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# June 2015 - House Numbers

Mr.A, Mr.B, Mr.C, and Mr.D live in the same street with house numbers ranging from 1 to 99. All four of them are perfect logicians and none of them share a house. Mr.A, Mr.B, and Mr.C are always truthful. Mr.D sometimes lies, although Mr.A and Mr.B do not know that yet. Mr.A, Mr.B, and Mr.C's house numbers form an arithmetic sequence with common difference 2, in any order. Mr.C knows Mr.D's house number. Otherwise, no one knows anyone else's house number.

Mr.A and Mr.B are trying to find out Mr.D's house number. One day Mr.A meets Mr.D and asks him what the first digit of his house number is. Mr.D answers. Mr.A then asks him whether his house number is prime or not. Mr.D answers. Mr.A knows he can figure out Mr.D's house number but also knows that if Mr.B had the same conversation, he wouldn't be able to tell. However, when he gets there, he finds that he is wrong.

The next day, unaware of Mr.A's conversation, Mr.B meets Mr.D and asks him whether his house number is square. Mr.D answers. Mr.B then asks him whether his house number is greater than 50. Mr.D answers. Again, Mr.B thinks he knows Mr.D's house number. Again, when he gets there, he is wrong.

On the next day, Mr.A, Mr.B, and Mr.C all meet together. They each tell their house numbers and then Mr.A and Mr.B tell their conversation. Mr.C then tells them that, in both cases, Mr.D had answered the first question falsely and the second question truthfully. Mr.C also mentions that the difference between his and Mr.D's house number is a multiple of 8. Mr.A and Mr.B are now able to correctly find Mr.D's house number. What is it?

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