Article by Joe Celko (71062.1056@CompuServe.COM)
Role playing and simulation games have introduced dice in noncubic shapes, but the traditional spotted cube has endured the test of time. In fact, dice are credited by several historians as the oldest gambling device invented by man. I would like to take some time to give the humble cube another, more detailed, look. This is not so much an article as a collction of interesting facts.
I will assume that every gamester knows that the opposite faces of traditional dice total to seven (16, 25, 34). But did you also know that there are two ways that the spots can be arranged on the cubes, so that you have right and left handed dice. One arrangement is the mirror image of the other. All Western dice are now made with he same handedness; if you place a die on the table with the one spot face up and turn it until the two spot is toward your left, the three will be to the right.
Now get some Japanese, Chinese and Korean dice. The first thing that you will notice about Asian dice is that the one spot is very big and it is painted red. Chinese and Korean dice also paint the four spot red. Japanese dice are handed in the same direction as Western dice, but Chinese dice are not. Western dice put the two spot on opposite corners of the face, Asian dice stack the pair of spots on each other along the middle axis of the face. Asian dice are also usually smaller and are very often used in sets of three, instead of in pairs.
The first principle of cheating is that you don't have to win all the time; you just want to win most of the time. Loaded dice have weights inside them which favor the dice landing on the side with the weights. To prevent this, modern dice are made from clear plastic which will allow you to see that they are made of uniform material. To get around this, modern crooks use clear weights or add the weights (which can be lighter or heavier than the dice material) to the solid colored spots.
Another trick is to use dice that are not perfect cubes, so that they will tend to land on the largest face; these are called "shapes" for obvious reasons.
Ever wonder why you see people in crap games in the movies blowing on the dice before they shoot them? Another method for cheating is to paint the dice with a clear sticky substance which is activated by the moisture in your breathe.
Since you can see only three faces of a die at one time, you will find crooked dice called "horses" (before you ask, I have no idea where that name comes from) which are not spotted correctly. This means that certain combinations are impossible to throw, and other totals are more likely than they should be.
An interesting idea for making "honest horses" comes from Col. George Sicherman of Buffalo, NY. He found an irregular arrangement of spots which has the same probability distribution of its totals as a regular pair of dice. One die is spotted (1, 3, 4, 5, 6, 8) and the other is spotted (1, 2, 2, 3, 3, 4). Here is a table of throws and totals for traditional and Sicherman dice:
Traditional Dice  Sicherman Dice  



You can also use a regular die and one spotted (0, 0, 0, 6, 6, 6) to get an uniform distribution of numbers from 1 to 12.
While the distribution of totals on traditional dice is "kinda bell shaped", it is not a binomial probability distribution. Without making you get a statistics book, let's just say that the binomial probability distribution is the normal probability distribution for discrete values. Or you could say that the normal distribution is the binomial distribution for continuous variables. You can almost approximate the normal distribution with a pair of dice EACH spotted (2, 3, 3, 4, 4, 5). The totals look like this:
total  occurs  Binomial 

4  1  0.5 
5  4  3.0 
6  8  7.5 
7  10  10.0 
8  8  7.5 
9  4  3.0 
10  1  0.5 
We are used to the idea that relations are usually transitive. That means that if a is greater than b, and b is greater than c, then we know that a is greater than c. But in the real world, not all relations are transitive. Make a set of dice which are spotted:
A = (1, 1, 4, 4, 4, 4) B = (3, 3, 3, 3, 3, 3) C = (2, 2, 2, 2, 5, 5)
You let your victim pick one of the three dice and you then pick one. You are going to play a game of "high roll". The normal assumption would be that the first player has an advantage because he can pick the "best die" from the set. But look at the real odds in the game.
If the first player picks A, you pick C and win 2/3 of the time. C would lose 1/3 of the time.
If the first player picks B, you pick A and win 2/3 of the time. A would lose 1/3 of the time.
If the first player picks C, you pick B and win 2/3 of the time. B would lose 1/3 of the time.
Dice games can be classified as:
1) Total games: The winner is determined by the total of the spots he shoots. A simple highest throw is such a game.
2) Combination games: Certain rolls have special scores before the game begins. These scores may or may not be related to their totals. Poker dice is an example of this family. While dice with playing cards faces (A, K, Q, J, 10, 9) are often used, the game can be played with regular dice.
3) Point games: The shooter determines a winning or losing throw for himself by rolling the dice. Craps is the best known game in this family.
4) Compound games: The game is a mixture of two or three of the above principles. Ship, Captain and Crew (sometimes called Ship, Captain and Mate) is such a game. This was a popular bar game played for drinks with five dice and throwing cup. Each player gets three throws. He has to get a 6 (ship), a 5 (captain) and a 4 (crew), in that order, before he can make a point. As he throws each of these special numbers, those dice are removed from play. This is the combination phase of the game.
The total of the two dice left over is his point and the other player has to beat that total (the total part).
The gimmick that the three dice combination of Ship, Captain and Crew has to be built in order. That is, the shooter must throw the 6 before or at the same time as the 5 and the 5 before or at the same time as the 4. These last two dice cannot be set aside one at a time, but have to be thrown as a pair.
If the shooter makes his Ship, Captain and Crew on his first or second throw, he can use the other throws to try to get a high point. Obviously, the best situation is to get Ship, Captain and Crew on the first throw, so you have two chances to throw a high point.