Key idea. The second player can restore equal piles after every move.

1. The piles start equal: 10 stones each.

2. Whenever the first player removes a number of stones from one pile, the second player removes the same number from the other pile.

3. This reply is legal because the piles were equal just before the first player’s move. Afterwards they are equal again, so the rule can be repeated.

4. Eventually the first player empties a pile. The second player empties the equally large other pile and takes the last stone.

Remember. Symmetry is useful when you can restore it after every opponent move.