Proof that there exists a 7-sided polyhedron dice

Question

We know that we can build a polyhedron shaped dice with $2n$ faces using two regular pyramids with n-sided bases, but how can we build a fair polyhedron dice with 7 faces ? Can we generalize the technique to build any $(2n+1)$ faced fair polyhedron dice ?

Edit:

I'm aware that there is a polyhedron with 7 faces (pentagonal prism), I'm interested in if there is away to proof (even if a non-constructive proof) that it can be used as fair dice.

Answer 1

My answer will probably get deleted because it's not a real answer, but I just wanted to list a few resources that might provide inspiration for your problem, and it would be awful trying to add this list as a comment.

Edward Taylor Pegg, A Complete List of Fair Dice

http://www.mathpuzzle.com/MAA/37-Fair%20Dice/thesis/thesis7.html

The thesis addresses the problem of asymmetric dice and provides some mathematical models, so it seems relevant to your question.

Antonio Recuenco-Munoz, The Physics of Dice

https://www.geocities.ws/dicephysics/Recuenco-Munoz_Presentation.pdf

Just a power point with some useful information.

Strzalko et al., Can The Dice Be Fair By Dynamics

https://www.researchgate.net/profile/Andrzej_Stefanski/publication/220265123_Can_the_Dice_be_Fair_by_Dynamics/links/00b49516fc20417349000000/Can-the-Dice-be-Fair-by-Dynamics.pdf

We consider the dynamics of the three-dimensional model of the die which can bounce with dissipation on the table. It is shown that for the realistic values of the initial energy the probabilities of the die landing on the face which is the lowest one at the beginning is larger than the probabilities of landing on any other face.

Let me know if it's helpful.

Answer 2

Start with a septagon then add pyramids on the ends so it can't land on either end. This will be fair if you only consider the 7 sides that it can possibly land on while ignoring the 14 impossible sides.

Regarding the pentagonal prism, by definition it cannot be a fair die. However, my experimental data suggests that if the ratio of the height to the radius is about 1.582, all 7 sides are equally likely if you drop it on a hard surface.