On April 16, 2015, Tanya Khovanova wrote in her blog:
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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?
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Please comment.
1 7
2 14
3 21
4 28
5 35
6 42
7 49
8 56
9 63
10 70
11 77
12 84
13 91
14 98
Digits 7, 4, 1 and 8 are represented twice as unit digits.
Tens digits 2 (possible 21, 28), 4 (possible 42,49), 7 (possible 70, 77) and 9 (possible 91, 98) don't let you know what the number is.
Alice (knows the units digit): You (Bob) don't know the number.
Bob (knows the tens digit): I know the number.
What units digit did Alice see so she know Bob had 2, 4, 7 or 9?
If she saw a 1 she would know it was either the ambiguous 2 or the ambiguous 9, but the exact same would be the case if she saw an 8. So whether Bob saw a 2 or a 9, Bob would still not know if Alice saw a 1 or an 8.
If Alice saw a 2 she'd know the ambiguity between 42 and 49. But the same would be true if she saw a 9. So a 4, if given to Bob, would still be ambiguous.
OK, what about Alice's seeing 3, 4, 5, 6 or 7?
If Alice knew 3, 4, 5 or 6 she'd know Bob saw no ambiguity and wouldn't have said what she said.
7 is an odd case; let's talk about it later.
If Alice knew 0 or 7 she'd say Bob doesn't know, so if Bob saw 7 he wouldn't know if it was 70 or 77.
What does that leave?
Perhaps Alice saw a 7 and made the judgement that Bob saw a 7 also, but also perhaps, Bob was told either that there was no tens digit or that the tens digit was zero. Maybe despite being very logical, Alice overlooked this possibility when she made her statement.
If so, the number is 7.

Posted by Charlie
on 20160510 14:00:22 