»  Solutions to puzzles in my National Review Online Diary

  July 2003


In my July diary I posed the following brain-teaser:

Of two unknown integers, each between 2 and 99 inclusive, a person P is told the product and a person S is told the sum. When asked whether they know the two numbers, the following dialogue ensues:

P:  "I don't know them."
S:  "I knew that already."
P:  "Then I now know the two numbers."
S:  "Then I now know them, too."

What are the two numbers? Prove that your solution is unique.



This is an exceptionally elegant puzzle. You start by noticing that if the numbers were two primes (7 and 11, for example), then P would know them at once on being given the product (77). Since he begins by telling us he doesn't know them, they are not both primes.

Furthermore, none of the prime factors of either number can be bigger than 47. Why not? Well, consider a case where one of them is; let's say, it's 59. The smallest number this is a factor of is 59, the next smallest is 118, and so on. Since we have been told that our unknown integers are both between 2 and 99 inclusive, 59 must be one of the numbers. The other one must be the product, divided by 59. P could work all this out from just the product, once he'd spotted that factor 59. Since he didn't, there can't be any prime factor like this —i.e. bigger than 50.

All that is deduced from just the first one of those statements — the statement, by P, that after being given the product, he does not know the numbers!

You might want to pick up the logic and work it from there. For a complete solution, see Prof. Dijkstra's paper here.