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(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.

 

“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

 

3,2,1,1,1,0

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

6,5,4,4,3,3

Non-uniform

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

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

 

5,4,4,3,3,2

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

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.

 

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

 

Hexahedron

English

words

Hexahedron

German

Words

 

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

 

260,240,220,200,180,160

K=1 a=20, b=140

250,225,200,175,150,125

K=1 a=25, b=100

200,180,160,140,120,100

K=1 a=20, b=80

 

64,32,16,8,4,2

K=1, 2^X

Backgammon

60,50,40,30,20,10

K=1, a=10, b=0

30,29,28,27,26,25

K=1, a=1, b=24

 

24,23,22,21,20,19

K=1, a=1, b=18

18,17,16,15,14,13

K=1, a=1, b=12

13,12,9,8,4,3

Rummy Die

 

13,12,8,7,4,3

Rummy Die

13,10,9,5,4,1

Rummy Die

 

12,11,10,9,8,7

K=1, a=1, b=6

 

 

12,12,11,11,10,10,

9,9,8,8,7,7

2x(12..7)

K=2, a=1, b=6

Formula Dé

12,11,8,7,3,2

Rummy Die

12,11,7,6,3,2

Rummy Die

 

11,10,6,5,2,1

Rummy Die

11,10,7,6,2,1

Rummy Die

10,9,8,7,6,5

K=1, a=1, b=4

 

10,9,8,7,6,0

10,8,8,8,7,7,6,6,5,5,5,3

D12, non-uniform

P(10)=1/12, P(8)=1/4,

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

P(5)=1/4, P(3)=1/12

3 in a star, 5s in a square

Golo Golf Par 5 Die

9,8,7,6,5,4

K=1, a=1, b=3

 

9,8,8,8,7,7,6,6,5,5,5,4

D12, non-uniform

P(9)=1/12, P(8)=1/4,

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

P(5)=1/4, P(4)=1/12

4 in a circle, 5s in a square

Golo Golf Par 5 Die

9,8,5,4,3,1

Non-transitive die

Miwin

9,7,6,5,2,1

Non-transitive die

Miwin

 

8,7,6,5,4,3

K=1, a=1, b=2

8,7,7,7,6,6,5,5,4,4,4,3

D12, non-uniform

P(8)=1/12, P(7)=1/4,

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

P(4)=1/4, P(3)=1/12

3 in a circle, 4s in a square

Golo Golf Par 4 Die

8,7,6,4,3,2

Non-transitive die

Miwin

 

8,7,6,2,1,0

Could be a die from an everlasting calendar

8,6,5,4,3,1

Sicherman die

Grand Illusions

8,6,5,4,3,1

Sicherman die

GameStation

 

8,6,6,6,5,5,4,4,3,3,3,1

Same numbers as Sicherman die

D12, non-uniform

P(8)=1/12, P(6)=1/4,

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

P(3)=1/4, P(1)=1/12

1 in a star, 3 in a square

Golo Golf Par 3 Die

7,6,5,4,3,2

K=1, a=1, b=1

7,6,6,6,5,5,4,4,3,3,3,2

D12, non-uniform

P(7)=1/12, P(6)=1/4,

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

P(3)=1/4, P(2)=1/12

2 in a circle, 3 in a square

Golo Golf Par 3 Die

 

6,5,4,3,2,0 (blank)

+5,+4,+3,+2-1,-2

6,4,2,-1,-3,-5

Odd Negative

 

5,3,1,-2,-4,-6

Even Negative

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

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

-6.,-5,-4,-3,-2,-1

K=1, a=1, b=-7

(or a=-1, b=0)

 

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.

0..6

Not fair

P(0)=1/16, P(1)=1/8, P(2)=3/8,

P(3)=1/2, P(4)=3/8, P(5)=1/8, P(6)=1/8

Y2K Die

 

Range 1..7

Elongated

heptagonal prism

Abraham Neddermann

Pentagonal

Prism.

Not fair

 

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

Octahedron

Modified

8-sided

Antiprism

Crystal Caste

Deltoidal Icositetrahedron

D24, 3 x 1-8

K=3

Numbers in diamond

D-Total by

A. Simkin /

GameScience

 

Wanted: D8 pipped

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

Others

128,64,32,16,8,4,2,1

2^(X-1)

Doubling die

9,8,7,6,5,4,3,2

K=1, a=1, b=1

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).

 

Elongated

nonagonal prism

Abraham Neddermann

9,8,7,6,5,4,3,2,0,0

Non-uniform

P(9)=P(8)=…=P(2)=1/10

P(0)=1/5

Cheater, 1 replaced by 0

 

Missing in my collection

Heptagonal prism

Not fair

 

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.

 

Pentagonal

Trapezohedron

numbered

Pentagonal

Trapezohedron

pipped

Modified

10-sided

Antiprism

Crystal Caste

Icosahedron

D20, 2 x 0-9

K=2

Rhombic Triacontahedron

D30, 3 x 0-9

+9..+0,9..0,-9..-0

K=3

 

Truncated Octahedron

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).

Pentagonal

Trapezohedron

Modified

10-sided

Prism

Bear Cub Machine

10 sided spinner

 

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.

 

900000,…,100000,000000

K=1, a=100000,

b=-100000

 Koplow

90000,…,10000,00000

K=1, a=10000, b=-10000

Koplow

9000,…,1000,0000

K=1, a=1000, b=-1000

Koplow

 

900,…,100,000

K=1, a=100, b=-100

Koplow

90,80,…,10,00

K=1, a=10, b=-10

90,80,…,10,00

K=1, a=10, b=-10

Crystal Caste

 

30,30,30,29,29,29,…,21,21,21

3x(30…21)

K=3, a=1, b=20

Formula Dé

20,20,19,19,...,11,11

2x(20…11)

K=2, a=1, b=10

Formula Dé, Truant

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).

 

Elongated hendecagonal prism

Abraham Neddermann

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).

 

Pentagonal

Dodecahedron

Rhombic

Dodecahedron

AskAstro

Modified

12-sided

Antiprism

Crystal Caste

Deltoidal Icositetrahedron

D24, 2 x 1-12

K=2

Numbers in pentagon

D-Total by

A. Simkin /

GameScience

 

Others

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

55,50,…5,0

K=1, a=5, b=-5

M=13

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

Elongated tridecagonal prism

Abraham Neddermann

 

M=14

This heptagonal trapezohedron is numbered 1..14.

Heptagonal

Trapezohedron

 

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.

Elongated pentadecagonal prism

Abraham Neddermann

M=16

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

Octagonal

Bipyramid

Octagonal

Bipyramid

Hexi Die

M=19

There are two icosahedral cheater dice:

20,20,19,18,…3,2

Non-uniform

P(20)=1/10,

P(19)=…=P(2)=1/20

average 239/20=11.95

Cheater, 1 replaced by 20

Chessex/Koplow/Truant

19,18,…3,2,1,1

Non-uniform

P(1)=1/10,

P(19)=…=P(2)=1/20

average 201/20=10.05

Cheater, 1 replaced by 20

Truant

M=20

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

Icosahedron

Modified

20-sided

Antiprism

Crystal Caste

 

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

57,51,50,46,45,43,38,35,32,29,24,20,19,16,13,11,07,04,02,00

 

M=24

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

Tetrakishexahedron

Deltoidal Icositetrahedron

Franck Dutrain

Deltoidal Icositetrahedron

Numbers in center of face

D-Total by

A. Simkin /

GameScience

 

M=30

Rhombic triacontahedron, numbered 1..30:

Rhombic

Triacontahedron

M=34

Dekaeptagonal Trapezohedron, numbered 1..34:

Dekaeptagonal
Trapezohedron

Chessex

M=50

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

trapezohedron and a flattened sphere:

Eikosipendegonal

Trapezohedron

GameScience

Big 50 Topper

Flattened Sphere

Not fair

Alan Davies

 

 

M=100

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

Zocchihedron

Not fair

GameScience

 

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:

 

1,5/6,2/3,1/2,1/3,1/6

K=1, a=1/6, b=0

1/1,1/2,1/3,1/4,1/5,1/6

X^(-1)

11/12,7/8,5/6,3/4,2/3,1/2

 

3/4,2/3,2/4,1/2,1/3,1/4

1/2,1/3,1/4,1/4,1/6,1/6

Non-uniform

1/2,1/3,1/4,1/5,1/6,1/8

 

1/2,1/3,1/4,1/6,1/8,1/12

1/2,1/4,1/4,1/8,1/8,1/8

Non-uniform

1/3,1/6,1/6,1/12,1/12,1/12

Non-uniform

 

1,.75,.67,.50.,33,.25

1.00,0.75,0.67,0.50,0.33,

0.25

1.00,0.50,0.25,0.10,0.05,

0.01

 

1,7/8,3/4,5/8,1/2,3/8,1/4,

1/8

K=1, a=1/8, b=0

Fractional D8

10/10,9/10,…,1/10

K=1, a=1/10, b=0

Fractional D10

0.9,0.8,…,0.1,0.0

K=1, a=0.1, b=-0.1

Koplow

 

0.09,0.08,…,0.01,0.00

K=1, a=0.01, b=-0.01

Koplow

0.009,0.008,…,0.001,0.000

K=1, a=0.001, b=-0.001

Koplow

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.

 

1,1,1,1,0,0,0,0

D2 (binary die)

Exile Game Studio

(Ubiquity Dice)

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

2D2 (sum of 2 binary dice)

Exile Game Studio

(Ubiquity Dice)

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

3D2 (sum of 3 binary dice)

Exile Game Studio

(Ubiquity Dice)

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.

2D4: Y2K Die

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:

2D6

Double spinner

2D6

Double roller

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:

3D6, triple roller

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

Left 4,3,3,2,2,1

Right 8,6,5,4,3,1

Grand Illusions

Left 8,6,5,4,3,1

Right 4,3,3,2,2,1

GameStation

 

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

Red 4,4,4,4,4,1

Blue 6,3,3,3,3,3

Green 5,5,5,2,2,2

Red beats blue with probability 25/36,

blue beats green with probability 21/36,

and green beats red with probability 21/36.

Grand Illusions

Blue 9,7,6,5,2,1

Orange 9,8,5,4,3,1

Black 8,7,6,4,3,2

Miwin Dice. Each die beats

the one in clockwise direction

with probability 17/36

(lose 16/36, draw 3/36)

Miwin

 

Top 6,6,2,2,2,2

Right 5,5,5,1,1,1

Bottom 4,4,4,4,0,0,0

Left 3,3,3,3,3,3

Efron’s Dice. Each die beats

the one in clockwise direction

with probability 2/3

Grand Illusions

 

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.

 

Center 6,5,4,3,2,1 (regular D6), average 21/6=3.50

Left 5,4,3,2,1,1 (6 replaced by 1), average 16/6=2.67

Right 6,6,5,4,3,2 (1 replaced by 6) , average 26/6=4.33

Chessex / Koplow

 

Right 9,8,7,6,5,4,3,2,1,0 (regular D10) , average 45/10=4.50

Left 9,8,7,6,5,4,3,2,0,0 (1 replaced by 0), average 44/10=4.40

Chessex / Koplow

 

Left regular D20 , average 220/20=11.0

Right 1 replaced by 20, average 239/20=11.95

Chessex / Koplow

 

From left to right (top and bottom pictures show the same die from front and back)

2 x 11-20, average = 310/20=16.5

1 replaced by 20, average 239/20=11.95

regular D20 , average 220/20=11.0

20 replaced by 1, average 201/20=10.05

2 x 0-9, average 90/20=4.5

Truant

 

Loaded Dice, throw 7 or 11

Left: 6,6,6,2,2,2 Right 5,5,5,5,5,5

 

Loaded Dice

Left: 6,6,5,5,4,4 Right 5,5,3,3,1,1

Both dice: Best bet is 7 followed by 5 or 9

2 x 5,5,3,3,1,1 dice - Best bet is the 6 followed by 4

Koplow

 

Loaded Dice,

Left: 6,6,5,5,1,1 Right 5,5,4,4,3,3

Will roll only points - 4,5,6,8,9,10,11, best bets are 9 & 10
Craps is 2,3,7 or 12 on the first roll, & 7 thereafter.
Koplow

 

Loaded Dice

Left: 6,6,3,3,2,2 Right 5,5,4,4,1,1

Both dice: Best bet is 7
2 x 5,5,4,4,1,1 dice: Best bets are 5,6,9
Koplow

 

6,6,6,6,6,6

Loaded die

5,5,5,5,5,5

Loaded die

4,4,4,4,4,4

Loaded die

 

3,3,3,3,3,3

Loaded die

3,3,3,3,3,3

Loaded die

0,0

Loaded die

 

Loaded Dice

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

 

wanted

wanted

Loaded die

“1”

Loaded die

“2”

Loaded die

“3”

Loaded die

“4”

Loaded die

“5”

Loaded die

“6”

 

Loaded die “6” (left), regular die (right)

Both dice are hollow

 

Loaded die, filled with liquid.

Can be “loaded” within seconds

to roll any number with high probability

 

 

Place Value Dice

Set of 9 D10 dice

Can throw any number from 0 to 999’999.999 in steps of 0.001

Trick Dice

Trick Dice

Yellow: 971,872,773,377,278,179

Green: 960,762,663,564,366,168

Blue: 954,855,756,657,558,459

Black: 840,741,642,543,345,147

Red: 780,681,483,384,285,186

 

*        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

 

500,200,200,100,50,50

500,200,200,50,50,?

400,300,250,200,200,200
Type 1

 

400,300,250,200,200,200
Type 2

400,300,250,200,150,?

150,150,100,100,100,100
bulb

 

150,150,100,100,100,100
tap

2x policeman, 4x blank

 

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