Monday, 11 September 2017

Paired and Self-Complementary Magic Tori of Order-4

In his paper "Self-complementary magic squares of 4x4," Mutsumi Suzuki has shown that there are 352 self-complementary magic squares of order-4. Although Suzuki does not use Frénicle standard form, his examples are essentially different, and the total of these 352 self-complementary squares can therefore be compared with the grand total of the 880 essentially different magic squares of order-4 previously identified by Bernard Frénicle de Bessy.

On his page "Self-Similar Magic Squares," Harvey Heinz has studied Mutsumi Suzuki’s work, and points out that the 352 self-complementary magic squares include 48 Dudeney Type III associated semi-pandiagonal, 96 Dudeney Type VI semi-pandiagonal, and 208 Dudeney Type VI “simple” magic squares. The Dudeney Types referred to by Harvey Heinz come from Henry Dudeney’s classification of the 880 Frénicle magic squares into 12 types, each of which corresponds to a different complementary number pattern.

Today we know that the 880 Frénicle indexed magic squares are partial viewpoints of 255 essentially different magic tori of Order-4 (OEIS sequence A270876). A previous post "Complementary Number Patterns on Fourth-Order Magic Tori" has shown that some of the 12 Dudeney complementary number patterns become redundant on fourth-order magic tori, and this has resulted in the creation of a new reduced set of complementary number pattern types which are better adapted:
New order-4 magic torus complementary number patterns types I to II replace former Dudeney patterns for magic squares.
New order-4 magic torus complementary number patterns types III to VIII replace former Dudeney patterns for magic squares.

Therefore, with less complementary number patterns, what are the implications for the totals of self-complementary and paired complementary magic tori of order-4? In the study that follows, all of the 255 magic tori of order-4 are listed, and the details of their complements are given. In the final observations, the different cases of self-complementarity or paired complementarity are analysed, together with the respective complementary number patterns.

Please note that if you click on the button that appears at the top right hand side of the pdf viewer below, a new window will open and full size pages of the paper "Self-Complementary and Paired Complementary Magic Tori of Order-4" will then be displayed, with options for zooming.

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