Logic Puzzles from multiple points of view 4/5

Over the next few days I’ll be posting a group of five puzzles, related by a “multiple points of view” theme.

None of these puzzles belong to me. And I don’t know where to extend credit. I think they’ve just sort of existed, forever.

Prisoners with Hats

Three perfectly logical 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, but execute any man guessing incorrectly.

They look back at him in stunned 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 26.″

And he went free. What were Alan and Graham’s numbers?  Explain your reasoning. Why did it take five questions to get an answer?

Place questions/clarifications below. To submit proposed solutions, click here.