Here is a epistemic logic puzzle that I wrote for my students in the undergraduate logic course I have been teaching this semester at the College of Staten Island at CUNY. We had covered some similar puzzles in lecture, including Cheryl’s Birthday and the blue-eyed islanders.
Bonus Question. Suppose that Alice and Bob are each given a different fraction, of the form
, where is a positive integer, and it is commonly known to them that they each know only their own number and that it is different from the other one. The following conversation ensues.
JDH: I privately gave you each a different rational number of the form
. Who has the larger number? Alice: I don’t know.
Bob: I don’t know either.
Alice: I still don’t know.
Bob: Suddenly, now I know who has the larger number.
Alice: In that case, I know both numbers.
What numbers were they given?
Give the problem a try! See the solution posted below.
Meanwhile, for a transfinite epistemic logic challenge — considerably more difficult — see my puzzle Cheryl’s rational gifts.
Solution.
When Alice says she doesn’t know, in her first remark, the meaning is exactly that she doesn’t have
Many of the commentators came to the same conclusion. Congratulations to all who solved the problem! See also the answers posted on my math.stackexchange question and on Twitter:
Epistemic logic puzzle: Still Don’t Know.
Another question from the logic final exam. Epistemic logic. https://t.co/UBpQPTyVPc
— Joel David Hamkins (@JDHamkins) December 20, 2017
Joel–By A’s 1st comment, it is known that she doesn’t have 1/1. By B’s response it is known that he does not have 1/2. A’s response makes it clear she also does not have 1/3. B’s sudden knowledge attests to his having 1/4. A now knows this as well, and she also clearly knows what her own number is. (I don’t, but it must be 1/n for n greater than or equal to 5.)
does any of that make sense?
> B’s sudden knowledge attests to his having 1/4.
Not necessarily right, B could also be 1/3, since that would still be greater than whatever Alice could have, but only in one of those situations can Alice be sure what he has.
B’s sudden knowledge attests to his having a 1/3 or 1/4. A then would only know B’s with certainty if she had a 1/4. Therefore A’s number is 1/4 and B’s 1/3. Otherwise the analysis above is correct.
That sounds right to me, Jerome. Thank you very much.
–Charlie Sitler