  We examine the open problem of finding the shortest string that contains each
of the n! permutations of n symbols as contiguous substrings (i.e., the
shortest superpermutation on n symbols). It has been conjectured that the
shortest superpermutation has length $\sum_{k=1}^n k!$ and that this string is
unique up to relabelling of the symbols. We provide a construction of short
superpermutations that shows that, if the conjectured minimal length is true,
then uniqueness fails for all n >= 5. Furthermore, uniqueness fails
spectacularly; we construct more than doubly-exponentially many distinct
superpermutations of the conjectured minimal length.
