A magic square

Magic square

In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...

 is in Frénicle standard form, named for Bernard Frénicle de Bessy

Bernard Frénicle de Bessy

Bernard Frénicle de Bessy , was a French mathematician born in Paris, who wrote numerous mathematical papers, mainly in number theory and combinatorics. He is best remembered for Des quarrez ou tables magiques, a treatise on magic squares published posthumously in 1693, in which he described all...

, if the following two conditions apply:

1. the element at position [1,1] (top left corner) is the smallest of the four corner elements; and
2. the element at position [1,2] (top edge, second from left) is smaller than the element in [2,1].

Frénicle's posthumously published book of 1693 described all the 880 essentially different order-4 magic squares.

This standard form was devised since a magic square remains "essentially similar" if it is rotated or transpose

Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

d, or flipped so that the order of rows is reversed — there exists 8 different magic squares sharing one standard form. For example, the following magic squares are all essentially similar, with only the final square being in Frénicle standard form:

8 1 6 8 3 4 4 9 2 4 3 8 6 7 2 6 1 8 2 9 4 2 7 6
3 5 7 1 5 9 3 5 7 9 5 1 1 5 9 7 5 3 7 5 3 9 5 1
4 9 2 6 7 2 8 1 6 2 7 6 8 3 4 2 9 4 6 1 8 4 3 8

Generalising the concept of essentially different squares

For each group of magic squares one might identify the corresponding group of automorphism

Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

s, the group of transformations preserving the special properties of this group of magic squares. This way one can identify the number of different magic square classes

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...

.



Example: From the perspective of Galois theory

Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

 the most-perfect magic square

Most-perfect magic square

A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 with two additional properties:# Each 2×2 subsquare sums to 2s, where s = n2 + 1....

s are not distinguishable. This means that the number of elements in the associated Galois group

Galois group

In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

is 1. Please compare A051235 Number of essentially different most-perfect pandiagonal magic squares of order 4n. with A000012 The simplest sequence of positive numbers: the all 1's sequence.