## June 2015 - House Numbers (Solution)The answer is 71. Consider the conversation between Mr.B and Mr.D. As Mr.B only knows his own house number, he could only figure out Mr.D's house number if there were at most two options (If there were two, Mr.B would have to occupy one of the options to rule it out). Now, the only way this could happen from the conversation is if Mr.D answered Yes to both questions. Then the only options are 64 and 81 so Mr.B occupies one of them. Now consider the conversation between Mr.A and Mr.D. If Mr.D answered No to the second question, there would be no way for Mr.A to deduce Mr.D's house number. Thus Mr.D answered Yes to the second question. A table of possibilities for each answer to the first question is helpful.
Now, there must be at most two possibilities. Thus Mr.D said either 6, 8, or 9. However, if Mr.D said 9, then no matter what Mr.A or Mr.B's house numbers are, Mr.B could have figured that out. Thus Mr.D said 6 or 8 and Mr.A's house number is one of 61, 67, 83, or 89. Since Mr.A's house number is odd and Mr.A, Mr.B, and Mr.C's house numbers form an arithmetic sequence with common difference 2, Mr.B's house number is also odd and is therefore 81. Now,the difference between Mr.A and Mr.B's house numbers is either 2 or 4 so Mr.A's house number is 83 and Mr.D answered 8 to Mr.A's first question. Now consider the conversation between Mr.A, Mr.B, and Mr.C. Mr.C reveals that Mr.D's house number does not begin with 8, is prime, is not square, and is greater than 50. Mr.C's house number could have been 79 or 85. If it had been 85 then Mr.D's house number could have been 53 or 61. If it had been 79 then Mr.D's house number would have to be 71. Since Mr.A and Mr.B could figure it out with just that information, Mr.C's house number is 79 and Mr.D's house number is 71.
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