Donald Aucamp offered this problem in the Puzzle Corner department of MIT Technology Review in October 2003. Three logicians, A, B, and C, are wearing hats. Each of them knows that a positive integer has been painted on each of the hats, and each of them can see her companions’ integers but not her own. They also know that one of the integers is the sum of the other two. Now they engage in a contest to see which can be the first to determine her own number. A goes first, then B, then C, and so on in a circle until someone correctly names her number. In the first round, all three of them pass, but in the second round A correctly announces that her number is 50. How did she know this, and what were the other numbers?

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Howard Haber gave this solution in the MarchApril 2004 issue. The key observation to bear in mind is that if one of the logicians sees that both of her opponents’ hats bear the same number, she can conclude immediately that her own number is twice this value.
Each of the logicians knows that her own number is either the sum or the positive difference of the other two. A, going first, reasons that the three numbers A, B, C must stand in one of two ratios, 1:2:3 or 5:2:3. She can’t decide between these possibilities, so she remains silent. But in the second round A has the benefit of knowing that each of the other logicians also remained silent in the first round. Now, suppose that the ratio had been 1:2:3. Then, when B took her turn in the first round, she would have concluded that the ratio must be either 1:2:3 or 1:4:3. Like A, she would have remained silent, unable to decide between these possibilities. But C, going next, would have determined that the ratio must be either 1:2:3 or 1:2:1 — and she could immediately have excluded the latter because in that case B would have been able to win the game by employing the key observation above. Noting that B didn’t do this would have allowed C to win the game herself by announcing her only remaining possibility, 1:2:3. The fact that C didn’t do this tells A that the ratio can’t be 1:2:3, and she concludes that it must be the other ratio she’d been considering, 5:2:3.
“For the problem as stated, just multiply all numbers above by 10.”
