Statement

Dóra, a mathematics teacher, has just taught her students how to compute the greatest common divisor (GCD) of multiple numbers. Now, she wants to challenge them with an interesting game. She writes the numbers 2, 3, …, N on the board in some order of her choice. The students must then erase the numbers following these rules: (1) In each step, they can remove one or more consecutive numbers from either the left or the right end of the sequence. (2) The group of numbers selected for removal can only be erased if their GCD is greater than 1. Suppose N = 6 and Dóra writes the following sequence on the board: 2, 4, 6, 3, 5 In the first step, the students could erase the following: (1) 2 (since \gcd(2) = 2), (2) 2, 4 (since \gcd(2,4) = 2), (3) 2, 4, 6 (since \gcd(2,4,6) = 2), or (4) 5 (since \gcd(5) = 5). They cannot erase 3 and 5 together, since \gcd(3,5) = 1. They cannot erase 6 and 3 together, as they are not at the left or right end of the sequence...