Sally and Sue have a strong desire to date Sam. They all live on the same street yet neither Sally or Sue knows where Sam lives. The houses on this street are numbered 1 to 99.

Sally asks Sam, "Is your house number a perfect square?". He answers. Then Sally asks "Is it greater than 50?". He answers again. Sally thinks she now knows the address of Sam's house and decides to visit. When she gets there, she finds out she is wrong. This is not surprising, considering Sam answered only the second question truthfully.

Sue, unaware of Sally's conversation, asks Sam two questions. Sue asks "Is your house number a perfect cube?". He answers. She then asks "Is it greater than 25?". He answers again. Sue thinks she knows where Sam lives and decides to pay him a visit. She too is mistaken as Sam once again answered only the second question truthfully.

If Sam's house number is less than the numbers of the houses where Sue and Sally live, and that the sum of all three of their numbers is a perfect square multiplied by two, what are Sally's, Sue's, and Sam's house numbers?

Since both girls only asked one other question after their first, Sam must have said "yes" to the first question in each case. (There are too many possibilities, otherwise)

For the same reason, he must have answered "yes" to both second questions

There are still two possibilities for a square greater than 50 (64 and 81), so Sally had to be able to eliminate one of them. The only obvious way is if she already knew who did live there. That probably means it was her house. So Sally lives at either 64 or 81.

Similar logic shows that Sue must live at either 27 or 64.

Since Sam's number is greater than 50 and less than Sue's she can't live at 27, so she must live at 64. Sally either lives with her at 64 (which violates the spirit of the problem) or she lives at 81.

64 + 81 + 50 = 195
64 + 81 + 64 = 209

The only number that is double a square between 195 and 209 is 200, so Sam must live at 55

Posted by TomM on 2002-06-14 07:42:35