CAT Data Interpretation-Interesting Question
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Matchstick problem
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The Matchstick problem is a cult problem, which is frequently appearing in TIME-AIMCATs and its variants appear in lot of competitive exams like CAT,XAT,MAT, Campus placement papers….This problem appears in various forms,like two people going for a drive taking turns while driving for a max. & min. kilometer limits.
The variants of same question came 2-3 times in original CAT papers.

Mainly 2 variants are there for this problem, we will discuss the type 1 variant in this post.

Type 1.
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There are n matchsticks, and 2 players A and B. One person should take minimum of 1 stick and can take maximum of 5 sticks at a time. The person who takes the last stick is the loser. Each player will play intelligently in order to win.
a.If there are 10 matchsticks and A has to play next, how many sticks he has to take to ensure that he wins?
b.If there are 21 matchsticks, and B is to play, Is there a chance for B to win?

Explanation:-

Consider the situation of 8 matchsticks and A has to play.
A can take max 5 and min 1.
Consider the sequence of steps that will follows if A plays intelligently to win

1. A will take 1 and remaining is 7.
2. Now B can take a max of 5, suppose he takes 5,then ,2 will remaining.
3. A will take 1 and 1 will remain.
4. B has to take min. 1.so he will lose and A will win.

Consider another sequence where in step 2,B takes min 1.

1. A will take 1 and remaining is 7.
2. Now B takes 1,then ,6 will remaining.
3. A will take 5 and 1 will remain.
4. B has to take min. 1.so he will lose and A will win.
So in both these extreme cases, irrespective of B’s play, A is winning.

This leads us the conclusion that, in this problem,
‘Winner is decided by initial number of matchsticks and the first play.”

Also, the aim of the player who plays first will always be to leave
{(min. limit + max. limit)+1}
ie here {(1+5)+1}=7 numbers of matchsticks to his opponent.
The same scenario will occur even if he leaves the multiples of this number.
ie in our question, the player who plays first, should aim to leave either 7 or a multiple of 7 matchsticks to his opponents so that irrespective of the future moves he is sure about his success.
So, in a game with the above min and max limits, the first player should aim at leaving the number of matchsticks as 7,14,21,28,35……..

The aim of the player who plays first will always be to leave
{(min. limit + max. limit) +1} numbers of matchsticks to his opponent.

Note:
Here if initially the number of matchsticks is a multiple of 7, then the first player will always lose.

Variant of Type 1:
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In the above case the min. limit was 1.
Suppose the min. limit has been increased to a higher number,say2.
Then also we need to apply our general formula,
ie the aim of the player who plays first will always be to leave
{(min. limit + max. limit) +1} or {(min. limit + max. limit) +2} numbers of matchsticks to his opponent.

So in this case, the player who plays first, should aim to leave either 8 or 9 or a multiple of 8 or 9 matchsticks to his opponents so that irrespective of the future moves he is sure about his success

Type 2 --will be discussed in coming posts...