A, B and C start playing a match-stick game with a total of 50 sticks distributed among themselves, with each having a natural number of match-sticks. As per the game, A starts by distributing match-sticks to B and C such that the number of sticks with B and C doubles respectively. A continues to do so for 4 rounds, after which she has only 2 sticks left and can no longer distribute to keep up the doubling trend for both B and C. If (b, c) represents the possible number of match-sticks that B and C could have at the beginning of the game, then how many combinations of (b, c) are possible?Correct answer is '2'. Can you explain this answer?

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A, B and C start playing a match-stick game with a total of 50 sticks distributed among themselves, with each having a natural number of match-sticks. As per the game, A starts by distributing match-sticks to B and C such that the number of sticks with B and C doubles respectively. A continues to do so for 4 rounds, after which she has only 2 sticks left and can no longer distribute to keep up the doubling trend for both B and C. If (b, c) represents the possible number of match-sticks that B and C could have at the beginning of the game, then how many combinations of (b, c) are possible?

Correct answer is '2'. Can you explain this answer?

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A, B and C start playing a match-stick game with a total of 50 sticks ...

Solution: Let the number of match-sticks with A, B and C initially be a, b and c respectively. After the 1st round, B and C’s match-sticks double to 2b and 2c respectively, while A’s match- sticks reduce to (a - b - c). Continuing similarly for the first 4 rounds, we have,

After 4 rounds, A is left with only 2 sticks. This means that B and C together had 50 - 2 = 48 sticks after 4 rounds i.e. 165+ 16c = 48 b + c = 3

Possible values for (b, c) are (0, 3), (1, 2), (2, 1) and (3, 0).
However, as per the question, the players begin the game with a natural number of match-sticks; thus (0, 3) and (3, 0) are not valid values.
Answer: 2

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A, B and C start playing a match-stick game with a total of 50 sticks ...

Introduction:
In this match-stick game, A starts with a total of 50 sticks and distributes them among B and C in such a way that the number of sticks with each of them doubles after each round. After 4 rounds, A is left with only 2 sticks and can no longer distribute to maintain the doubling trend for both B and C. We need to find the possible combinations of match-sticks that B and C could have at the beginning of the game.

Approach:
To solve this problem, we need to find all the possible combinations of match-sticks that B and C could have at the beginning. We can use a systematic approach to calculate the number of sticks with B and C after each round and check if it satisfies the given conditions.

Step 1: Possible combinations of match-sticks for B and C:
Let's assume that B starts with 'b' match-sticks and C starts with 'c' match-sticks. We need to find all the possible combinations of 'b' and 'c' such that B and C can have a natural number of match-sticks.

Step 2: Distributing match-sticks in each round:
In each round, A distributes match-sticks to B and C in such a way that the number of sticks with each of them doubles. Let's calculate the number of sticks with B and C after each round.

Round 1: B = 2b, C = 2c, A = 50 - (2b + 2c)
Round 2: B = 4b, C = 4c, A = 50 - (4b + 4c)
Round 3: B = 8b, C = 8c, A = 50 - (8b + 8c)
Round 4: B = 16b, C = 16c, A = 50 - (16b + 16c)

Step 3: Checking the conditions:
After 4 rounds, A is left with only 2 sticks and can no longer distribute to maintain the doubling trend for both B and C. We can set up the following equations to represent this condition:

A = 50 - (16b + 16c)
A = 2

Solving these equations will give us the values of 'b' and 'c' that satisfy the given conditions.

Step 4: Solving the equations:
Substituting the value of A in the first equation, we get:

2 = 50 - (16b + 16c)
16b + 16c = 48
b + c = 3

Step 5: Finding the possible combinations:
We need to find all the possible combinations of 'b' and 'c' such that b + c = 3. The possible combinations are:

b = 1, c = 2
b = 2, c = 1

Therefore, there are 2 possible combinations of (b, c) that satisfy the given conditions.

Conclusion:
After analyzing the match-stick game and applying the given conditions, we find that there are 2 possible combinations of (b, c) that satisfy the doubling trend and the remaining number of match-sticks.

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A, B and C start playing a match-stick game with a total of 50 sticks distributed among themselves, with each having a natural number of match-sticks. As per the game, A starts by distributing match-sticks to B and C such that the number of sticks with B and C doubles respectively. A continues to do so for 4 rounds, after which she has only 2 sticks left and can no longer distribute to keep up the doubling trend for both B and C. If (b, c) represents the possible number of match-sticks that B and C could have at the beginning of the game, then how many combinations of (b, c) are possible?Correct answer is '2'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A, B and C start playing a match-stick game with a total of 50 sticks distributed among themselves, with each having a natural number of match-sticks. As per the game, A starts by distributing match-sticks to B and C such that the number of sticks with B and C doubles respectively. A continues to do so for 4 rounds, after which she has only 2 sticks left and can no longer distribute to keep up the doubling trend for both B and C. If (b, c) represents the possible number of match-sticks that B and C could have at the beginning of the game, then how many combinations of (b, c) are possible?Correct answer is '2'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A, B and C start playing a match-stick game with a total of 50 sticks distributed among themselves, with each having a natural number of match-sticks. As per the game, A starts by distributing match-sticks to B and C such that the number of sticks with B and C doubles respectively. A continues to do so for 4 rounds, after which she has only 2 sticks left and can no longer distribute to keep up the doubling trend for both B and C. If (b, c) represents the possible number of match-sticks that B and C could have at the beginning of the game, then how many combinations of (b, c) are possible?Correct answer is '2'. Can you explain this answer?.

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