# Non-Symmetric Fair Dice

Probability Level 2

Any die is modeled by some polyhedron. If the polyhedron is completely symmetric in the sense that any face can be taken to any other face via a rigid motion, then the die will be fair; when the die is rolled, the probability of landing on any face will equal the probability of landing on any other face.

Do there exist fair dice that are not completely symmetric?

Hint: Start with a prism $P$ whose cross-sections are regular $n$-gons. Now consider the dual polyhedron $P^{\star}$, the polyhedron whose vertices are the centers of the faces of the original prism. This $P^{\star}$ looks like two pyramids with regular $n$-gon cross-sections that have been glued together at their bases. Now, can you modify $P^{\star}$ to obtain a fair die that isn't completely symmetric?

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