Answer:
Ann has little chance to win if she is presented with 4 counters.
Ann can always win from a pile of 6 counters.
(both are explained below)
Step-by-step explanation:
If Ann is presented with 4 counters, and
1. if she takes out 3, she will lose since the opponent will pull out 1 and the last one.
2. if she takes 2 her opponent will take out 1 and she can't pull out the last 1 since her opponents last move was to pull out 1 counter so she will lose.
3. If she takes out 1 and her opponent takes out 3 in the next move she loses.
but if instead of 3 her opponent takes out 2 and in the last move Ann takes out the last 1 then she will win.
So, If Ann is presented with 4 counters she has little chance to win provided in the move just before, her opponent didn't move 1 counter.
Now,
if there is 6 counters to Ann, and
1., if Ben's previous move was 1 then Ann can win if she takes out 3 or 2.
If she takes out 3 Ben can take out 1 or 2 and in the last move she will take out 2 or 1 (respectively) and winning the game.
If she takes out 2 Ben can take out 1 or 3 and in the last move Ann wins by pulling out 3 or 1 respectively.
2. if Ben's previous move was 2 then Ann can win if she takes out 1 or 3.
If she takes out 1 Ben can take out 2 or 3 and in the last move she will take out 3 or 2(respectively) and winning the game.
If she takes out 3 Ben can take out 1 or 2 and in the last move Ann wins by pulling out 2 or 1 respectively.
2. if Ben's previous move was 3 then Ann can win if she takes out 1 or 2.
If she takes out 1 Ben can take out 2 or 3 and in the last move she will take out 3 or 2(respectively) and winning the game.
If she takes out 2 Ben can take out 1 or 3 and in the last move Ann wins by pulling out 3 or 1 respectively.
