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.

6,6,6,6,6,6

K=6, a=6, b=0

Loaded die

5,5,5,5,5,5

K=6, a=5, b=0

Loaded die

4,4,4,4,4,4

K=6, a=4, b=0

Loaded die

3,3,3,3,3,3

K=6, a=3, b=0

Loaded die

3,3,3,3,3,3

K=6, a=3, b=0

Non-transitive die

Grand Illusions

0,0

Lord of the Rings

Free Peoples

Games Workshop

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.

Hexahedron

3 x 0-1

K=3

Octahedron

4 x 0-1

K=4

Ubiquity Die

Not yet found: D2 (coin), D4 2x(0-1); D10 5x(0-1), 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.

“Coin”

1-2

Bear Cub Machine

Tetrahedron

2 x 1-2

K=2

Formula Dé

Hexahedron

3 x 1-2

K=3

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.

150,150,100,100,100,100

Non-uniform

P(150)=1/3, P(100)=2/3

Monopoly Dice Game

6,6,6,2,2,2

3x(6,2)

K=3, a=4, b=-2

Loaded die

6,6,2,2,2,2

Non-uniform

P(6)=1/3, P(2)=2/3

Non-transitive die

Grand Illusions

6,3,3,3,3,3

Non-uniform

P(6)=1/6, P(3)=5/6

Non-transitive die

Grand Illusions

5,5,5,2,2,2

3x(5,2)

K=3, a=3, b=-1

Non-transitive die

Grand Illusions

5,5,5,1,1,1

3x(5,1)

K=3, a=4, b=-3

Non-transitive die

Grand Illusions

4,4,4,4,4,1

Non-uniform

P(4)=5/6, P(1)=1/6

Non-transitive die

Grand Illusions

4,4,4,4,0,0

Non-uniform

P(4)=2/3, P(0)=1/3

Non-transitive die

Grand Illusions

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.

2,2,1,1,1,1,0,0

Non-uniform

P(2)=1/4, P(1)=1/2, P(0)=1/4

2D2 (sum of 2 binary dice)

Exile Game Studio

(Ubiquity Dice)

2,1,1,0,0,0

Non-uniform

P(2)=1/6, P(1)=1/3, P(0)=1/2

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.

Elongated

triangular prism

HABA

Elongated

triangular prism

Abraham Neddermann

Rounded-off

triangular prism

GameScience

Rounded-off

triangular prism

Crystal Caste

Hexahedron

2 x 1-3

K=2

Hexahedron

2 x 1-3

K=2

Deltoidal

Icositetrahedron

D24, 8 x 1-3

1..3 pips

K=8

D-Total by

A. Simkin /

GameScience

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

This is an unfair die:

3,3,2,2,2,1

Non-uniform,
P(3)=1/3, P(2)=1/2, P(1)=1/6

Dragonland Game?

Others

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

6,6,5,5,1,1

2x(6,5,1)

K=2, non-linear

Loaded die

Koplow

6,6,4,4,2,2

2x(6,4,2)

K=2, a=2, b=0

Loaded die

Koplow

6,6,3,3,2,2

2x(6,3,2)

K=2, non-linear

Loaded die

Koplow

5,5,4,4,3,3

2x(5,4,3)

K=2, a=1, b=2

Consecutive numbers

Loaded die

Koplow

5,4,2

K=1, non-linear

HABA Schleckermaul

5,5,4,4,1,1

2x(5,4,1)

K=2, non-linear

Loaded die

Koplow

5,5,3,3,1,1

2x(5,3,1)

K=2 a=2, b=-1

Loaded die

Koplow

4,3,2

K=1, a=1, b=1

Consecutive numbers

HABA Schleckermaul

4,4,4,3,3,2

Non-uniform,
P(4)=1/2, P(3)=1/3, P(2)=1/6

Formula Dé

Jeux Descartes

Also used in Dragonland

game?

4,3,1

K=1, non-linear

HABA Schleckermaul

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.

3,3,2,1,1,0

Non-uniform

P(3)=1/3, P(2)=1/6

P(1)=1/3, P(0)=1/6

3,2,2,1,1,0

Non-uniform

P(3)=1/6, P(2)=1/3

P(1)=1/3, P(0)=1/6

The Golf Rules Game

3,2,1,1,1,0

Non-uniform

P(3)=1/6, P(2)=1/6

P(1)=1/2, P(0)=1/6

Babylon 5

Defense Grid Die

3,2,2,2,1,1,1,0

Non-uniform

P(3)=1/8, P(2)=3/8

P(1)=3/8, P(0)=1/8

3D2 (sum of 3 binary dice)

Exile Game Studio

(Ubiquity Dice)

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.

Tetrahedron

Isosceles

Tetrahedron

Modified

square prism

Crystal Caste (left)

Bear Cub Machine (right)

Octahedron

D8, 2 x 1-4

K=2

Dodecahedron

D12, 3 x 1-4

K=3

Koplow

Deltoidal Icositetrahedron

D24, 6 x 1-4

K=6

Numbers in triangle

D-Total by

A. Simkin / GameScience

Spherical D4

This die is not

numbered, but it

could be used to

generate numbers

1..4

Spinner D4

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

Unfair dice generating 1..4 (Sicherman):

4,3,3,2,2,1

Non-uniform

P(4)=1/6, P(3)=1/3

P(2)=1/3, P(1)=1/6

Sicherman die

GameStation

4,3,3,2,2,1

Non-uniform

P(4)=1/6, P(3)=1/3

P(2)=1/3, P(1)=1/6

Sicherman die

Grand Illusions

Others

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

500,200,200,100,50,50

Non-uniform,
P(500)=1/6, P(200)=1/3

P(100)=1/6, P(50)=1/3

Monopoly Dice Game

400,300,250,200,200,200

Non-uniform,
P(400)=1/6, P(300)=1/6

P(250)=1/6, P(200)=1/2

Monopoly Dice Game

9,9,8,7,6,6

Non-uniform
P(9)=1/3, P(8)=1/6

P(7)=1/6, P(6)=1/3

8,8,7,7,6,6,5,5

2x(8…5)

K=2, a=1, b=3

Formula Dé

8,7,7,6,6,5

Non-uniform

P(8)=1/6, P(7)=1/3

P(6)=1/3, P(5)=1/6

The Golf Rules Game

6,5,4,4,3,3

Non-uniform

P(6)=1/6, P(5)=1/6

P(4)=1/3, P(3)=1/3

5,5,4,4,3,2

Non-uniform

P(5)=1/3, P(4)=1/3

P(3)=1/6, P(2)=1/6

The Golf Rules Game

5,4,4,3,3,2

Non-uniform
P(5)=1/6, P(4)=1/3, P(3)=1/3, P(2)=1/6

Averaging die

This die has the same average as a “regular” D6, namely 3.5

5,4,3,3,2,2

Non-uniform

P(5)=1/6, P(4)=1/6

P(3)=1/3, P(2)=1/3

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.

Pentagonal

Trapezohedron

D10, 2 x 0-4

K=2 a=1, b=-1

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.

Elongated

pentagonal prism

Abraham Neddermann

Pentagonal

Trapezohedron

D10, 2 x 1-5

K=2 a=1, b=0

Not yet found: D20, 4x1-5

This die is not fair:

Triangular

Prism.

Not fair.

GameScience

Others

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

50,50,40,40,30,30,20,20,10,10

K=2, a=10, b=0

Sumator

GameScience

40,40,30,30,20,20,10,10,00,00

K=2, a=10, b=-10

GameScience

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

20,15,10,5,5,0

Non-uniform
P(20)=1/6, P(15)=1/6, P(10)=1/6,

P(5)=1/3, P(0)=1/6

6,6,5,4,3,2

Non-uniform
P(6)=1/3, P(5)=1/6, P(4)=1/6,

P(3)=1/6, P(2)=1/6

Cheater, 1 replaced by 6

6,5,4,2,0,0 (blank,blank)

Non-uniform

P(6)=1/3, P(5)=1/6, P(4)=1/6,

P(3)=1/6, P(2)=1/6

5,4,3,2,1,1

Non-uniform

P(5)=1/6, P(4)=1/6, P(3)=1/6,

P(2)=1/6, P(1)=1/3

Cheater, 6 replaced by 1

5,3,2,1,1,0

Non-uniform

P(5)=1/6, P(3)=1/6, P(2)=1/6,

P(1)=1/3, P(0)=1/6

Babylon 5

M=6

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

Range 0..5

Hexahedron

numbered

Hexahedron

pipped

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

Hexahedron

pipped

Hexahedron

numbered

Modified

6-sided

Antiprism

Crystal Caste (left)

Hasbro / Monopoly (right)

Football

6-sided prism

Hasbro

(Monopoly)

Dodecahedron

D12, 2x1-6

K=2

Hasbro

(Monopoly)

Deltoidal Icositetrahedron

D24, 4 x 1-6

K=4

Numbers in square

D-Total by

A. Simkin /

GameScience

Spinner D6

Other shapes:

Sphere

Flattened

Sphere

Concave
a.k.a “Bones”

Crooked

Not fair

Jumping

Not fair

Elliptical

Not fair

Tactile

Crystal pips*

Holes

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.


971,872,773,377,278,179 Trick die	960,762,663,564,366,168 Trick die	954,855,756,657,558,459 Trick die

840,741,642,543,345,147 Trick die	780,681,483,384,285,186 Trick die	300,280,260,240,220,200 K=1 a=20, b=180