If you toss a coin, it rarely lands on its edge.

Not again!!

This question came up about a year ago, and degenerated to a
discussion about which physical phenomena one should take into account
and what the assumptions are.  It appears the answer depends on factors
like:

  - Rotation or not of "coin" as it is thrown.

  - The elasticity of the surface and "coin".

  - The friction of the coin/surface interaction.

  - The kinetic energy of the coin as it lands.

  - Whether the coin falls vertically or not.

In conclusion, I say the problem is ill-defined, and no authoritative
solution exists.  The reason we trust normal dice is because the faces
are indistinguishable, apart from the markings.  The fairness is then
obvious by symmetry.

The five platonic solids have this property, as has many others.  The
ten-sided die used in D&D and other games is an example of a
non-platonic fair die: the faces are indistinguishable, but the
corners are not.  There are surprisingly few ways to make such
"face-regular" objects.  The ten-sided die can easily be generalized
to any even number of faces above 4 (the cube is an instance of this
family).  Another family is obtained by taking a regular n-sided
polygon and connecting the vertices of this with a point above the
polygon and with a point below, giving 2n triangular faces (n>=3).
This includes the regular octahedron [Editor: "octagon" in the
original].  The remaining platonic solids have 4, 12 and 20 faces.
Apart from these (and the two infinite families) there are only
thirteen other possibilities: 2 with 12 faces, 4 with 24 faces, one
with 30 faces (seen in game shops), one with 48 faces, 4 with 60 faces
and one with 120 faces.

You can make other "fair" dice by using non-flat surfaces or making
some surfaces impossible to balance on.

-- Torben AEgidius Mogensen <torbenm@diku.dk> (June 1994)

>Torben AEgidius Mogensen <torbenm@diku.dk> wrote:
>>[re. the problem of a three-sided coin, and how thick it should be]
>>In conclusion, I say the problem is ill-defined, and no authorative
>>solution exists. The reason we trust normal dice is because the faces
>>are indistinguishable, apart from the markings. The fairness is then
>>obvious by symmetry.

I did some experiments with a three-sided coin, made of a number of
coins (British 10p pieces, which have a D:T ratio of 13:1) taped
together.  My experiments showed that the 'fair' shape lies somewhere
between D/T=1.3 and 1.86.  However, Torben's point about symmetry is
very important.  I found that while it was not possible intentionally
to bias the coin in favour of heads or tails, it was easy to throw in
such a way as to get more or fewer sides.  (It's a matter of how much
roll as opposed to tumble you give the coin.)  Clearly, then, this
shape of coin is not suitable for gaming use.

-- Robert <ierbb@csv.warwick.ac.uk> (June 1994)