Key idea. The first move can create two identical halves with a gap between them.

1. With 20 plates, the first player takes the bananas on plates 10 and 11, the two middle plates. Two identical rows of nine bananas remain.

2. With 21 plates, the first player takes the banana on plate 11. Two identical rows of ten bananas remain.

3. In either case, after every opponent move, the first player makes the reflected move in the other half. A move on one plate is copied on its mirror plate; a move on two adjacent plates is copied on their two mirror plates.

4. This is always legal: before the opponent moves, both halves look identical, so the mirror bananas are still there. The empty middle plate or plates prevent any move from crossing between the halves.

5. The first player can therefore reply whenever the opponent can move. Since each move removes bananas, the game finishes, and the first player must take the last banana.

Remember. To prove a mirror strategy, explain why the reply stays available and cannot overlap the opponent’s move.