Instructions

In a game played by two people there were initially N match sticks kept on the table. A move in the game consists of a player removing either one or two matchsticks from the table. The one who takes the last matchstick loses. Players make moves alternately. The player who will make the first move is A. The other player is B.

Question 9

The smallest value of N (greater than 5) that ensures a win for B is

Solution

When total matchstick is 1, B will surely win as A is the one who is picking up matchstick first.

When total matchstick is 2, A will surely win as A will pick one matchstick then B has to pick last one.

When total matchstick is 3, A will surely win as he takes up a strategy so that B will have last matchstick to pick up. So A will pick 2 matchstick.

When total matchstick is 4, B will surely win as if A picks up one matchstick, remaining matchsticks will be 3 and we know that when total 3 matchsticks are there then the one, who is picking up matchstick first, is winning surely.

When total matchstick is 5, A will surely win as he takes up a strategy so that B will have last matchstick to pick up. So A will pick 1 matchstick first, then 4 matchsticks will be there and we know that the one, who picks up matchstick second now, will win. Hence A will win.

When total matchstick is 6, A will surely win either he takes up a strategy so that B will have last matchstick to pick up. So A will pick 2 matchstick first, then 4 matchsticks will be there and we know that the one, who picks up matchstick second now, will win. Hence A will win.

Hence it is making a pattern when number of matchsticks are 1,4,7,10......N then B is winning the game.

So here minimum value of N greater than 5 will be 7.

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