List of the 880 Frénicle Magic Squares of Order-4

Bernard Frénicle de Bessy was the first to determine that there were 880 essentially different magic squares of order-4, and his findings were published posthumously in the "Table Générale des Quarrez Magiques de Quatre," in 1693.

Title of Frénicle's study of magic squares published in 1693

The "Table Générale des Quarrez Magiques de Quatre" published in 1693

It was more than 300 years later, that it was discovered that these 880 magic squares of order-4 were displayed on 255 magic tori of order-4. At first listed by type in 2011, the 255 magic tori of order-4 were later given additional index numbers in a "Table of Fourth-Order Magic Tori" in 2012. In 2019 some of the magic tori of order-4 were found to be extra-magic, with nodal intersections of 4 or more magic lines and / or knight move magic diagonals. Other studies of the magic tori of order-4 have included sub-magic 2 x 2 squares (in 2013), magic torus complementary number patterns (in 2017), and even and odd number patterns (in 2019). Then after a gap of 5 years, the magic tori have been found to belong to 137 plus or minus groups (in 2024).

It has become increasingly important to provide an easily accessible document that recapitulates these different findings. I have therefore compiled the following "List of the 880 Frénicle Indexed Magic Squares of Order-4," with their corresponding Dudeney types and also full details of the magic tori:

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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.

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