Track / Stage 4 / 260 of 340 #520 of 1964

Problem 520

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Find the answer

[ Winning and losing positions ]

There are 300 matches in a box. On a turn, it is allowed to take no more than half of the matches in the box. The player who cannot make a move loses.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

The first player wins. Winning positions are those in which there are $2^{n}$ - 1 matches left in the box.

The first move is to leave 255 matches.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.