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?
(In reply to re: could be
by Ady TZIDON)
I posted that because if Alice received a 0 she was able to deduce that Bob's number was a 7 (as the only number under 100 divisible by 7 and finishing in 0 is 70), but also that Bob with that seven could not know if the number was 70 or 77.
But when Alice speaks, Bob can understand that Alice is sure that he is not able to determine the number. This give him a new information, because he can exclude number 77. He then knows that number is 70.
Bob can exclude 77 because if Alice should have got a 7 she shouldn't be able to be sure that Bob do not know the number. She should has been in doubt about if the number of Bob was a 7 or a 0 (in the second case - a 0 - but non in the first -a 7 - Bob should have been able to deduce Tanya's number).
For the other possibilities or both know the number and know that the other knows, or both do not know it and know that the other do not know or ( 42 and 49) Alice know and know that Bob does not know and Bob can't do any other thing that confirm Alice.
Edited on May 10, 2016, 4:37 pm
Posted by armando
on 2016-05-10 16:23:45