Sorry, I've been offline and missed all the follow-up questions.

After the jailor has laid out the problem, everyone knows the the total is in the range from 1+2+3=6 to 7+8+9=24.

After A and B have had their questions answered, everyone knows that the only possible totals are the composite numbers between 6 and 24 -- 9, 15, or 21.

C sees his 5 and realizes that if the total is not 15, there are only two possibility for the holdings of A and B, 1 and 3 (the only two numbers that can be added to 5 to equal 9); or 7 and 9 (the only only two numbers that add with 5 to 21).

C sees that A or B, holding either 1 or 3 will realize that there is no way the other two prisoners have numbers that will bring the total to 21, so will correctly guess the total. Similarly, if they hold a 7 or 9, they will realize that the total can't be 9.

Larry raised a good question about whether C needs to have a 5. I think the same logic works if C has a 4 or 6. Outside that range, it's still solvable, but a little different. If C has a 3 for example, he will know that the only possibilities are 9 and 15, and he can ask about either--the answer will tell him which is the total.

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