Question

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.

The largest value of N (less than 50) that ensures a win for B is?​

Answer 1

Given:

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.

To find:

The largest value of N (less than 50) that ensures a win for B is?

Solution:

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

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

When the total number of the matchstick is 3, A will win as A will pick 2 matchsticks then B has to pick the last one.

When the total number of the matchstick is 4, B will win as A picks up 1 matchstick, remaining matchsticks will be 3 and we know that from above.

Similarly, the series for B to win follows:

1, 4, 7, 10.........N.

a = 1, d = 4-1 = 3

we use the formula, Tn = a + (n - 1)d

Since we need to the largest value of N (less than 50)

let Tn = 49

49 = 1 + (n - 1)3

49 = 1 + 3n - 3

49 = 3n - 2

51 = 3n

n = 17

Therefore, the 17th term of series 1, 4, 7, 10.........N is 49 < 50 is the largest value of N that ensures a win for B.