Fair and unfair N-sided dice in principle.

[Spinnor]

In principle does classical physics allow us to make a fair N-sided die where N ≥ 3?

If I give you a set of N positive numbers n_i whose sum is 1 can a N-sided die be made in principle such that each face of the N-sided die comes up on average with a probability n_i?

Can N above be reduced to N≥2?

Thanks for any help!

Happy New Year!

 

[D H]

It's not physics that's the issue. It's geometry. N=2 is possible; it's called a coin. You just need to give it razor-sharp edges so the coin can't land on edge. The only other possibilities are the Platonic solids with N=4 (tetrahedron), N=6 (cube), N=8 (octahedron), N=12 (dodecahedron), and N=20 (icosahedron).

 

[Vanadium 50]

DH is right. However, there is a better solution: make your "dice" rod-shaped: like a pencil, where the cross-section is a regular polygon. That allows arbitrary N > 2.

 

[jbriggs444]

DH is right. However, there is a better solution: make your "dice" rod-shaped: like a pencil, where the cross-section is a regular polygon. That allows arbitrary N > 2.

In addition to the rod-shaped solution, there are a number of convex polyhedra which are not regular, but which do have a symmetry so that all faces are alike. For instance, if one takes a cube and turns each face into a shallow square pyramid then one can end up with a 24-sided convex polyhedron with all 24 faces identical to one another.

Convex polyhedra which do not have a symmetry arrangement so that every [stable] face is the same as every other are problematic unless the testing procedure is carefully defined. When such a symmetry exists, it is easy to argue that the launching conditions are irrelevant as long as the launching conditions share the corresponding symmetry. Without such a symmetry, it is possible for launching details to bias the results in favor of a particular face. For instance, a low-energy drop onto a energy-absorbing surface may make the result strongly dependent on initial orientation while a high-energy drop onto an elastic surface may make the result strongly dependent on the way the die is loaded.

It follows that there is no theoretically sound way to get an arbitrary discrete probability distribution from a single roll of a single die just by engineering the die correctly.

 

[Hassan2]

I think the OP's question is more difficult to answer than it seems at the beginning.

For example a homogenous cube is certainly fair for six equal probabilities set to each face. How if we set different probabilities to the faces? Given the desired probabilities, can we implant a very dense particle within the cube to move the center of mass to a desired location to make the die fair? It is physics in combination with geometry.

 

[jbriggs444]

For example a homogenous cube is certainly fair for six equal probabilities set to each face. How if we set different probabilities to the faces? Given the desired probabilities, can we implant a very dense particle within the cube to move the center of mass to a desired location to make the die fair? It is physics in combination with geometry.

My answer appears in post #4 above. No, it is not possible.

 

[Hassan2]

jbriggs444, now I understand your post a bit better. So, calculating how much degree we can nudge the cube on each face before it get out of balance, is not enough to calculate the expectation of landing on that face.

Thanks.