Empty one pile of balls and win

Question

There are two piles of balls. One contains 5 balls and the other contains 18 balls.
A and B are playing the following game:
In each round, a player has to move balls from the larger pile to the smaller pile such that the number moved was a non-zero multiple of how many were previously in the smaller pile. A starts, and A and B alternate until a pile of balls is empty. The person who emptied it is the winner.
Who will win this game?

Zerris · Accepted Answer

A moves 5 balls over, which leaves piles of 10-13.

The only legal move for B is moving 10, leaving 20-3.

A responds by moving 9, leaving 11-12.

The only legal move for B is moving 11, leaving 22-1.

A moves all 22 balls, leaving 0-23.

Nautilus · Answer

The total number of balls always remains 23, a prime number, meaning:

Being left with only 1 ball in the smaller pile is the only way to immediately win.
Scenario 1 can only be forced on the loser right away if he/she's left with 11 balls in the smaller one and 12 in the bigger one.
2's condition can be forced on the loser right away only if (s)he leaves 2, 3, 4 or 6 (factors of 12 save 1 and itself) balls in the small pile, leading to a 12-ball pile.
The loser can be forced to leave a 3-ball pile if left with a 10-ball one, which can be done in the very first move by A.

Empty one pile of balls and win

2 Answers

Add your own answers!

Ask a Question