IMC 2019 Q12
12. The game of Rorrim2 is played on a 4 × 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
Intermediate Mathematical Challenge 2019, Question 12. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 8
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.
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).