16 and 8
Initially, each person thinks his hat can be the sum or difference of the other two numbers. A person can identify on his turn iff (the other two people have equal numbers, or if one of his two possibilities would have another player identifying on a previous turn).
Had the narrarator been honest, for Arthur to identify his immediately, the hats would have to be in proportion
2/1/1
Since he didn't, Benjamin knows that case does not hold. For Benjamin to identify on his turn, they would have to be in proportion
1/2/1 or 2/3/1
Since Benjamin didn't identify immediately, either, Charlie knows neither of those cases apply. He can identify in the cases
1/1/2, 2/1/3, 1/2/3, or 2/3/5
Since Charlie thinks he knows, Arthur and Benjamin must have numbers in ratio 1/1, 2/1, 1/2, or 2/3. Since Arthur thinks Charlie might be correct, Benjamin and Charlie must be in proportion 1/2, 1/3, 2/3, or 3/5. Since Benjamin thinks there's a problem, Arthur and Charlie are not in proportion 1/2, 2/3, 1/3, or 2/5.
1 1 2 not possible, narrator is truthful
1 1 3 not possible, Benjamin has acceptable ratio
2 2 3 not possible, Benjamin has acceptable ratio
3 3 5
2 1 2 not possible, Benjamin would have identified
2 1 3 not possible, narrator is truthful
4 2 3
6 3 5
1 2 4
1 2 6
1 2 3 not possible, narrator is truthful
3 6 10
2 3 6 not possible, Benjamin has acceptable ratio
2 3 9
4 6 9
2 3 5 not possible, narrator is truthful
For Charles to have 12, the numbers would have to be
16 8 12
3 6 12 (Charles can't rule out 3/6/18)
2 4 12 (Charles can't rule out 2/4/8)
For Charles to be able to identify his number and think it is 12, the numbers can only be
16, 8, and 12.
