Pandiagonality of the Squared Matrices of Associative and Related Magic Squares

The present study is inspired by Miguel Angel Amela's paper dated September 2016, entitled "Powers of Associative Magic Squares," and by Francis Gaspalou's paper dated 7th October 2016, entitled "Note on the features of the associative magic squares which give magic solutions when raised at an even power."

Both of these studies refer to an earlier paper by Charles K. Cook, Michael R. Bacon, and Rebecca A. Hillman; “The Magicness of Powers of Some Magic Squares,” published in The Fibonacci Quarterly, Volume 48, Number 4; 2010.

Pandiagonality of the Squared Matrices of Associative and Related Magic Squares

In the present study, which approaches the subject from a magic torus viewpoint, the use of formulae for generation enables an analysis of comparable magic tori throughout the different doubly-even and odd-orders. Please note that there are no associative magic squares of singly-even-orders, as demonstrated by C. Planck  in his "Pandiagonal magics of orders 6 and 10 with minimal numbers," published in "The Monist," Volume 29, N° 2 (April 1919), pages 307-316.

The pages that follow show that both associative and non-associative matrices can yield magic solutions when squared. Although the study was initially intended for associative magic squares only, other types of magic squares, displayed on related magic tori, are now also included for the sake of comparison.

The research also reveals that recurring pandiagonal patterns appear on the squared matrices, throughout the higher-orders of magic tori generated by a same formula.

1/4 pandiagonal squared matrix of an associative magic square viewpoint of an order-12 semi-pandiagonal magic torus

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