Table of Contents About the Author What's new Abstract Main Results The Transformation Method The Permutation Method The Intermediate Square Method Enumeration Programs Some References Notations, Definitions & Conventions An Example of Enumeration Programs Magic Squares of Order 4 Magic Squares of Order 5 Magic Squares of Order 6 Magic Squares of Order 7
Structure of Magic and Semi-Magic Squares,
Methods and Tools for Enumeration
by Francis Gaspalou

I give the main notations, definitions and conventions; it is not an exhaustive list.

1. Notations

n Order of magic or semi-magic square
S Magic constant
S = n(n+1)/2 for normal squares, made with numbers from 1 to n.
S = Σ + σ with:
Σ = A+B+C+... (capital letters)
σ = a+b+c+... (lower-case letters)
k Number of basis (a,b,c...;A,B,C...) in the decomposition of a square with capital and lower-case letters
k is a function of n
N Number of parameters
N=(n-1) for semi-magic squares of order n
N=(n-1)-2 for magic squares of order n
Note that the dimension of the vector space is N + 1 (one condition is given here: S is fixed in function of n; in other words, I deal with normal squares)
A1  A2  A3  ...  An
Cells of magic or semi-magic squares (or matrix of transformation)

2. Definitions
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Magic square Rows, columns and the two diagonals are magic: the sum of numbers is constant (this sum is the "magic constant")
Semi-magic square Rows and columns are magic, but not the two diagonals
Pandiagonal square Rows, columns and broken diagonals are magic
Semi-pandiagonal Specific square of order 4 (A2+B1+C4+D3=34 and A3+B4+C1+D2=34)
Basis (a,b,c...;A,B,C...) Set of n values for lower-case letters a,b,c,... and of n values for capital letters A,B,C,.... allowing to build a table of addition
Caution! It is different from the basis in a vector space.
Intermediate square Square with capital and lower-case letters
Isomorphic With the same structure
Regular Each (capital or lower-case) letter occurs only one time in each line
Semi-regular For each line, the sum of capital letters is Σ (and then the sum of lower-case letters is σ)
Irregular For one line, the sum of capital letters is different from Σ (or the sum of lower-case letters is different from σ)
Line Row, column or diagonal
Latin square Square with n letters a, n letters b, etc. Each letter occurs only one time in each row and each column
Gallic square Square with n letters a, n letters b, etc. It is not compulsory, as in a Latin square, that each row (or column) has only one a letter; it is sufficient that a occurs n times in the square. Idem for b, etc.
Group Set of elements having special properties (in mathematics, you can define a rule which is associative, there is an identity element and each element has an inverse)
Abstract definition
of a group
Relations of definition of a group (relations on generators)
Transposition Specific transformation made with permutations of rows and columns symmetrically located from the centre of the square

3. Conventions
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I deal only with "normal" magic or semi-magic squares, i. e. with squares made with the numbers from 1 to n.

I don't divide by 8 the number of magic squares as it is usually done in the literature.
For example, I speak about 7 040 squares of order 4 and not about 7 040/8 = 880 squares.
As a matter of fact, for a given square, I think you have to consider all the variations of group G and not only the 8 classical variations (rotations and reflections) of octic group.

For counting the number of squares given by an intermediate square (square with capital and lower-case letters),
I don't take the transformations of this intermediate square.

Regular squares are for me a subset of semi-regular squares.