Unfair Dice

My friend and ex-colleague, Scott Ogawa, was musing about different kinds of dice. He wondered if it might be possible to construct a die where the probability of any face coming up would be proportional to its value. So, the probability of a 1 showing would be 1/21, a 2 would be 2/21, etc.

Now this, I’m afraid, is more a problem for experimental physics than for mathematics. I doubt that an answer can even be found analytically. Aside from getting the right geometry, the probabilities might very well depend on the material of the die, the surface on which it’s rolled, and other factors.

In response, I proposed a simpler problem that does yield to mathematics. Can you design a die where the area of each face is proportional to its value? Let’s be more specific. Can you construct a hexahedron with the same topology as a cube (three pairs of opposing quadrilateral faces), where the areas of the faces are in a ratio of 1:2:3:4:5:6? To keep things interesting, let us also insist that the pairings are the same as for an ordinary die – that is: 1 is opposite 6, 2 is opposite 5, and 3 is opposite 4. If such a polyhedron is possible, give an example; if not, explain why not.

Extra credit: Same problem with other polyhedra.