Question 23
Alice and Bob are given an integer $n \ge 2$ and a token to play the following game. Initially, Alice chooses an order of the numbers $1, 2, \dots, n$, and writes them down in a row, in that order, on a sheet of paper. Next, Bob chooses one of these numbers and places the token on that number. Continuing, Alice moves the token on one of the neighbouring numbers, then Bob moves the token on one of the neighbouring numbers of the current position, and so on and so forth in turns. For each $k$ in the range $1, 2, \dots, n$, the token may be placed on number $k$ at most $k$ times; Bob's first move is counted. The player who can no longer move the token loses. Determine the values of $n$ for which Alice has a winning strategy.
Mathematical Olympiad Rioplatense, 2010, Level 3
Mathematical Olympiad Rioplatense, 2010, Level 3