Product and Sum (unsolved)

(Seen in a presentation by Rineke Verbrugge, in NVTI Theory day, 2007)
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 dialog 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.

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5 Comments on “Product and Sum (unsolved)”

  1. simpple Says:

    Note that this is a puzzle that needs time and something to write, and is not an ideal “friday puzzle”.

  2. vic Says:

    I established a way to verify whether a pair of number is the answer, and was able to verify the answer I found online to be correct. But I have no proof. Is there a proof? I really wanna know.

  3. simpple Says:

    There is a proof, and even variations of this problem. The simplest explanation is probably in wikipedia, under “Impossible Puzzle”. By googling “Sum-Product Problem Freudenthal Sprows” I also found some relevant links, which suggest that you can dig even deeper with mathematical papers from the 70s.

    • vic Says:

      Thanks! I’ll look it up sometime. I had to google for answer in order to stop thinking about it! It’s preventing me from finishing my homework! lol.
      I guess you’re the owner of the blog right? This blog is nice! Although it’s not updated recently, the contents are still very interesting. I’ll drop by again when I am free.

      Also here’s one problem that puzzled me for several years (since high school). I never figured out the answer.
      You can take a look! I think it’s not in your blog but maybe you’ve seen it. It probably won’t qualify as a Friday Puzzle either given its difficulty.
      (Yup.. it’s just a google search away, but I never figured it out “myself”)

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