Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS

Abstract

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.

Linear Type-p Most-Perfect Squares

“…The tasks of counting and constructing classical most-perfect squares were first approached by McClintock [6] and culminate in the work of Ollerenshaw and Bree [8], which gives a count of the classical most-perfect squares for any doubly even order n, along with a construction method for all such squares. Also, classical most-perfect squares are useful in constructing Franklin magic squares (see [11] and [7], and [9] for historical background).…”

Pandiagonal Type-p Franklin Squares

“…We construct such squares in prime power orders. Our construction is motivated by a relationship, first noted in [10] and further explored in [6], between classical most-perfect magic squares of triply even order and pandiagonal classical Franklin squares.…”

“…The third category concerns construction and enumeration: One example is [1], in which Hilbert bases for polyhedral cones are used to place an upper bound on the number of Franklin squares. Another example is [10], in which an involution on arrays is used to define an injection from the set of most-perfect squares of order 8 to the set of pandiagonal Franklin squares of order 8, thus giving a reasonable lower bound on the number of order-8 Franklin squares. Importantly, this latter work was generalized in [6] to squares of order 8k for any k ∈ Z + .…”

“…The tasks of counting and constructing classical most-perfect squares were first approached by McClintock [5] and culminate in the work of Ollerenshaw and Bree [7], which gives a count of the classical most-perfect squares for any doubly even order n, along with a construction method for all such squares. As mentioned above, classical most-perfect squares are used in [10] and [6] for constructing Franklin squares. When p ≥ 2, a linear construction of type-p most-perfect squares of order p r (r ≥ 2) is given in [4].…”

“…Inspiration for these results comes chiefly from [10] and [6], where the authors introduce an involution θ that maps classical most-perfect squares into pandiagonal classical Franklin squares. This involution may be generalized (see Section 2) so that it applies to type-p most-perfect squares, examples of which exist in all orders p r with r ≥ 2 by [4].…”