## 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.”

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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September 6, 2009 at 10:23 pm

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

May 30, 2013 at 7:35 am

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.

May 30, 2013 at 8:12 am

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.

May 30, 2013 at 8:25 am

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.

http://www.mathsisfun.com/pool_balls.html

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”)

May 30, 2013 at 8:31 am

Thank you for your suggestion.

We discussed the same puzzle, using coins instead of balls:

– https://fridaypuzzle.wordpress.com/2007/12/15/the-fake-coin/

That puzzle was a quite nice one, and easier to discuss with a group since it does not really require pen and paper.