  We show that the hard-square lattice gas with activity z= -1 has a number of
remarkable properties. We conjecture that all the eigenvalues of the transfer
matrix are roots of unity. They fall into groups (``strings'') evenly spaced
around the unit circle, which have interesting number-theoretic properties. For
example, the partition function on an M by N lattice with periodic boundary
condition is identically 1 when M and N are coprime. We provide evidence for
these conjectures from analytical and numerical arguments.
