SolUtion
7
As the number of cells to be coloured is odd, one of those must map onto itself when the rectangle is rotated through $180^{\circ}$ about its centre. So the central cell in the rectangle must be coloured. The diagram shows that there are seven pairs of cells which may be chosen as the second and third cells to be coloured so that the coloured grid has rotational symmetry of order 2. So Aston can complete the grid in seven ways.
A5. Aston wants to colour exactly three of the cells in the grid shown so that the coloured grid has rotational symmetry of order two. Each of the cells in the grid is a square. In how many ways can Aston do this?