On April 16, 2015, Tanya Khovanova wrote in her blog:

<begin>
Here is my new logic puzzle.
I thought of a positive integer that is below 100 and is divisible by 7. In addition to the public knowledge above, I privately tell the units digit of my number to Alice and the tens digit to Bob. Alice and Bob are very logical people, but their conversation might seem strange:

Alice: You do not know Tanya’s number.
Bob: I know Tanya’s number.

What is my number?
<end>

Please comment.

For Alice to say that Bob did not know Tanya's number, she would have to know that the tens digit was either 2, 4, 7, or 9:

21 or 28
42 or 49
70 or 77
91 or 98

However, if it was 2 or 9, then Bob would not be able to know what the number was since the same ones digits matched for either 2 or 9:

21 or 91
28 or 98

That leaves 42, 49, 70, and 77.

We can eliminate the first two, because if with Alice holding either a 2 or 9, Bob would have to be holding a 4 with no way of knowing what Alice held.

So Alice must have had a 0 or a 7.  If she held a 7, then she would not have been able to conclude that Bob did not know the number because if Bob had a 0, he would already know.  Thus, Alice held a 0 and Bob  held a 7.

Tanya's number is 70.

Posted by hoodat on 2016-06-28 04:43:30