What are magic squares?  They are squares of order n with n² numbers within the square’s n x n matrix such that the numbers in each column, row and diagonal add up the same number.   For example, the magic square for order n=3 looks like this:

Look carefully at the rows, columns and diagonals – they each add up to the same number – 15!  Now this magic square is a special case; it is the only magic square for order 3.  Sure you can rotate it and make reflections, but this is the only arrangement for an n=3 magic square.

Magic squares exist for n >= 3.  There exists a trivial magic square for n=1, and for n=2, there is no magic square.  The number for which the columns, rows, and diagonals add up to is called the magic constant.  The value of the magic constant for a square of order n is determined by the formula:

So for n=3, the magic constant (C_m) = 15.  For n=4, C_m = 34.  For n=5, C_m = 65, etc.

How many solutions are there for magic squares for n>3?  There are many!  For n=4, there are 880.  For n=5, there are 275305224.  How many are there for n=6?  I have seen an estimate for over 1.7745×10¹⁹!

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