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

Ann and Ben are playing a game with a pile of counters.

The rules are:

(a) at each turn a player may remove 1, 2 or 3 counters

excepting the number removed by the opponent on the

previous turn;

(b) the player who makes the last legal move wins;

Rule (b) automatically means that if there is one counter

left and the player can't move, then the opponent wins.

Can Ann win if she is presented with 4 counters?

Show that Ann can always win from a pile of 6 counters

regardless of Ben's previous move.

Question

Birdy

Mathematics

8 November, 19:32

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

Ann and Ben are playing a game with a pile of counters.

The rules are:

(a) at each turn a player may remove 1, 2 or 3 counters

excepting the number removed by the opponent on the

previous turn;

(b) the player who makes the last legal move wins;

Rule (b) automatically means that if there is one counter

left and the player can't move, then the opponent wins.

Can Ann win if she is presented with 4 counters?

Show that Ann can always win from a pile of 6 counters

regardless of Ben's previous move.

+5

Answers (1)

Know the Answer?

Winston Santana · Answer 1

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.