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10

Problem 10

Solution

This type of problem just requires working out all of the possibilities and eliminating the cases that are excluded.

Out of all of the ways to assign numbers, there are only eight scenarios that Albert could have seen that would have left uncertainty to him about what number was on his forehead:

  1. B = 1, C = 2 (A = 1 or 3), B must be uncertain
  2. B = 2, C = 1 (A = 1 or 3)
  3. B = 1, C = 3 (A = 2 or 4)
  4. B = 3, C = 1 (A = 2 or 4), B must be uncertain
  5. B = 1, C = 4 (A = 3 or 5)
  6. B = 4, C = 1 (A = 3 or 5)
  7. B = 2, C = 3 (A = 1 or 5)
  8. B = 3, C = 2 (A = 1 or 5)

Essentially, there are four observable pairs {(1,2), (1,3), (1,4), (2,3)} that would leave uncertainty. Only in scenarios 1 and 4 would Albert know with certainty that Bernard does not initially know (note that Albert’s statement immediately reveals to Bernard what his number was). In the other scenarios, it is possible that Bernard observes a pair that makes his number certain.

Expanding these scenarios, we have four total cases left:

  1. B = 1, A = 1, C = 2, (excluded, since C would have known she had 2)
  2. B = 1, A = 3, C = 2 (Cheryl was initially uncertain, could have been 4)
  3. B = 3, A = 2, C = 1 (Cheryl was initially uncertain, could have been 5)
  4. B = 3, A = 4, C = 1, (excluded, since C would have known she had 1)

However, cases 1 and 4 are excluded since Cheryl would have known before Albert spoke. In the two remaining cases, Cheryl does not have the largest number, so she wrote “no” on the paper.

The information in the bonus question reduces the possibility to only case 3:
A = 2, B = 3, C = 1
since in case 2, it is not possible to move just one person to the front to get the numbers in ascending order.

Another puzzle

The “blue-eyed islanders” puzzle is really neat

xkcd presentation: https://xkcd.com/blue_eyes.html
solution: https://xkcd.com/solution.html

Terence Tao wrote a detailed blog post: https://terrytao.wordpress.com/2011/04/07/the-blue-eyed-islanders-puzzle-repost/
Further discussion: https://terrytao.wordpress.com/2011/05/19/epistemic-logic-temporal-epistemic-logic-and-the-blue-eyed-islander-puzzle-lower-bound/

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