Question 9
Players $A$ and $B$ play a game as follows. Initially $A$ arranges the numbers $1, 2, \dots, n$ in a row as he wishes; $n$ is a given positive integer. Next, $B$ chooses one number and puts a stone on it. Then $A$ moves the stone to an adjacent number, $B$ does the same and so on. The stone can be placed on number $k$ at most $k$ times, $k = 1, \dots, n$; the initial move of $B$ is counted. The one who cannot move loses. For each $n$ determine who has a winning strategy.