Games With Special Numbered Or Pipped Dice

There is a large number of games with special dice. The best overview is available in various geeklists on http://www.boardgamegeek.com/:

        Dice - A Dicey Love Affair currently the most extensive list

        Games with specially numbered die/dice six sided dice

        okay, I admit it: I love specialty dice

        Games with Really Cool Dice

 

I am interested in numbered or pipped dice only, no symbols, and no stickers.

 

This section contains a list of dice from games as well as special dice sets (e.g. cheaters, trick dice, Sicherman dice and non-transtive dice)

 

To do: Include & check on Dice - A Dicey Love Affair

Half-Time Football

Nin-Gonost

Break the safe

Phase 10

Real Math - A Sampler of Games

Number Please!

Mission Command Air

Screaming Eagles Game

13 Dead End Drive

Outburst

Babylon 5

Push Over

GridIron Master

GOLO

Catz, Ratz & Batz

Shark

 

Equations

Peg Poker

Showdown Poker

The Train Game

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Special Dice Sets

Sum of two to five 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.

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.

 

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.

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:

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:

Quintuple Roller

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

 

 

 

 

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

http://mathworld.wolfram.com/SichermanDice.html

They are the only such alternate arrangement if face values are required to be positive. However, if faces are permitted to have zero value, then three additional possible equal-odds pairs of dice are possible: Two are obtained by subtracting one from each face on either of the two dice and adding one to each face the other, yielding to 5,4,4,3,3,2 / 7,5,4,3,2,0 and 3,2,2,1,1,0 / 9,7,6,5,4,3. The third one is obtained by performing the same +1 / -1 operation with a regular pair of D6, yielding to 7,6,5,4,3,2 / 5,4,3,2,1,0, as used in Magic’s Double D6:

 

If negative values are permitted, there are an infinite number of equal-odds 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

 

 

 

 

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

 

 

The original set of 5 dice, invented by Royal V. Heath in 1927, was extended to 8 dice by Stephen Bradd of Clinton, IL. The rule is the same as above, but subtract the sum from 80 instead.

Place Value Dice

 

Lotto Dice

7 dice, numbers 1-35. All red numbers appear twice.