  Understanding the distribution of digits in the expansions of perfect powers
in different bases is difficult. Rather than consider the asymptotic digit
distributions, we consider the base-10 digits of a restricted sequence of
powers of two. We apply elementary methods to show that this sequence of powers
of two can be constructed to preserve trailing digits while locally maximizing
the number of zeroes between nonzero digits. We also provide a heuristic
description of the trailing digits of these powers of two.
