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© Copyright 2002, Jim Loy
Let's define a die (singular of dice) as a cube with the numbers 1
through 6 on the faces (one number per face). And 1 is opposite 6; 2 is
opposite 5; and 3 is opposite 4. Ignoring the many ways that each number can be
rotated on one face, is the above a complete definition? In other words, can
there be more than one configuration of numbers according to that definition?
If you have never considered that, you might want to think about it.
Answer: Since the only number that is not adjacent to 1 is 6, then 1 is adjacent to both 2 and 3. Similarly, 2 is adjacent to 3. So we can orient the die so that we are looking at 1, 2, and 3, at the same time, as in the picture above left. The sequence 1, 2, 3 is either clockwise (which we might call right-handed) or counter-clockwise (which we might call left-handed). That means that there are two (and only two) possible kinds of dice, right-handed and left-handed, as the other three numbers (4, 5, and 6) cannot be rearranged after we decide where to put 1, 2, and 3. The one that I drew is right-handed. Well, are the dice out there in the real world left-handed or right-handed? From what I read, real dice are sometimes right-handed, and sometimes left-handed. Dice from one manufacturer may be all right-handed or all left-handed, however. These two orientations are an example of mathematical parity (see Dominoes On a Checker Board).
Above, I suggested that we ignore the many ways of rotating each
number on its face. If we consider such rotations (each face being a set of
dots in the normal dice configurations, as 3 is three dots along one diagonal
of a face), how many dice are there? 1, 4, and 5 do not change when rotated. 2,
3, and 6 each have two possible orientations. So there are eight combinations
(2x2x2) for right-handed dice, and eight combinations for left-handed dice, for
a total of 16. I think that in the real world, 2 and 3 are always oriented
either as I have shown in the above picture or like the die on the right. I
assume that for either of these two orientations of 6 can be used. In that
case, there are four combinations (2x2) for right-handed dice (in the real
world), and four for left-handed dice, for a total of eight.
We have all heard of loaded dice, dice which are weighted on one side (or lightened on the opposite side) so the same number comes up almost every time. There are also ways to shape the dice, by perhaps filing some of the edges, so that some numbers are more likely than others. You might ask, are the dice that can be bought in toy stores (or in the casino) fair? Do each of the six numbers come up with equal probability. Nothing is perfect, so the answer would have to be that they are not perfectly fair, but they are probably very very close to being fair. The number 6 would tend to be lighter than 5, which would be lighter than 4, etc. But I assume the manufacturers compensate for this tendency, somehow.
Well, can a person cheat, when using unloaded dice. Yes, absolutely. There are ways to force any number you want, with or without a dice cup. And that is another reason to avoid gambling, in my opinion.
The definition of left-handed and right-handed is arbitrary. We might start at the center and follow the right-hand rule of vectors, and then we get the opposite results. Or we might just call our orientation clockwise or counterclockwise, but then we have the same problem. It appears to be traditional to define left-handed and right-handed as I have done above. Most sources claim that modern western dice may show either handedness. Martin Gardner says they are always right-handed. Some of the dot patterns on the faces can be rotated as well, so there can be many different distinct dice.