*semi-magic*squares - 3x3 grids filled with integers 1,2,...,9 in such a way that each cell contains a different integer and the sum of the numbers in each row and each column is the same. It is easy to see that for a 3x3

*semi-magic*square this sum (sometimes called the

*magic constant*) should be 15. The definition of a

*semi-magic*square does not require (but does not preclude) that the sum of the numbers in each diagonal should also be equal to the

*magic constant*. If that's the case a

*semi-magic*square becomes a

*magic*square. Here are a few examples of 3x3

*semi-magic*squares. The middle square is not only a

*semi-magic*square, but also a

*magic*square.

*semi-magic*squares, so I wrote a little program to generate my own. If you ever need a list of all these squares, - here it is, arranged in lexicographical order.

It turned out, there are 72 different 3x3

*semi-magic*squares. But how "different" are they really? It is easy to see that each of the 8 geometric transformations of a square (identity, 90°, 180°, 270° counterclockwise rotations around the center, reflections across the horizontal axis, the vertical axis, and each diagonal) transforms a

*semi-magic*square into another

*semi-magic*square.

We can therefore split 72

*semi-magic*squares into 9 non-intersecting equivalency classes

*.*Members of same class are reflections or rotations of each other; members of different classes are not. Below are representatives of each of the 9 classes . Every

*semi-magic*square in a class is a geometric transformation of its representative. All 72

*semi-magic squares*can be produced out of these 9 by applying geometric transformations. From this perspective there are only 9 "substantially different"

*semi-magic*squares.

But why stop at just geometric transformations? What other transformations turn one

*semi-magic*square into another? Column swaps, row swaps (and sequences of them) do the trick. There are 6 transformations (including identity) done by row permutations and 6 transformations done by column permutations. The picture below shows the 6 row permutations.

It turns out the 6 row permutations are independent from the 6 column permutations and taken together as sequences of row/column permutations, they constitute 36 different transformations. Based on this transformation group we can split 72

*semi-magic*squares into just 2 non-intersecting equivalency classes

*.*Members of the same class are sequences of row/column permutations of each other, members of different classes are not. Here are the 2 classes representatives. All other

*semi-magic squares*are produced out of these two by series of row/column permutations.

*semi-magic*squares are the result of applying row/columns permutations and geometric transformations to just one

*semi-magic*square. And we a free to choose any one of them as "the one". Let's actually make it a

*magic square*, just for fun!

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