The King’s Game

A king challenged his advisor to a game. “We will take turns removing stones from a pile of fifteen. You may take one, two, or three on your turn. Whoever takes the last stone wins.”

The advisor paused. He didn’t think about his first move. He thought about the last.

“If I leave the king with four stones, he must leave me at least one. If I leave him with eight, I can always leave him with four. If I leave him with twelve...”

The advisor worked backwards from the end to the beginning, and his first move was determined by the final position he wanted.

He took three stones and won the game on his fourth turn, as the king was forced to mirror him down to zero.