Multiplicative Magic Tori

Because the present-day definition of complementary and self-complementary magic squares can be over-restrictive when studying magic tori, an alternative modular arithmetic interpretation is therefore proposed.

This approach suggests a new direction for research on magic squares (or magic tori), using modular multiplication, modular addition, and modular exponentiation.

Multiplicative Magic Tori (MMT) are examined throughout the orders 1 to 4. (Although there is no magic torus of order-2, there is a Multiplicative Diagonally Semi-Magic Torus (MDSMT)). The study then continues with a partial inspection of various examples from higher-orders, including some bimagic MMT of order-8.

A typical Multiplicative Magic Torus (MMT), or Multiplicative Magic Square (MMS) of order-4

The results include a new census of the Multiplicative Magic Tori (MMT) and Multiplicative Magic Squares (MMS) of orders 1 to 4. A detailed classification of the 82 Multiplicative Magic Tori (MMT) and 220 Multiplicative Magic Squares (MMS) of order-4 is given, together with explanatory graphics that highlight the main relationships and links.

In addition, it is shown that the diagonally magic, diagonally semi-magic, and lesser-magic tori, which are also present on the MMT, have special orthogonal sums. Some other interesting characteristics of modular exponentiations are also examined and commented.

The conclusions are presented in the form of integer sequences. Any input that might confirm, correct, or usefully append these findings, would be much appreciated

New Developments! 

Posted on the 24th April 2024, a new article entitled "Plus or Minus Groups of Magic Tori of Order 4" now demonstrates that the 255 magic tori of order 4 come from 137 ± Groups!

Although "Plus or Minus Groups of Magic Tori of Order 4" proposes an alternative way of looking at things, the "Multiplicative Magic Tori" represents valid intermediary research, and therefore remains available below.

Multiplicative Magic Tori (outdated by findings in the 137 ± Groups)

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 "Multiplicative Magic Tori" will then be displayed, with options for zooming.