Divide and conquer

Rules — Divide and conquer

• We start with two piles of stones (for example 31 and 30).
• On your turn you do two things: you remove one whole pile and you split the other pile into two non-empty parts.
• Turns alternate. After every move there are again exactly two piles.
• The player who cannot split loses — that only happens when both piles have a single stone (1 and 1), because a pile of one cannot be divided.

Advanced level (misère): the rule flips — whoever cannot split wins.

Strategy (careful, spoiler)

It is a game of winning and losing positions. The key is the parity of the two piles. Can you find which positions are traps for the player about to move?