Question 38
Given a set of several non-negative integers, a legal move consists in selecting a positive integer $a$ out of the set and perform one of these operations:
If $a$ is odd, it is replaced by $a-1$.
If $a$ is even, it is replaced by either $a-1$ or $a-2$.
Two players, A and B, make legal moves in turns, starting from a set with the numbers $1, 2, \ldots, n$; A plays first. One player wins if after their move, a set of $n$ zeroes is obtained (so there are no more possible legal moves). For each $n$, determine which player has a winning strategy.
If $a$ is odd, it is replaced by $a-1$.
If $a$ is even, it is replaced by either $a-1$ or $a-2$.
Two players, A and B, make legal moves in turns, starting from a set with the numbers $1, 2, \ldots, n$; A plays first. One player wins if after their move, a set of $n$ zeroes is obtained (so there are no more possible legal moves). For each $n$, determine which player has a winning strategy.