Probability Surfaces and Puzzles
Consider a six-sided die, in that if it is shaped as a
perfect cube and tossed upon a surface (level and smooth) the probability
of any particular face settling directly opposite of the surface is 1 in
6. The constraints that determine the outcome may be classified into
two general areas; physical and geometric, or shape. The physical
includes the laws (or rules) by which the die and the surface may attract
and combine with each other to produce and outcome. The geometric
(or shape) refers to the dimension, shape, form of the objects themselves,
i.e., the shape and form of the die and the surface.
The value of the outcome of a toss is determined by the
value (arbitrary) placed upon the surface chosen as representing the
outcome according to the rules of interaction of the die and the
surface. What happens when more die are added to the interaction is
simple to predict (in the normal sense).
Consider next the probabilities if the rules of
interaction are changed such that the surface of interaction is only the
boundaries of the form or shape of the individual die themselves.
That is, consider an arbitrary number of die, randomly forced away from a
point source and (in the absence of any other external forces) allowed to
be "mass attracted" back towards one another into a cluster; the
shape and size of which are determined by initial velocities and
directions in 3-space. (The rules of orbital mechanics may apply
here.)
The "value" of the resultant cluster is, lets
say, the value sums of the expo...
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