WLOG $x \geq y \: mod. 2$ 1. case: $x \equiv_2 y \equiv_2 1 \Rightarrow$ A wins trivially 2. case: $x \equiv_2 y \equiv_2 0 \Rightarrow$ B wins trivially 3. case: $x \equiv_2 1 \land y \equiv_2 0$ 3.1 case: $x \equiv_4 1$ A performs 2 on his first move. B has to perform 1 on two even piles. Afterwards A always performs 1 on the two odd piles. Since B has to perform 1 on 2 even piles, there always exist two odd piles after move B and the smallest odd natural number is $1 > 0$. Therefore A cannot loose. $\Rightarrow$ A wins. 3.2 case: $x \equiv_4 3$ 3.2.1 case: $y \equiv_4 2$ If A performs 1, B wins by 3.1. If A performs 2, B always performs 1 on the only two odd piles and B wins, similarly to 3.1. $\Rightarrow$ B wins. 3.2.2 case: $y \equiv_4 0$ A wins by performing 1 and then 3.2.1 case. $\Rightarrow$ A wins.