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.

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.

3rd box is the answer

step-by-step explanation:

pentagon

step-by-step explanation:

the coordinates   a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.