Alice and Bob are two infinitely intelligent logicians. Each has a number drawn on their forehead. Each can see the other’s number but not their own. Each knows that both numbers are positive integers. An observer tells them that the number 50 is either the sum or the product of the two numbers. Alice says to Bob, “I do not know my number,” and Bob replies, “I do not know my number either.” What is Alice’s number?

| SelectClick for Answer | | --- | | If Bob’s number isn’t a divisor of 50, then 50 must be the sum and Alice will know her number. If Bob’s number is 50, then 50 must be the product and again Alice should know her number. She doesn’t, so Bob’s number must be a proper divisor of 50, that is, 1, 2, 5, 10, or 25.By the same reasoning, if Alice’s number isn’t a proper divisor of 50 then Bob should know his number. Also, after Alice’s utterance Bob can infer that his own number is a proper divisor of 50, following her thinking above. So each number is a proper divisor of 50.Now, if 50 is the sum of the two numbers, then both numbers are 25 (as 25 + 25 is the only way to combine two of the candidate divisors to get 50). This means that if Alice’s number isn’t 25, then Bob can conclude that 50 is a product and can infer his own number. The fact that he can’t do this shows that Alice’s number must be 25 (and Bob’s number is either 2 or 25).(Via Matvey Borodin et al., “It’s Common Knowledge,” Recreational Mathematics Magazine 6:12 [December 2019], 9-32.) |

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