There is a 3×3 matrix. You have 2 colours, red and blue. In how many ways can…

Concept: When a shape must look identical after a symmetry operation such as a 180° rotation, that operation groups the shape's cells into orbits — sets of positions that get swapped or left fixed by the operation. For the colouring to survive the operation unchanged, every cell within one orbit must carry the same colour. If there are k independent orbits and c colours are available, the fundamental counting principle gives c^k total valid colourings, since each orbit's colour is chosen independently of the others.

Application: Number the 3×3 grid's cells 1 to 9 in the usual left-to-right, top-to-bottom order (as shown below), and observe how a 180° rotation moves each cell.

A 180° rotation sends the cell in position i to position (10 − i), so it pairs up the cells as follows, with the centre cell mapping to itself:

1. The four pairs (1, 9), (2, 8), (3, 7), and (4, 6) each swap places under the rotation, so both cells in a pair must be the same colour.

2. The centre cell (5) maps to itself, so its colour can be chosen independently of every pair.

3. This gives 5 independent groups of cells in total — 4 swapping pairs plus the self-mapped centre — each with 2 possible colours (red or blue).

4. By the fundamental counting principle, the total number of valid colourings is 2 × 2 × 2 × 2 × 2 = 25 = 32.

Cross-check: Without any restriction, a 3×3 grid with 2 colours has 29 = 512 possible colourings in total. The rotation-invariant colourings are exactly those that stay constant on each of the 5 orbits identified above, so their count is 25 = 32 — consistent with the value obtained by direct counting.