[ExI] tumbling pyramids again

[BillK]

On 16 February 2017 at 19:13, spike wrote:
> Hi BillK, thanks.  My reasoning is based on the intermediate value theorem.
> If h = 1, we can easily see that it will land on that face more than 20% of
> the time.  If h=2, then most would agree it will land there less than 20% of
> the time.  My reasoning holds that there must be some value for h between
> the limits of 1 and 2 where that pyramid lands square down 20% of the time.
> My calcs suggest that number is 1.58 but it is a mathematical model.
>
> Please sir, what do you calculate or estimate for h?
>


The answer is yes, but.......

This problem is mentioned at the foot of this page -
<http://www.mathpuzzle.com/Fairdice.htm>

To quote the relevant paragraph:
Can a non-isohedral fair die exist?  Consider a pyramid made from 4
isosceles triangles and a square.  If the pyramid is short and fat,
the square face will be landed upon more than a fifth of the time.  If
the pyramid is tall and thin the square face will be landed upon less
than a fifth of the time.  Is there a height where the square face
will be landed upon exactly one fifth of the time??? Yes, for a given
set of conditions.  If you knew the height, force, elasticity, and
throwing method, you could find the right height.  However, once the
conditions changed, the die would no longer be fair.
--------

I think he is saying that if you made your 1.58 die, then people would
quickly find tricky ways of throwing the die so that they could change
the odds.

BillK