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. The numbers on each die give the sum of 30 and have an arithmetic mean of five.
Description
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 (gold, silver).
First set of Miwin's dice: III, IV, V
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:
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 (III, IV, V, III) 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.
Second set of Miwin's dice: IX, X, XI
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:
The probability that a given die in the sequence (XI, X, IX, XI) 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.
Third and fourth set of Miwin's dice
Third set:
fourth set:
The probability that a given die in the first sequence (5, 3, 1, 5) or the second sequence (6, 4, 2, 6) will roll a higher number than the next in the sequence is 5/9; a lower number, 4/9.
Mathematical attributes
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.
Probabilities
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.
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.
Cumulative frequency
Reversed intransitivity
Removing the common dots of Miwin's Dice reverses intransitivity.
III:IV = 17:16, III':IV' = 4:5 → IV' > III'
analog: III/V, IV/V
Equal distribution of random numbers
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 (to obtain random numbers between 8 and 16, inclusive, follow the 1 – 9 instructions below, and add seven to the result of each roll).
1 – 9: 1 die is rolled (chosen at random): P(1) = P(2) = ... = P(9) = 1/9
0 – 80: 2 dice are rolled (chosen at random), always subtract 1: P(0) = P(1) = ... = P(80) = 1/9² = 1/81
Other distributions
0 – 90 (throw 3 times)
Governing probability: P(0) = P(1) = ... = P(90) = 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 (possible scores: 10, 20, 30, 40, 50, 60, 70, 80, 90) 1st throw is not 9: gives 10 times 1st throw, plus 2nd throw 1st throw is equal to the 3rd throw: gives 2nd throw (possible scores: 1, 2, 3, 4, 5, 6, 7, 8, 9) All dice equal: gives 0 All dice 9: no score
Sample:
This gives 91 numbers, from 0 – 90 with the probability of 8 / 9³, 8 × 91 = 728 = 9³ − 1
0 – 103 (throw 3 times)
Governing probability: P(0) = P(1) = ... = P(103) = 7/9³ = 7/729
This gives 104 numbers from 0 – 103 with the probability of 7 / 9³, 7 × 104 = 728 = 9³ − 1
0 – 728 (throw 3 times)