Problem #19

Consider all possible ways of coloring the squares of an n x n checkerboard so that if n is even, the number of black squares equals the number of white ones, and if n is odd, the number of black squares is one more than the number of white ones. [Note that the standard checkerboard colorings satisfy this condition.] Two colorings are considered to be equivalent if one can be obtained from the other by rotation or reflection of the board. For example, there are 2 inequivalent colorings of a 2 x 2 board (see figure).

i. How many inequivalent colorings are there for a 3 x 3 board?

ii. For a 4 x 4 board?

iii. For an n x n board?

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