I rose this question in my discrete math class (the unit on probability) not too long ago. For instance, a two-sided shape (like a coin) can be one with any geometrical shape as its "side," such as a circle (as with a coin) or a square, or any other connected shape in $\mathbb{R}^2$, and in tossing this shape there is a $1/2$ "probability" it will land on one side as opposed to another. A cube (like a dice) is a shape with $6$ sides, all congruent flat squares, and upon tossing this object into the air, there is a $1/6$ chance of landing on any given side.
Now suppose I wished to toss a $3$-sided object such that all facets of this shape are congruent, as in the penny or dice, resulting in a $1/3$ chance of landing on any given side.
Can there exist solids in $\mathbb{R}^3$ (or elsewhere) with $3,4$ and $5$ flat facets, such that all facets are congruent? Naturally these shapes may exist with curved sides, but I'm more concerned with solids of flat facets. I'm certain one of $3$ sides cannot exist, but perhaps some of you can enlighten me on how I might prove these cases?
What about a solid with $n$-faces? Perhaps this is related to packing problems (and hence I will add that tag)?
One of the dies you want you can make. The others you can fake.
The tetrahedron is the first platonic solid. It has four triangular faces.
The platonic solids give you fakes for probabilities of 1/3 (label the faces of a cube with 1, 2, and 3 each twice) and 1/5 (label the faces of an icosahedron with 1 through 5, each four times).
You're right that a real 3 is impossible. I'm pretty sure 5 is too. Maybe I'll come back to this with proofs, if noone else provides them.
For some larger values of n, check out the wikipedia page for deltahedra: http://en.wikipedia.org/wiki/Deltahedron