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Given that, B asks A when C is not present, 2 questions:
1) Is your house number a perfect square?
2) Is your house number greater than 50?
Assuming A's answers to be correct, B is able to infer house number of A from answers given by A.
Since B is able to infer house number of A from answers given by A the only possibility is that A has answered in 'yes' to both the questions. So that there are two possibilities: 64 and 81. And He can infer the house number of A only if his own house number is one of these two numbers.
Given that, C asks A when B is not present, 2 questions:
1) Is your house number a perfect cube?
2) Is your house number greater than 25?
Assuming A's answers to be correct, C is able to infer house number of A from answers given by A.
Since C is able to infer house number of A from answers given by A the only possibility is that A has answered in 'yes' to both the questions(since A answered 'yes' in reply to B's second question, so he must answer 'yes' to C's second question). So that there are two possibilities: 27 and 64. And He can infer the house number of A only if his own house number is one of these two numbers.
Given that A's house number is less than B and C's house number and A's house number is greater than 50, this implies that C's house number is 64 and B's house number is 81.
For getting house number of A, we will first use the following information:
Sum of house numbers of 3 individuals is perfect square multiplied by two.
Given that B's House Number is 81, C's House Number is 64, let A's house number be x. Also we know that 100 > x > 50. Now, x+81+64 is a perfect square multiplied by 2. This implies that (x+81+64)/2 = (x + 145)/2 is a perfect square. Since 100 > x > 50 this implies that
245/2 > (x + 145)/2 > 195/2. Now we look for a perfect square in the interval (195/2, 245/2)
=(97.5, 122.5)
Two Possibilities are 100, 121
Solving, (x + 145)/2 = 100 give us x = 55
and (x + 145)/2 = 121 give us x = 97
Now using the condition that A's house number is less than house number of both B and C we rule out x = 97
Hence, A's house number is 55.
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