A Chapter in the Scientific Studies of Magical Squares

Peter Loly

University of Manitoba

The enumeration of eighth-order Frankin squares by Schindel, Rempel, and Loly will be published in the Proceedings of the Royal Society A, in July or August 2006. While classical magic squares with the entries 1::n2 must have the magic sum for each row, column, and the main diagonals, there are some interesting relatives for which these restrictions are increased or relaxed. These include serial squares with sequential filling of rows that are always pandiagonal (having all parallel diagonals to the main ones on tiling with the same magic sum, also called broken diagonals), pandiagonal logic squares derived from Karnaugh maps [Loly and Steeds, Int. J. Math. Ed. Sci. Tech. 36(4), 2005, 375–388, with an application to Chinese patterns, Loly, The Oracle - The Journal of Yijing Studies, 2(12), January 2002, 2–13)], and Franklin squares that are not required to have any diagonal properties, but have equal half row and column sums and two-by-two quartets, as well as magical bent diagonals.

We modifed Walter Trump’s backtracking strategy for other magic square enumerations from GB32 to C++ to perform the Franklin count [a datafile of the 1,105,920 distinct squares is available], and also have a simplifed demonstration of counting the 880 fourth-order magic squares using Mathematica [a draft notebook]. Our early explorations of magic squares considered as square matrices used Mathematica to study their eigenproperties.

We have also studied the moment of inertia and multipole moments of magic squares and cubes (treating the numerical entries as masses or charges), finding some elegant theorems [Rogers and Loly, Am. J. Phys., 72(6), 786–9, June 2004, and European J. Phys. 26 (2005) 809–813], and have shown how to easily compound smaller squares into very high-order ones, e.g. 12,544 (= 28 x 72) th order [Chan and Loly, Math. Today, 38(4), 113–118, August 2002]. Brée and Ollerenshaw have a patent on using relatives of Franklin squares for cryptography, while a group at Siemens in Munich using pandiagonal logic squares has another pending. Other possible applications include dither matrices for image processing and providing tests for developing CSP (constraint satisfaction problem) solvers for difficult problems.

This presentation will be based on a spectacular 3’-by-4’ poster of the Franklin work.