Sunday, January 10, 2016

Freudenthal's Impossible Problem

An email from a friend arrived just as I was cramming myself into a middle seat for a cross-country flight:

I am thinking of two whole numbers greater than 1 whose sum is less than 100. I tell this to Jason and Brendan, and give Jason the product and Brendan the sum.

Jason: I don't know what the two numbers are.
Brendan: I knew Jason wouldn't know what the two numbers are.
Jason: Ah, now I know what the two numbers are.
Brendan: Ah, now I know what they are as well.

What are the two numbers? Assume Jason and Brendan are perfectly logical, honest, smart, etc.

Talk about good timing! The hours of the flight passed quickly as I scribbled figures in my notebook. (I had the answer by the time we landed, but it was not easy.)

This beautiful puzzle is justly famous; you can read about it online by searching 'Freudenthal's Impossible Problem.'


Evgeny Milyutin said...

Hi Jason!

I was surfing and ended up wandering into your blog -- lots of cool puzzles and this one kept me absorbed all of Sunday evening!

It reminds me of a task I was given in school as a 3rd-grader (which I solved after a couple days of trying!) - could be a good one for your Saturday school or to post in this blog. See it below:


E. said: I have 3 sons

J. asked: How old are they?

E. answered: The product of their ages is 36.

J.: I need more information

E.: The sum of their ages is equal to the number of your house

J.: Hmmm.. I need more information anyway

E.: The oldest one has blond hairs

After that J. understood how old are the sons of E.

How old are they?


Educationally yours,

Jason Zimba said...

Yes, now you mention it, that puzzle does resemble the Freudenthal problem! And, now you mention it, I remember hearing the puzzle at a family dinner once (somebody else's family). I did some googling and will post the result (or share it on Google Plus). Thanks for the comment.

Evgeny Milyutin said...

Thank you for your answer! You have such a math driven family dinners, reminds me my childhood :)

I'm working on some other some think-outside-the-box puzzles -- what's the best email to share it with you? (those are not public yet). My email is

I thought you would enjoy them and may be it's a good fit for your Saturday school!

Jason Zimba said...

ZimBlog email is