Question

Extension question (provide a full explanation of your method(s):

Answer 1

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.