In this page, the main results of my paper are presented, without (or with very little) explanations. These explanations are detailed in specific developments further in the text. Most of these results seem original.
It is well known for mathematicians that magic squares have a vector space structure. But it is less well known that magic squares have also a Gset structure (a group G operates on the set of magic squares; transformations on magic squares have a group structure).
The order of group G is the number of variations of a given square. Then enumeration can be reduced to a set of "elementary squares" or "essentially different squares".
For semimagic squares of order n, the order of group G is 2 (n!)
^{2}.
For magic squares of order n=2p or n=2p+1, the order of groups G is 2
^{p+2} (p!) and these two groups are isomorphic; they have the same abstract definition. The first values are:
Order of squares  Order of group G 
3  8 
4  32 
5  32 
6  192 
7  192 
8  1 536 
Some results on group G ( note: I don't divide by 8 the number of squares, as it is usually done in the literature)
Set  Total number of squares  Order of group G  Number of essentially different squares  Total number of isomorphic sets (with initial set included) 
Magic squares of order 4  7 040  32  220  1 
Magic squares of order 5  2 202 441 792  32  68 826 306  1 

Semimagic squares of order 4  549 504  1 152  477  1 

Semipandiagonal squares of order 4  3 456  1 152  3  1 
Pandiagonal squares of order 4  384  384  1  11^{(1)} 
Symmetrical squares of order 4  384  384  1  11^{(1)} 
...  ...  ...  ...  ... 

Symmetrical squares of order 5  388 352  128  3 034  3 
Pandiagonal squares of order 5  28 800  28 800  1  2 
de la LOUBERE's squares of order 5  5 760  5 760  1  8 
BACHET's squares of order 5  1 152  1 152  1  4 
...  ...  ...  ...  ... 

Symmetrical pandiagonal squares of order 7  161 525 472  96  1 682 557  ? 
(1) there are 9 semipandiagonal + 2 other sets
I found the exact mathematical definition of some groups G, as:
 the group G of order 32 for magic squares of order 4 is G 32 n° 7 of BURNS and has for "abstract definition":
s_{1}^{4} = s_{2}^{4} = s_{3}^{2} = s_{4}^{2} = s_{1}^{3} s_{2 }s_{1 }s_{2} = s_{3 }s_{1 }s_{3 }s_{1}^{3} = s_{2}^{3} s_{1 }s_{2 }s_{1 }
= (s_{4 }s_{1})^{2} = (s_{4 }s_{2})^{2} = (s_{4 }s_{3})^{2} = (s_{3 }s_{2})^{2} = 1.
 the group G of order 1 152 for semipandiagonal squares of order 4 is the only group of order 1 152 of degree 8 and has for "abstract definition":
s_{1}^{12} = s_{2}^{2} = (s_{1}^{4} s_{2})^{2}(s_{1}^{8} s_{2})^{ 2} = (s_{1}^{3} s_{2})^{2}(s_{1}^{9} s_{2})^{ 2} = (s_{1}^{3} s_{2 }s_{1}^{4} s_{2})^{ 2} = 1
and I identified examples of generators.
A same set of squares has a group G (transformations on the position of cells) and a group G' (permutations on the contents of cells). E.g.:
Set  Order of group G (transformations)  Order of group G' (permutations) 
7 040 magic squares of order 4  32  2 
3 456 semipandiagonal of order 4  1 152  384 
384 pandiagonal squares of order 4  384  384 
If the set has a group structure, there is probably equality between the order of group G and the order of group G' (and the dimension of the set).
The transformation and permutation methods can be combined to enumerate the squares in a given set.
E.g.: if you apply the transformation method on the 7 040 magic squares of order 4, you find 220 elementary squares. But these 220 elementary squares can be reduced to 166 if you apply afterwards the permutation method. 166 is the minimum number of elementary squares you can have.
The order of group G is naturally less than the dimension of the set of squares. There is equality for pandiagonal squares of order 4 and 5, but not for superior orders:
Order  Number of pandiagonal squares  Order of group G  Order of the subgroup of transpositions 
4  384  384  128 
5  28 800  28 800  800 
6  0  irrelevant  irrelevant 
7  ? (estimated by Walter TRUMP)  2 352?  2 352 
The intermediate square method which consists to decompose a square into two squares (capital letters and lowercase letters) is very fruitful to represent some sets of squares, as regular and semiregular squares.
If a GrecoLatin square of order n can be built, then it is possible to associate k (n!)² semimagic squares to it, k being the number of orderly basis. This number k is a function of n:
The k (n!)² squares in the associated set are not only semimagic. Their real properties depend on those of the original GrecoLatin square. For example:
 the 3 456 semipandiagonal squares of order 4 are generated from one GrecoLatin square which is semipandiagonal,
 the 28 800 pandiagonal squares of order 5 are generated from one GrecoLatin square which is pandiagonal,
 ...
The 2 (n!)² squares given by a GrecoLatin square of order n (with one basis and its corresponding basis) have probably a group structure.
I calculated the first sets of regular squares (with all the basis):
 Order 4  Order 5  Order 6  Order 7 
Number of regular squares (with all the basis)
 3 456 (semipandiagonal)
 57 600 (pandiagonal and one other isomorphic set)  0
 2 438 553 600 (pandiagonal and 7 other isomorphic sets) 
(For order 7, I am not absolutely sure that pandiagonal squares with their isomorphic sets are the only regular squares).
For magic squares of order 6, I found orthogonal squares to Latin diagonal lowercase squares. I calculated the total number of such squares (for all the basis), as well as the equivalent numbers for orders 4 and 5:
 Order 4  Order 5  Order 6 
Number of orthogonal squares to Latin diagonal squares  3 456 (semipandiagonal)  410 880
 56 005 447 680

Among these orthogonal squares of order 6, I found an intermediate square that generates 483 840 squares, which is for the moment the largest set of squares that can be generated by an intermediate square of order 6.
For magic squares of order 6 too, I found squares with 8 regular lines (6 rows and 2 diagonals for example), which seems to be the maximum possible number.
For symmetrical pandiagonal squares of order 7, I found a new transformation W of order 2 allowing to define a group G of order 96 (and then to reduce by 2 the computation time for enumeration).
For symmetrical pandiagonal squares of order 7, I found squares with 24 irregular lines, which is the maximum possible number.
All the magic squares and semimagic squares can be generated from elementary capital and lowercase letter squares. I drove such a task for order 4 (for superior orders, it is naturally impossible to do it up to the very end). For example, I found that all the 7 040 squares of order 4 can be generated from 22 elementary squares (with the first basis).
I showed that magic squares of order 4 and semimagic of order 4 can be decomposed in basic sets, each set being a part of another set before, with a graph of filiation and with isomorphic sets. I think  and it is only a conjecture  it is true also for order n.