I present to you another epistemic puzzle, my personal favourite:
Problem #12: A Rational Problem
Albert and Bernard are being tested by their teacher once more. He gives them both a little note with a number on it, and he tells them both numbers are different from each other and of the following form for n and k natural numbers larger than 0:
1/2n + 1/(2k+1)×2n
Then the following conversation happens:
| Teacher: | "Who of you has the smallest number?" |
| Albert: | "I don't know." |
| Bernard: | "I also don't know." |
| Albert: | "I still have no clue." |
| Bernard: | "No, nothing for me either." |
| Teacher: | "You two can keep going like this for a long time, but you will never find the answer like this." |
| Albert: | "Ah, that's very interesting information, yet still I don't know if I'm the smallest." |
| Bernard: | "Neither do I." |
| Teacher: | "Again, continuing like this will not lead any of you two to know their number." |
| Albert: | "That is truly remarkable, but I do not know if I have the smallest number." |
| Bernard: | "Nor do I." |
| Albert: | "Ah! But now I suddenly know who is smaller!" |
| Bernard: | "Fantastic, then I know both of our numbers!" |
Which numbers are written on Albert’s and Bernard’s notes?