An interesting class of purely pandiagonal, i.e. non-magic, whole number (integer) squares of orders (row/column dimension) of the powers of two which are related to Gray codes and square Karnaugh maps has been identified. Treated as matrices these squares possess just two non-zero eigenvalues. The construction of these squares has been automated by writing Maple code, which also performs tests on the results. A rather more trivial set of pandiagonal non-magic squares consisting of the monotonically ordered sequence of integers existing for all orders has also been found.