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There are 15 colours for one line of symmetry, 5 by 5, as 3 styles of column with all different colours and as 3*5, number of rows, is 15, 15 colours are needed.

There are 28 colours for one line symmetry, 7 by 7, as 4 styles of column with all different colours and as 4*7, number of rows, 28 colours is needed.

The formula in terms of n, when n is an odd number of rows is, (n+1)/2 * n (this formula used for explanation) or n2+n/2
(simplified).

This works as the number of styles of column with different colours is (n+1)/2. EG (5+1)/2 (5by5 pattern) is 3(the number of styles of column with different colours).

And the n you times, is the no. of rows in the square.