Track / Stage 4 / 304 of 340 #564 of 1964

Problem 564

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Find the answer

Vasya strikes the strings of a 6-string guitar from 1 to 6 and back. Each subsequent strike hits the adjacent string. On which string number will the 2000th strike fall? (The order of striking the strings: $1-2-3-4-5-6-5-4-3-2-1-2-\ldots$ )

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Answer. 2.

Solution. The first string receives the strikes $1,11,21,31, \ldots, 2001$ . Therefore, the 2000-th strike will fall on the 2-nd string.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.