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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