Solution
Rotating a Three-by-Three Grid
Show the problem again
Each cell of a 3 × 3 square tile is coloured black or white. Two tiles are considered the same if one can be rotated in the plane (by 90°, 180° or 270°) to match the other; flipping the tile over is not allowed. How many different tiles are there? Give an integer.
Worked solution
The answer is 140. The group is the four rotations. The identity fixes all $2^9 = 512$ colourings. A 90° or 270° rotation fixes the centre and splits the other 8 cells into two 4-cycles (corners and edges), so it fixes $2^3 = 8$ colourings. The 180° rotation fixes the centre and pairs the other cells into 4 two-cycles, fixing $2^5 = 32$. Burnside gives $(512 + 8 + 32 + 8)/4 = 560/4 = 140$.
Source: Standard application of Burnside's lemma; no traceable original source. Statement written for AxiomIQ.