Puzzling Asked by Tharindu Sathischandra on May 8, 2021

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?

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.

Correct answer by Zerris on May 8, 2021

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.

Answered by Nautilus on May 8, 2021

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