Answer: Option B
:
B
Since Sally thinks she has enough information, we deduce that Sam answered his house number was a perfect square greater than 50 (answering Yes to both). There are two of these {64, 81} and Sally must live in one of them in order to have decided she knew where Sam lives. Sam answered only the second question truthfully, so his house number is greater than 50, but not a perfect square.
Since Sam answered Sue’s second question truthfully, he had to have answered yes to “Is it greater than 25?” Sue was able to deduce Sam’s number, so he also must have said it was a perfect cube. Cubes greater than 25: {27, 64}. Sue must live in one of these houses to deduce Sam’s number.
Since Sam’s number is greater than 50 and is less than Sue’s number, she must live in 64. Since Sue and Sally are not roommates (we’re told there are three numbers), Sally must live in 81.
Given fact: the sum of their numbers is a perfect square multiplied by two.
Sue + Sally + Sam = 2p (for p an integer)
Or, 64+81+Sam = 2p2
Applying the constraint that Sam’s number is greater than 50 and less than 64, it looks like Sam = 55 (p=10).
In summary,
Sam = 55, Sue = 64, Sally = 81
Hence Answer option B