[extropy-chat] alt dot fair dice

[Spike]

> Spike
> Subject: [extropy-chat] alt dot fair dice

> 
> Can other shapes be made such that there is
> equal probability of any face downward?  I can
> think of one: a five sided pyramid shaped
> solid (four triangular faces and one square
> face).  If the pyramid is tall and skinny, it
> is less likely to land on the square face.  If
> it is short and flat, the square face is more
> likely to end downward.  So (I think) the
> intermediate value theorem demands that there
> is an aspect ratio somewhere between short and
> tall that would make the square face equally 
> likely to land downward, even if the surface
> area of the square face is different from
> the triangular... spike

Wait, I am now realizing this whole problem is a
lot more complicated than I first imagined.  With
the above example, the fairness of the square-base
pyramid would depend on having a level surface
upon which one rolls the die.  Imagine a family
of pyramid shaped dice with progressively more
triangular faces.  As the number of faces increases,
the probability of landing on any given triangular
face goes down, so the probability of landing on
the polygonal base must be adjusted downward, so
the thing gets taller.  Right?  So if one imagines
rolling a tall die on an inclined surface, it is
easy to see that it might be exactly *impossible*
to make it stand on its base, yet it could still
land with equal probability on any triangular face.

spike