Describing A Game: In the Game there are 2 players (player 1 and player 2) and there is a board with a number of ball in it $n$.There is also a set $A =${$A1...Am$} which hold the amount of ball each player is allowed to remove in one move.
progress of the game: each player at his turn remove $a\in A$ balls from the board.
loser: is a player who cannot remove $a\in A$ form the board. Meaning the number of balls on the board is smaller than $a,$$\forall a \in A$ .
For example $A = ${$2,3$} and $n=6$. the first player remove 3 balls then the second player remove 2 balls and we are left with one ball on the board so the second player win.
Another example $A = ${$2,3$} and $n=6$. the first player remove 2 balls then the second player remove 2 balls.then the first player remove 2 balls then the second player is left with zero balls on the board so the first player win.
I can see that for each $A$ and $n$ there are 2 option: Player one has as Winning Strategey,or Player two has as Winning Strategey.
I can see it via a tree that I draw that represent a game. which spread from the root (player 1) to the second level (player 2) based on $a,$$\forall a \in A$ and so on.
Edit: I need to prove that for each $n$ and a set $A =${$A1...Am$} there exist one of the following situation:
palyer 1 has a winning strategy.
player 2 has a winning strategy.
But I find it hard to proof (via induction).