Construct a die as a perfect sphere. Mark the dots with some technique
that doesn't protrude from the surface nor move the center of mass away
from the center of the sphere. Now "load" this die by drilling into the
"one" side and refilling the drilled-out hole with something more dense
than the rest of the sphere, but preserve the spherical surface.
Roll this "die" onto a hard, smooth, level, clean surface with hard
smooth walls so it can roll and bounce around for a while. The point of
the hard smooth surfaces is we want essentially no friction other than
air friction and miniscule rolling friction, and we want no bumps or
dirt or low spots on the surfaces. The idea is that when air friction
and miniscule rolling friction eventually stop the ball, there won't be
any residual torque on the ball.
As described, this loaded spherical die will always stop with "six"
showing. That is, the probability of rolling "six" is 1. "Six" is the
only stable position.
Now begin to "morph" this sphere into a cube by slightly flattening the
sides/regions where the dots are. At first the "cubeness" is slight;
the die is still nearly spherical. At any degree of flatness, both
"one" showing and "six" showing are stable, but "one" is barely stable.
The probability of rolling "six" is no longer 1, but close to 1 because
the "one" position has a narrower region of stable rotational
equilibrium than the "six" position. With further flattening, the "two"
"three" "four" "five" positions become stable. As we continue to
flatten the faces and give the non-"six" positions a wider range of
stable rotational equilibrium, the probably of rolling "six" keeps going
down.
But the probability of rolling "six" won't go as low as 1/6 unless the
die is fair (not loaded). So the probability of rolling "six" on a
loaded real die (that is more dense on the "one" side) is something less
than 1 but more than 1/6. The more drastically it is loaded, and the
more rounded its corners/edges, the higher the probability of rolling a
"six." The less it is loaded and the less rounded its corners/edges,
the closer the probability of rolling "six" approaches 1/6.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu