – 90, all three dice are rolled, one at a time, in a random order. The result is calculated based on the following rules:
  • 1st throw is 9, 3rd throw is not 9: gives 10 times 2nd throw
  • 1st throw is not 9: gives 10 times 1st throw, plus 2nd throw
  • 1st throw is equal to the 3rd throw: gives 2nd throw
  • All dice equal: gives 0
  • All dice 9: no score
Sample:
1st throw2nd throw3rd throwEquationResult
99not 910 times 990
91not 910 times 110
84not 8 + 484
13not 1 + 313
7877 = 7, gives 88
444all equal0
999all 9-

This gives 91 numbers, from 0 – 90 with the probability of 8 / 9³, 8 × 91 = 728 = 9³ − 1. In the 0 – 103 distribution, the governing probability is P = P =... = P = 7/9³ = 7/729. This gives 104 numbers from 0 – 103 with the probability of 7 / 9³, 7 × 104 = 728 = 9³ − 1
In the 0 – 728 distribution, the governing probability is P = P =... = P = 1 / 9³ = 1 / 729. This gives 729 numbers, from 0 – 728, with the probability of 1 / 9³. This system yields this maximum: 8 × 9² + 8 × 9 + 8 × 9° = 648 + 72 + 8 = 728 = 9³ − 1. One die is rolled at a time, taken at random. Create a number system of base 9:
  • 1 must be subtracted from the face value of every roll because there are only 9 digits in this number system
  • × 81 + × 9 + × 1
Examples:
1st throw2nd throw3rd throwEquationResult
9998 × 9² + 8 × 9 + 8728
4723 × 9² + 6 × 9 + 1298
2411 × 9² + 4 × 9 + 0117
1340 × 9² + 3 × 9 + 330
7776 × 9² + 6 × 9 + 6546
1110 × 9² + 0 × 9 + 00
4263 × 9² + 1 × 9 + 5257

Games

Since the middle of the 1980s, the press wrote about the games. Winkelmann presented games himself, for example, in 1987 in Vienna, at the "Österrechischen Spielefest, Stiftung Spielen in Österreich", Leopoldsdorf, where "Miwin's dice" won the prize "Novel Independent Dice Game of the Year".
In 1989, the games were reviewed by the periodical "Die Spielwiese". At that time, 14 alternatives of gambling and strategic games existed for Miwin's dice. The periodical "Spielbox" had two variants of games for Miwin's dice in the category "Unser Spiel im Heft" : the solitaire game 5 to 4, and the two-player strategic game Bitis.
In 1994, Vienna's Arquus publishing house published Winkelmann's book Göttliche Spiele, which contained 92 games, a master copy for four game boards, documentation about the mathematical attributes of the dice and a set of Miwin's dice. There are even more game variants listed on Winkelmann's website.
Solitaire games and games for up to nine players have been developed. Games are appropriate for players over six years of age. Some games require a game board; playing time varies from 5 to 60 minutes.
In the 1st variant, two dice are rolled, chosen at random, one at a time. Each pair is scored by multiplying the first by nine and subtracting the second from the result: 1st throw × 9 − 2nd throw. This variant provides numbers from 0 – 80 with a probability of 1 / 9² = 1 / 81.
Examples:
1st throw2nd throwEquationResult
999 × 9 − 972
919 × 9 − 180
199 × 1 − 90
299 × 2 − 99
289 × 2 − 810
849 × 8 − 468
139 × 1 − 36

In the 2nd variant, two dice are rolled, chosen at random, one at a time. This variant provides numbers from 0 – 80 with a probability of 1 / 9² = 1 / 81. The pair is scored according to the following rules:
  • 1st throw is 9: gives 10 × 2nd throw − 10
  • 1st throw is not 9: gives 10 × 1st throw + 2nd throw − 10
Examples:
1st throw2nd throwEquationResult
9910 × 9 − 1080
9110 × 1 − 100
8410 × 8 + 4 − 1074
1310 × 1 + 3 − 103

In the 3rd variant, two dice are rolled, chosen at random, one at a time. The score is obtained according to the following rules:
  • Both throws are 9: gives 0
  • 1st throw is 9 and 2nd throw is not 9: gives 10 × 2nd throw
  • 1st throw is 8: gives 2nd throw
  • All others: gives 10 × 1st throw − 2nd throw
Examples:
1st throw2nd throwEquationResult
99-0
9310 × 330
841 × 44
595 × 10 + 959

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 as follows:
  • A: 2, 2, 14, 14, 17, 17
  • B: 7, 7, 10, 10, 16, 16
  • C: 5, 5, 13, 13, 15, 15
  • D: 3, 3, 9, 9, 21, 21
  • E: 1, 1, 12, 12, 20, 20
  • F: 6, 6, 8, 8, 19, 19
  • G: 4, 4, 11, 11, 18, 18
One can verify that A beats ; B beats ; C beats ; D beats ; E beats ; F beats ; G beats. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. Namely,
  • G beats ; F beats ; G beats ; D beats ; D beats ; F beats ;
  • A beats ; G beats ; A beats ; E beats ; E beats ;
  • B beats ; A beats ; B beats ; F beats ;
  • C beats ; B beats ; C beats ;
  • D beats ; C beats ;
  • E beats.
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
The colours are often as shown below
  • A: Red
  • B: Blue
  • C: Green
  • D: Yellow
  • E: Magenta
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 ;
  • A beats C beats E beats B beats D beats A.
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. Consequently, whatever dice the two opponents choose, the third player can always find one of the remaining dice that beats them both :

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.

Georgescu dice

In 2024, American scientist Nicholas S. Georgescu discovered a set of 23 dice which solve the four-player intransitive dice problem.
0406183105116158173203213234
1294689109119153175196226243
2415472113122148177189216252
330627894125143179205229238
442478498128138181198219247
5315590102131156183191209233
6436373106134151162184222242
7324879110137146164200212251
8445685114117141166193225237
933649195120159168186215246
1045497499123154170202228232
11345780103126149172195218241
12236586107129144174188208250
13355069111132139176204221236
1424587592135157178197211245
1536668196115152180190224231
16255187100118147182206214240
17375970104121142161199227249
18266776108124160163192217235
19385282112127155165185207244
2027608893130150167201220230
2139687197133145169194210239
22285377101136140171187223248

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/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 win cyclically against each other in a ratio of 35:34.
The miwin's dodecahedra win cyclically against each other in a ratio of 71:67.
Set 1:
D IIIpurple125679101114151618
D IVred134589101213141718
D Vdark grey234678111213151617

Set 2:
D VIcyan1234910111213141718
D VIIpear green12567891015161718
D VIIIlight grey345678111213141516

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 11grey to blue131729313743475367717383
PD 12grey to red131923294143475961677983
PD 13grey to green171923313741535961717379

Set 2: The numbers add up to 468.
PD 1olive to blue71119232937434753616771
PD 2teal to red71317193137414359616773
PD 3purple to green111317232931414753597173

Generalized Muñoz-Perera's intransitive dice

A generalization of sets of intransitive dice with faces is possible. Given, we define the set of dice as the random variables taking values each in the set with
so we have fair dice of faces.
To obtain a set of intransitive dice is enough to set the values for with the expression
obtaining a set of fair dice of faces
Using this expression, it can be verified that
So each die beats dice in the set.

Examples

3 faces

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.

4 faces

Again it can be verified that.

6 faces

Again. Moreover.