  The densest packings of N unit squares in a torus are studied using
analytical methods as well as simulated annealing. A rich array of dense
packing solutions are found: density-one packings when N is the sum of two
square integers; a family of "gapped bricklayer" Bravais lattice solutions with
density N/(N+1); and some surprising non-Bravais lattice configurations,
including lattices of holes as well as a configuration for N=23 in which not
all squares share the same orientation. The entropy of some of these
configurations and the frequency and orientation of density-one solutions as N
goes to infinity are discussed.
