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.)