Dice as Random Number Generators

The main purpose of pipped and numbered dice is the generation of random numbers. In this section we describe various such number generators based mainly on isohedral shapes. “Regular” N-sided dice typically generate random numbers in the range 1..N or 0..N-1, but other ranges are possible, as shown in the sequel.

 

Let M denote the number of different numbers on an N-sided die. In most cases, N=M, however there are dice where the same number appears more than once, in which case M<N.

 

In this section we start with a short math section, mainly for mathematically interested alealogists. We then show dice with M=1 up to M=100. For each value of M, we list dice according to the characteristics of the random numbers they generate:

        uniform distributions: rolling a die produces numbers which are equally likely

        0..M-1: the most common member of this family is the D10 with numbers from 0 to 9

        1..M: the most common member of this family is the “normal” cube, a hexahedron with numbers 1..6

        Consecutive numbers other than 0..M-1 or 1..M: These dice generate random numbers with an “offset”, e.g. a D6 numbered 7,8,9,10,11,12.

        Non-consecutive numbers: The numbers of such a die are not consecutive (i.e., there are some numbers “missing” in the sequence), but they still are rolled equally likely. The most prominent member of this family is the backgammon doubling D6 with numbers 2,4,8,16,32,64.

        non-uniform distributions: some numbers are more likely than others. These dice are specially designed to produce these numbers, e.g. as cheaters or for special games.

Towards the end some special dice sets, such as Sicherman dice, Non-transitive dice, cheater dice or trick dice are shown and explained.

Math

The main purpose of dice, apart from being collected, is to generate random numbers, or, in mathematical terms, a discrete random variable. The statistical properties of a die can be described by its probability density function. The most common die, a cube (D6) numbered 1,2,3,4,5,6 generates a random variable with discrete values 1,2,3,4,5, and 6. If the die is fair, each of these numbers is rolled with equal probability of 1/6.

 

Let us assume that a die has N faces with M different numbers. If all the numbers are different, then N=M. If all numbers appear K times, then N=K·M. Such dice are “fair” in the sense that the probability density function is uniform. These dice allow to make equal-probability random selections between a number of choices.

 

On most dice each number is printed only once, in which case K=1, as for the “regular” cube mentioned above. Other examples include the “binary cube”, a D6 numbered 0,0,0,1,1,1, hence K=3 because each number appears three times. The die with the largest K in my collection is the D-Total by A. Simkin / L. Zocchi, a deltoidal icositetrahedron with N=24, showing, among others, 8 x 1..3 pips.

 

For many common fair dice, its M different numbers can be derived from the numbers 1..M with a linear function aX+b. For a=1, the resulting random variable assumes values of consecutive integers with “offset” b.

 

Examples:

        a=1, b=0: This generates the sequence 1,2,3,..,M. The most common die in this family is the regular cube, numbered 1..6

        a=1, b=-1: This generates the sequence  0,1,2,..,M-1. The most common die in this family is the D10, numbered 0..9

        M=6, a=1, b=6: This is used by an educational die with numbers 7,8,9,10,11,12.

        M=6, a=10, b=0: This yields die with numbers 10,20,30,40,50,60.

        N=6, M=2, K=3, a=4, b=-2: This yields a loaded die numbered 2,2,2,6,6,6

        M=6, a=1/6, b=0: This yields a fractional die with numbers 1/6, 1/3, 1/2, 2/3, 5/6, 1

 

Another fairly common die is the backgammon doubling. Its probability distribution function is uniform (i.e., it is fair), and its numbers can be generated from the sequence 1..M with the function 2^(X-b), power of 2.

Examples:

N=6, M=6, K=1, b=0: D6 backgammon doubling die numbered 2,4,8,16,32,64.

N=8, M=8, K=1, b=-1: D8 backgammon doubling die numbered 1,2,4,8,16,32,64,128.

 

There are fractional dice that also use a power law, X^(-1)

N=6, M=6, K=1: D6 fractional die numbered 1,1/2,1/3,1/4,1/5,1/6.

 

There are some dice where some numbers are printed more often than others. Even if the original die is a perfect isohedron (i.e., a fair shape), the resulting random variable is not fair. In mathematical terms, its probability density function is non-uniform. Examples are cheaters, e.g. a D6 numbered 6,6,5,4,3,2. The “6” appears twice, but there is no “1”. Other dice with non uniform distributions include non-transitive dice, Sicherman dice and dice specially designed for particular games. Some of them are described in more detail at the end of this section.

M=1

These are dice that allow rolling a single number only, i.e. a degenerate case of a random number generator. They are mainly used as cheaters, one is a non-transitive die.

 

 

M=2

Binary dice.

Range 0..1

Fair binary dice, generating numbers 0 and 1. All shapes with an even number of faces are possible. Shapes in my collection: hexahedron and octahedron.

 

Not yet found: D2 (coin), D4 2x(1-2); D10 5x(1-2), etc

Range 1..2

Fair binary dice, generating numbers 1 and 2. All shapes with an even number of faces are possible. Shapes in my collection: coin, tetrahedron, and hexahedron.

 

 

Not yet found: D8, 4x(1-2); etc.

Others

Dice with only two numbers, cheaters and non-transitive dice and special dice designed for games.

 

 

 

 

M=3

Dice with three different numbers.

Range 0..2

There is no fair die in my collection that generates the numbers 0..2. However, there are some dice with non-uniform distributions.

 

Range 1..3

Fair ternary dice, generating numbers 1..3. All shapes with a number of faces which is divisible by 3 are possible. Shapes in my collection: triangular prism, hexahedron, and deltoidal icositetrahedron.

 

 

 

Not yet found: D12, 4x(1-3)

 

This is an unfair die:

 

Others

Dice with three numbers other than 1..3 are used as cheaters or in special games.

 

 

 

 

 

M=4

The tetrahedron is the most common die in this family.

Range 0..3

There is no fair die in my collection that generates the numbers 0..3. However, there are some dice with non-uniform distributions.

 

 

Range 1..4

Fair quaternary dice, generating numbers 1..4. All shapes with a number of faces which is divisible by 4 are possible. Shapes in my collection: tetrahedron, square prism, octahedron, dodecahedron, and deltoidal icositetrahedron.

 

 

 

 

 

Not yet found: D20, 5x(1-4)

 

Unfair dice generating 1..4 (Sicherman):

 

Others

Dice with four numbers other than 1..4 are used as cheaters or in special games.

 

 

 

M=5

Range 0..4

Fair dice generating numbers 0..4. All shapes with a number of faces which is divisible by 5 are possible. Shapes in my collection: pentagonal trapezohedron.

Range 1..5

Fair dice generating numbers 1..5. All shapes with a number of faces which is divisible by 5 are possible. Shapes in my collection: pentagonal prism, pentagonal trapezohedron.

 

Not yet found: D20, 4x1-5

 

This die is not fair:

Others

There are some D10 with all numbers printed twice. These dice are fair.

 

 

There are some hexahedra with one number printed twice, yielding an unfair die.

 

 

M=6

The most common shape of this family is the cube (hexahedron).

Range 0..5

 

Not yet found: D12, 2 x 0-5

 

Range 1..6

Fair dice generating numbers 1..6. All shapes with a number of faces which is divisible by 6 are possible. Shapes in my collection: hexahedron, 6-sided antiprism, 6-sided prism, dodecahedron, deltoidal icositetrahedron

 

 

 

Other shapes:

 

 

 

 

Others

There is a large number of D6 with numbers other than 1..6. These include trick dice, backgammon dice, educational dice, mathematical dice (non-transitive and Sicherman) and special dice designed for games. Most of these dice are hexahedra, and there are also a few dodecahedra.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Not yet found: 11,10,9,8,7,6,5. K=1, a=1, b=5

M=7

There are only very few dice with 7 different numbers.

Range 0..6

There is no fair die numbered 0..6 in my collection. However, there is a special die generating numbers 0..6 as sum of two quaternary dice. It consists of two halves, each with 0,1,2, or 3 dots. When the die is rolled, these halves can move (more or less) independently.

 

Range 1..7

 

Not yet found: D14, 2x(1-7)

M=8

There are many D8 on the market, mostly octahedral numbered 1..8. Besides that there are very few other dice.

Range 1..8

 

Wanted: D8 pipped

Not yet found: D16, 2x(1-8)

Others

M=9

There are only two dice with nine different numbers in my collection, a fair prism and a cheater. There is also a heptagonal prism (missing in my collection).

 

 

 

M=10

Range 0..9

In addition to the “classical” pentagonal trapezohedron, there are also a 10-sided antiprism, an icosahedron and a rhombic triacontahedron with numbers 0..9. There are also two types of flattened octahedra, which are not fair.

 

 

Range 1..10

Dice numbered 1..10 are less popular than those numbered 0..9 because with two of the latter dice numbers 1..100 can be generated (00=100).

 

Not yet found: D20, 2x(1-10)

Others

There are surprisingly many dice with 10 numbers other than 0..9 or 1..10. Most of them are so called place value dice, but there are also 10 sided antiprisms, icosahedra and rhombic triacontahedra.

 

 

 

M=11

Dice with 11 different numbers are very rare. There do not even seem to be cheaters (e.g. a D12 with the 12 replaced by a 1 or vice versa).

 

M=12

Range 1..12

There are many different shapes for dice numbered 1..12: pentagonal dodecahedron, rhombic dodecahedron, 12-sided antiprism, and deltoidal icositetrahedron (with each number printed twice).

 

 

Others

This die can be used to roll minutes from 0 to 55 in steps of 5:

M=13

Dice with 13 different numbers are very rare. This one is numbered 1..13.

 

M=14

This heptagonal trapezohedron is numbered 1..14.

 

There is also a 14 sided die based on the cuboctahedron. However, this is not a numbered die, but one with poker symbols (Card Dice)

M=15

Dice with 15 different numbers are very rare. This one is numbered 1..15.

M=16

The octagonal bipyramid with 16 faces exists both with “regular” and with hexadecimal numbers (0..F).

M=19

There are two icosahedral cheater dice:

M=20

The most popular die numbered 1..20 is the icosahedron, but there is also a 20-sided antiprism.

 

There is only one die in my collection with 20 different numbers other than 1..20:

 

M=24

There are two shapes of dice with 24 different numbers, tetrakishexahedron and deltoidal icositetrahedron.

 

M=30

Rhombic triacontahedron, numbered 1..30:

M=34

Dekaeptagonal Trapezohedron, numbered 1..34:

M=50

Two different shapes for dice numbered 0..49, the eikosipendegonal

trapezohedron and a flattened sphere:

 

M=100

The Zocchihedron, a spherical die numbered 1..100:

 

Fractions

The numbers on the dice presented so far were all integers. There are a few dice with fractions, mainly D6 but also some D8 and D10:

 

 

 

 

 

 

Special Dice Sets

Sum of two or three dice

In many games, two or three dice are used and their sum is used as a random number. If the individual dice yield a uniform distribution, the density function of their sum is no longer uniform. The example of dice with M=2, M=4 and M=6 random numbers is considered below.

M=2: Sum of two or three binary dice

Let us assume that there are three fair binary dice with numbers 0 and 1. A single die generates those numbers with probability 1/2 each. The sum of two dice has three possible outcomes: 0 (with probability 1/4), 1 (1/2), and 2 (1/4). For the sum of three dice, the possible outcomes are 0 (with probability 1/8), 1 (3/8), 2 (3/8), and 3 (1/8)

There is an interesting set of three dice representing these three cases, the so-called Ubiquity Dice by Exile Game Studio. They are all octahedra, the white one being a binary die, the red and the blue one generating the sum of two or three binary dice, respectively.

 

M=4: Sum of two quaternary dice

A fair quaternary die generates numbers 0, 1, 2, and 3 with probability 1/4 each. The sum of two such dice has seven possible outcomes: 0 (with probability 1/16), 1 (1/8), 2 (3/8), 3 (1/2), 4 (3/8), 5 (1/8), and 6 (1/8).

 

The so-called Y2K die is a special die that generates exactly this type of random numbers. It consists of two halves, each with 0,1,2, or 3 dots. When the die is rolled, these halves can move (more or less) independently.

M=6: Double Spinner / Roller

There are spinners and rollers that generates the same probability distribution as the sum of two D6: 2 (with probability 1/36), 3 (1/18), 4 (1/12), 5 (1/9), 6 (5/36), 7 (1/6), 8 (5/36), 9 (1/9), 10 (1/12), 11 (1/18), and 12 (1/36).

The two parts of that spinner move (more or less) independently:

M=6: Triple Roller

There is a unique roller that generates the same probability distribution as the sum of three D6. The three parts of that spinner move (more or less) independently:

M=6: Sicherman Dice

A fair “regular” cube generates numbers 1, 2, 3, 4, 5, and 6 with probability 1/6 each. The sum of two such dice has eleven possible outcomes: 2 (with probability 1/36), 3 (1/18), 4 (1/12), 5 (1/9), 6 (5/36), 7 (1/6), 8 (5/36), 9 (1/9), 10 (1/12), 11 (1/18), and 12 (1/36).

Sicherman dice are the only other pair of 6-sided dice bearing only positive integers which have the same probability distribution as a pair of normal dice. These dice were discovered by Colonel George Sicherman, of Buffalo, New York and were originally reported by Martin Gardner in a 1978 article in Scientific American. The numbers can be arranged so that all pairs of numbers on opposing sides sum to equal numbers, 5 for the first and 9 for the second.

http://en.wikipedia.org/wiki/Sicherman_dice

 

Non Transitive Dice

A set of nontransitive dice is a set of dice for which the relation "is more likely to roll a higher number" is not transitive. This situation is similar to that in the game Rock, Paper, Scissors, in which each element has an advantage over one choice and a disadvantage to the other.

 

Each die beats the one in clockwise direction

 

 

Three dice:

        3,3,5,5,7,7; 2,2,4,4,9,9; and 1,1,6,6,8,8 (Efron set. Uses all the numbers from 1 to 9, and the face value of each die sums to 30).

        1,1,1,13,13,13; 0,3,3,12,12,12; and 2,2,2,11,11,14 (the face value of each die sums to 42).

        1,4,4,4,4,4; 3,3,3,3,3,6; and 2,2,2,5,5,5 (no face has a number higher than 6, and each die has two different numbers. This is the red/blue/green pipped set shown above)

        1,2,5,6,7,9; 1,3,4,5,8,9; 2,3,4,6,7,8 (this is the wooden pipped set shown above, blue/orange/black). Minwin dice, invented in 1975 by Michael Winkelmann. No number appears twice on the same die, the face value of each die sums to 30, and the average is 5.

        1,1,1,13,13,13; 0,3,3,12,12,12; 2,2,2,11,11,14 (Schwenk's dice)

Four dice:

        6,6,2,2,2,2; 5,5,5,1,1,1; 4,4,4,4,0,0; and 3,3,3,3,3,3 (Efron set. No number is greater than 6. This is the white numbered set shown above)

        2,3,3,9,10,11; 0,1,7,8,8,8; 5,5,6,6,6,6; and 4,4,4,4,12,12 (Efron set).

        1,2,3,9,10,11; 0,1,7,8,8,9; 5,5,6,6,7,7; and 3,4,4,5,11,12 (Efron set)

        2,3,3,9,10,11; 0,1,7,8,8,8; 5,5,6,6,6,6; and 4,4,4,4,12,12;

 

References:

http://en.wikipedia.org/wiki/Nontransitive_dice

http://www.grand-illusions.com/magicdice.htm

http://miwin.com/

http://www.sciencenews.org/articles/20020420/mathtrek.asp

http://www.maa.org/mathland/mathtrek_04_15_02.html

http://plus.maths.org/issue41/features/hobbs/index.html

Cheater Dice

Cheater dice generate random numbers that differ from “fair” dice. Some dice generate on average higher numbers than fair dice, others lower numbers. Other sets generate only a few (down to a single) random numbers and are used to tweak results in games such as craps.

 

 

 

 

 

 

 

 

 

 

 

Loaded Dice

These dice are physically loaded, such that a particular number is rolled with very high probability

 

 

 

 

 

Place Value Dice

Trick Dice

 

        Trick: A magician can add the numbers of the five dice within seconds – the result will be different each time the dice are rolled.

        Solution: The answer will be a four digit number. Sum the final digits of the dice to get the final two digits of the answer; subtract this from 50 to get the first two digits.

        Explanation: http://www.mathpuzzle.com/dicetrick.txt

 

Formula Dé Game

Blue D30: 3 x (21-30)

Black D20: regular

Purple D20: 2 x (11-20)

Green D12: 2 x (7-12)

Red D8: 2 x (5-8)

Orange D6: 4,4,4,3,3,2

Yellow D4: 2 x (1-2)

 

Monopoly Dice Game

 

 

 

 

Monopoly Premier League

D12, 2x1-6

 

Rummykub Rummy Dice Game

Rummykub Rummy Dice Game 1995 Pressman

 

From left to right:

Top row: 13,12,9,8,4,3 ; 13,12,8,7,4,3 ; 13,10,9,5,4,1 ; 13,9,8,5,4,face

Bottom row: 12,11,7,6,3,2 ; 12,11,8,7,3,2 ; 11,10,6,5,2,1 ; 11,10,7,6,2,1 ; 10,9,6,5,1,face