  How many fair coin tosses to choose 1 of $n$ options with uniform
probability? Although a probability problem, the solution is essentially
number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and
the haupt exponent. We propose a bit-efficient scheme, prove optimality, derive
the expected number of coin tosses $e[n]$, characterize its fractal structure,
and develop sharp upper and lower bounds, both discrete and continuous. A minor
but noteworthy corollary, with real-world examples, is that any lottery or
simulation with finite budget of random bits will have a predictable pattern of
lucky and unlucky numbers.
