[extropy-chat] alt dot fair dice

[Mike Lorrey]

--- Spike <spike66 at comcast.net> wrote:

> We usually think of a cube when someone mentions
> a gaming die, but of course any of the five platonic
> solids can make a fair die.  By thought experiment,
> we can verify that each face is the same shape and
> the CG is the same distance from the table with
> any face downward.
> 
> 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.  Also the CG height is different
> with the square face down than with the triangular
> faces down.  (Is it?)

You might instead look at a solid of two triangles and three squares.
This might be easier to optimize by adjusting the squares into
rectangles. Has anyone done a physical analysis of the die-rolling
capabilities of non-perfect solids?

Gamers use a ten sided die to get 1 in 5 odds, btw.


> 
> Actually that suggests a class of non-platonic
> fair dice.  A six sided "pyramid" with five
> triangles and a pentagonal base also would
> hafta have an intermediate base to height ratio
> that would fair-ize it.  Right?  Wouldn't that
> argument apply to arbitrarily many triangular
> sides?  

Actually, what these sorts of die might be good at is to roll for
'gripping hand' type scenarios, where one of several options is far
more probable than the others.

> 
> One could even make a three-sided fair die, if 
> one did not demand the "sides" be flat planes: two 
> curved kinda triangular surfaces with a base that 
> looks sorta like an ellipse but with pointy ends.

Yeah, thought of that one long ago. Found a seed that was in that
configuration and used it in a game.

> 
> A two sided fair die is a coin, but that suggests
> another three sided die: a cylinder, like a
> really fat coin, equally likely to land on edge
> as on either side.  For that matter one could 
> grind arbitrarily many flat sides on a cylinder,
> so that the cylinder gets longer and thinner
> as the number of flat sides gets larger.
> 
> Other than machining a bunch of these oddball
> shapes, is there any way to mathematically
> prove that they would have equal probabilities 
> of landing on any face?

Why machine them? Get yourself a jack knife and some balsa wood. Better
yet, aren't you carving some pumpkins yet? 

=====
Mike Lorrey
Vice-Chair, 2nd District, Libertarian Party of NH
"Necessity is the plea for every infringement of human freedom.
It is the argument of tyrants; it is the creed of slaves."
                                      -William Pitt (1759-1806) 
Blog: http://www.xanga.com/home.aspx?user=Sadomikeyism


		
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