perplexus.info :: Numbers : The Dating Game

The Dating Game (Posted on 2002-06-14)

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?

Submitted by Happy

Rating: 4.2857 (14 votes)

Solution: (Hide)

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 is 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 = 2 p^2 (for p an integer)
64 + 81 + Sam = 2 p^2
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
(Source: Primrose Puzzles, who apparently got this from rec.puzzles :)

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
answer	K Sengupta	2007-03-08 12:10:49
re(2): Solution	Andrew	2004-10-21 17:08:04
re: Solution	Easy_to_find_out	2004-10-20 21:49:08
re(2): Solution	cges	2002-12-12 05:50:58
my solution	shaun	2002-06-28 00:40:35
re: Solution	TomM	2002-06-20 21:05:00
re(5): Now kids, bicker! :)	Happy	2002-06-18 11:46:24
re(4): Now, kids, don't bicker :)	levik	2002-06-18 10:24:30
re(3): Now, kids, don't bicker :)	levik	2002-06-15 11:36:05
re: re: re2: Now, kids, don't bicker :)	TomM	2002-06-15 10:46:04
re: re2: Now, kids, don't bicker :)	levik	2002-06-15 09:12:33
re2: Now, kids, don't bicker :)	Nick Reed	2002-06-14 15:18:09
re: Now, kids, don't bicker :)	TomM	2002-06-14 11:27:34
Now, kids, don't bicker :)	levik	2002-06-14 09:35:31
re4: re: Solution	Nick Reed	2002-06-14 07:59:50
re: re2: re: Solution	TomM	2002-06-14 07:55:48
re2: re: Solution	Nick Reed	2002-06-14 07:51:28
re: re: Solution	TomM	2002-06-14 07:47:11
re: Solution	Nick Reed	2002-06-14 07:43:36
Solution	TomM	2002-06-14 07:42:35
Thoughts	Nick Reed	2002-06-14 07:41:31

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