Three Numbers, One the Sum

Alice, Bob and Carol are perfect logicians. Each has a positive integer written on their forehead; each sees the other two numbers but not their own. It is common knowledge that one of the three numbers equals the sum of the other two. A referee asks them in the order Alice, Bob, Carol, Alice, Bob, Carol, ... "Do you know your own number?" Every answer is heard by all. You can see that Bob wears 5 and Carol wears 7, but not Alice's number. You observe that the first six answers are all "no", and that the seventh answer, Alice's second, is "yes". What number is on Alice's forehead?

• graph theory
• induction
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Source: Classic three-foreheads sum puzzle in wide circulation; no traceable original source. Statement written for AxiomIQ.