The Sum of Some Hats
Warning: this puzzle is hard. I shared it with a drinking Swede, and he recast it with names of Swedish politicians on his blog. I hope I got the details right!
Three prisoners are seated facing each other, with hats on their heads. Each hat has a counting number {1, 2, 3, 4, …} on it, and each prisoner can see the hat of the other two prisoners, but not his own.
The warden says, I will set free the man who can tell me his own number.
They look back at him in silence.
“OK, a hint. One of your numbers is the sum of the two. Alan?”
“I don’t know my number”
“Bert?”
“I don’t know my number”
“Graham?”
“I don’t know my number”
“Alan?”
“I still don’t know my number”
“Bert?”
“My number is 52”
And he went free. What were Alan and Graham’s numbers? (And how did you discover this?)
JD2718 
Alan 13
Bert 52
Graham 39
Reasoning goes like this:
Alan sees 52 and 39 he knows his number is 13 or 91
Bert sees 13 and 39 he knows his number is 26 or 52
Graham sees 13 and 52 he knows his number is 39 or 65
Alan can’t say his number
Bert can’t say his number
When Graham can’t say his number Bert thinks,
If I am number 26 Graham would see 26 and 13 and conclude that he was number 39 as 13 would be impossible(otherwise I would see two number 13 and be able to answer).
As Graham can’t answer I can’t be number 26 so I must be number 52.
Alan can’t still know his number on his second turn.
There is something wrong, but I am not having an easy time finding it. Where did you find the numbers 13 and 39?
I got 13 from the sequence 52,26,13 and started out testing some combinations. This sequence is interesting as combinations where two have the same number are very limiting.
Sorry so slow to respond. It appears that Alan would get his number on the second pass if it were 13 52 39.
I need to look more closely though, because I may have carelessly modified the problem and generated multiple answers. Give me through the weekend to reanalyze.
Would you mind posting your answer to this one, Jonathan?
Here’s a start.
When Alan says he doesn’t know, all three understand that he doesn’t see equal numbers on Bert and Graham’s hats. The ratio 1:1:2 is not correct, and all three know it.
When Bert says he doesn’t know, all three understand that the ratio 1:2:1 is out. Further, all know that Bert did not see 1:x:2, since he would have understood that the ratio was 1:3:2, and would have answered. So 1:3:2 is out.
OK, let’s go further. When Graham says he doesn’t know, he provides us with a ton of information.
If he saw 1:2:G, he knows 1:2:1 is out, so he would have known. 1:2:3 is out.
If he saw 1:3:G he would have known, so 1:3:4 is out.
Also, 1:1:2 is out (he would have known instantly)
I don’t think I’ve skipped any.
Anyone else ready to jump in?
Review, and fill in, correcting error. Discovering mistake.
Alan says no. Eliminates 2:1:1 (corrects mistake above)
Bert says no. Eliminates 1:2:1. 2:3:1 (corrects mistake above)
Graham says no. Eliminates 1:1:2. 2:3:5. 1:2:3
Alan says no. Eliminates 3:2:1. 4:3:1. 3:1:2. 8:3:5. 5:2:3
and I stand corrected and chagrined. I altered a number and made multiple answers possible.
Bert might see 3:B:1 and conclude 3:4:1
Bert might see 4:B:1 and conclude 4:5:1
Bert might see 3:B:2 and conclude 3:5:2
Bert might see 8:B:5 and conclude 8:13:5
Bert might see 5:B:3 and conclude 5:8:3
The first and fourth possibilities yield solutions: 39:52:13 (your answer) or 32:52:20 (my intent)
Apology? Of course. I goofed.
Fixable? Sure. Let Bert conclude his hat is 78.