Each square of a 4 × 4 gameboard is colored either red or black. Two colorings are considered equivalent if one can be obtained from the other by a rotation, but mirror images are not necessarily the same. How many different gameboard colorings are possible?

 Jun 27, 2024

Apply Burnside's Lemma: Number of distinct colorings = 1/4*(∣C_0 degrees​∣ + ∣C_90 degrees​∣ + ∣C_180 degrees​∣ + ∣C_270 degrees​∣)

For C_0 degrees: No squares are moved so it is 2^16

For C_90 degrees: Since there are 4 quadrants, it is 2^4

For C_180 degrees: Each square must match the one diametrically opposite to it. This creates 8 pairs of squares, where each pair must be the same color, so it is 2^8

For C_270 degrees: You do the same thing as C_90 degrees: 2^4

So the answer is 1/4*(65536 + 16 + 256 + 16) = 16448


(I'm not fully sure about this answer, but if it is wrong please tell me) :)

 Jun 29, 2024

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