Skip to main content
3 of 5
Added other logically-consistent positions for C.
Steve
  • 4k
  • 13
  • 35

[I'd got a draft partial I forgot to post before a busy weekend - now revised very slightly, but plan to revisit soon]

It seems particularly relevant that in step 3 [since edited to make this particular phrasing nonsense...]

the direction of the quarter turn (clockwise or anti-clockwise) from their initial position facing the warden (who by implication is to the side observing the whole line) is not stated. Some prisoners can be facing left and others right, which allows all hats to be observed.

Working from the last clue:

Earnest gains additional information from Dennis' comment, combined with what Earnest can see. If Dennis were behind Earnest, facing the same direction, then Dennis would see everything that Earnest does. The only new information that Earnest would gain from this would be which way Dennis was facing, which cannot be combined with any other information to learn Earnest's hat colour. Thus we must conclude that Earnest can see at least one hat which Dennis does not - i.e. Earnest must be facing Dennis.

Earnest cannot see Bob, but Carol can see both Bob and Earnest. Dennis can see Earnest (in order to know that Earnest cannot see Bob)

Given that

Dennis can see Earnest, and Earnest can see at least one person/hat that Dennis cannot (in order to gain additional information), we can conclude that Earnest and Dennis are facing each other, and can see each other.

In order for EVERYONE to know their hat colour

it is necessary that hats at BOTH ends of the line are observed, which is certainly the case as at least one of Dennis and Earnest sees them.

So far, we can conclude (without loss of generality) that the line looks something like this:

      ?       =       =       ?
 ...  C> ...  B  ...  E> ... <D  ...
or
      =       =       ?       ?
 ...  B  ...  E> ... <C  ... <D  ...
or
      =       =       ?       ?
 ...  B  ...  E> ... <D  ... <C  ...

At this point

Alice and any other prisoners' positions in the line are to be deduced, and we also need to determine all hat colours and which way Bob is facing, as well as which of the 3 logically consistent positions for Carol is in fact the case.

I also accidentally deleted a comment along the lines that I think that it will probably be the case that Alice and Bob must observe both ends of the line, and thus be facing opposing directions, but my initial "partial proof" of this was flawed, so it might not be the case.

Steve
  • 4k
  • 13
  • 35