25. I gave both Ria and Sylvie a piece of paper. Each piece of paper had a positive integer written on it. I then told them that the two integers were consecutive. Ria said "I don't know your number". Then Sylvie said "I don't know your number". Then Ria said "Ah, I now know your number". Which of these could be the integer on Ria's piece of paper?
A 1
B 2
C 4
D 7
E 11
Solution
B

If Ria's number were 1, she would know, since she and Sylvie have consecutive positive integers written on their pieces of paper, that the number on Sylvie's piece of paper is 2. Therefore, when Ria says she does not know Sylvie's number, Sylvie can deduce that Ria's number is not 1 . Similarly, when Sylvie says she does not know Ria's number, Ria can deduce that Sylvie's number is not 1. Also if Sylvie's number were 2, she would be able to deduce that Ria's number is 3 since she knows it is not 1 . Therefore, since Sylvie says she does not know Ria's number, Ria can deduce that Sylvie's number is not 2 .
Therefore, since Ria can now say that she knows Sylvie's number, it follows that either Ria's number is 2 and she can deduce that Slvie's number is 3, or that Ria's number is 3 and she can deduce that Sylvie's number is 4 . If Ria's number is greater than 3 she would not be able to deduce Sylvie's number.
Therefore, of the options given, the only possibility for Ria's number is 2 .