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