Enumerating the bent diagonal squares of
Dr Benjamin Franklin FRS
BY DANIEL SCHINDEL†, MATTHEW REMPEL AND PETER LOLY*
Department of Physics and Astronomy, The University of Manitoba, Winnipeg,
Manitoba R3T 2N2, Canada
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.
Keywords: Franklin squares; enumeration; counting; pandiagonal magic squares;
backtracking; polyhedral cones
The
via
*A
†Pr
Mic
Rec
Acc
1. Introduction
A resurgence of interest in the handful of Benjamin Franklin’s bent diagonal
squares of 8th and 16th order which he constructed in 1736–1737 (Pasles 2001,
2003; Ahmed 2004a,b) may be related to the tercentenary of his birth which
occurred on 17 January 2006. The same year also marks the 250th anniversary of
Franklin’s election as a Fellow of the Royal Society of London. While Franklin
squares must have the semi-magic property where the entries of all rows and
columns add to the same magic sum, they do not have to have the same sum
for the main diagonals which are required for fully magic squares. A distin-
guishing feature of Franklin’s famous squares is that they have ‘bent’ diagonals
with elements adding to the magic sum as indicated for the ‘right’ bends by
following the symbols in the following square (with allowance for tiling, wrap-
around or periodic boundary conditions, and with this square rotated for ‘down’,
Proc. R. Soc. A
doi:10.1098/rspa.2006.1684
Published online
electronic supplementary material i