12. The game of Rorrim 2 is played on a $4 \times 4$ board, starting with a counter in one corner, as shown.
At each turn, the player moves the counter to a cell that is the reflection of its current cell in one of the six dashed lines.
How many cells could the counter occupy after precisely three turns?
A 4
B 6
C 8
D 12
E 16
12. C The diagram shows the original diagram shaded as on a chess board. Note that as the counter starts on an unshaded cell then, in whichever line it is reflected, one reflection will move it into a shaded cell. Similarly, a second reflection will move the counter from a shaded cell into an unshaded cell. Finally, a third reflection will move the counter from an unshaded cell into a shaded cell. It is left to the reader to show that it is possible to reach each of the shaded cells in precisely three reflections. Hence the number of cells the counter could occupy after three moves is equal to the number of shaded cells, that is 8 .