Peter Loly

Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS

All 8th order Franklin bent diagonal squares with distinct elements 1, …, 64 have been constructed by an exact backtracking method. Our count of 1, 105, 920 dramatically increases the handful of known examples, and is some eight orders of magnitude less than a recent upper bound. Exactly one-third of these squares are pandiagonal, and therefore magic. Moreover, these pandiagonal Franklin squares have the same population count as the eighth order ‘complete’, or ‘most-perfect pandiagonal magic’, squares. However, while distinct, both types of squares are related by a simple transformation. The situation for other orders is also discussed.

A new class of pandiagonal squares

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.

The inertia tensor of a magic cube

Magic cubes are shown to have maximally symmetric inertia tensors if they are interpreted as rigid body mass distributions. This symmetry is due to their semi-magic property where each row, column, and pillar has the same mass sum. The moment of inertia depends only on this property and the number of point masses in each row, column, and pillar. Because magic cubes do not possess detailed cubic symmetry, other scenarios that result in maximally symmetric inertia tensors are discussed.

The electric multipole expansion for a magic cube

Recent studies of the rotational properties of magic cubes are extended to a corresponding electrical problem. While expecting the dipole moments of magic cubes to vanish, it was a surprise to find their quadrupole moments also vanished. These properties are shown to follow from the RCP (row, column, pillar) symmetry of the orthogonal line sums, namely the semi-magic property of the cubes, which opens up the conclusions to a much wider class of charge distributions.