26 Jul '03 21:04

Having nothing better to do with my time, I was wondering recently about how one would fashion a three-sided die (an unnecessary problem, as one can trivially assign two sides of a six-sided die to each of the three outcomes). The most obvious shape that occurred to me was a cylinder (essentially an elongated coin), though there are more exotic possibilities. The question is: What dimensions should that cylinder assume? Specifically, what should the ratio of the diameter of each (equally sized) face be to the length of curved side so that it is equally probable that when the die is cast the die lands on one face, the other, or the curved side? (Uniform material composition of the die is, of course, assumed.)

My preliminary thoughts were that the length of the diameter should be twice the length of the curved side. If you put such a cylindrical die in a sphere (or, in 2-D, a rectangular die in a circle) there would be, I believe, an equal chance of the die tending to fall *towards* one face, the other face, or the curved side, if the die were randomly placed in any of an infinity of possible positions. However, as intuition suggests, this does not solve the problem: the die is in a state of less stable equilibrium on its curved side, and may even always fall over with minimal lateral momentum. Oddly, a ratio of 1:1 seems about right. Can any more mathematically gifted site member demonstrate what the ratio must be? (There could be a big market among Dungeons and Dragons enthusiasts!)

Aiden

My preliminary thoughts were that the length of the diameter should be twice the length of the curved side. If you put such a cylindrical die in a sphere (or, in 2-D, a rectangular die in a circle) there would be, I believe, an equal chance of the die tending to fall *towards* one face, the other face, or the curved side, if the die were randomly placed in any of an infinity of possible positions. However, as intuition suggests, this does not solve the problem: the die is in a state of less stable equilibrium on its curved side, and may even always fall over with minimal lateral momentum. Oddly, a ratio of 1:1 seems about right. Can any more mathematically gifted site member demonstrate what the ratio must be? (There could be a big market among Dungeons and Dragons enthusiasts!)

Aiden