Saturday, 14 September 2019

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 from "Divers ouvrages de mathematique et de physique / par Messieurs de l'Académie Royale des Sciences [M. de Frénicle ... et al.]."
Title of Frénicle's study of magic squares published in 1693

This is the first page of the Table General of Magic Squares of Order-4 by Bernard Frénicle de Bessy
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. Then in 2018, it was found that the 255 magic tori came from 82 multiplicative magic tori of order-4. 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).

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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