Suppose that you have n colours and m mutually independent dice, each of
which has r sides. Each dice lands on any of its sides with equal
probability. You may colour the sides of each die in any way you wish, but
there is one restriction: you are not allowed to use the same colour more than
once on the sides of a die. Any other colouring is allowed. Let X be the
number of different colours that you see after rolling the dice. How should you
colour the sides of the dice in order to maximize the Shannon entropy of X?
In this article we investigate this question. We show that the entropy of X
is at most 21log(n)+O(1) and that the bound is tight, up to a
constant additive factor, in the case of there being equally many coins and
colours. Our proof employs the differential entropy bound on discrete entropy,
along with a lower bound on the entropy of binomial random variables whose
outcome is conditioned to be an even integer. We conjecture that the entropy is
maximized when the colours are distributed over the sides of the dice as evenly
as possible.Comment: 11 page