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This blog celebrates all things mathematical. Solutions to the problems posed here will eventually appear at

Mathematics Resolution

Hints will often appear in the comment section. Feel free to comment, but reserve solutions to the Mathematics Resolution blog comment section.

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My pen name (pronounced dow-groots) is an anagram of a famous mathematician and popularizer of paradoxes.

Wednesday, September 23, 2009

The Impossible Problem of Hans Freudenthal

Hans (Freudenthal) concocts a little game for his friends Sam and Patty. Choosing two numbers each greater than one but also less than 30, he secretly tells Sam the sum and he secretly tells Patty the product. Sam and Patty are not allowed to discuss the values that they are given and yet they are challenged to find the two original numbers.

You witness the following exchange:

Patty says, I can not find the numbers.

Sam then says, I knew that you could not find them.

Patty responds, Oh, then I know what the numbers must be.

Sam replies, Now I too know the numbers.
How is this possible? Of course, Patty and Sam have the advantage of knowing more than we do. Can you solve this problem with the little information given?

Start with the hint that the sum is less than 30.

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2 comments:

  1. Magnus DaugroutesSeptember 23, 2009 at 5:12 PM

    Since Patty says that she does not know the numbers, we may conclude that the two numbers can not both be primes.

    Sam's response means that she knew that the numbers could not both be primes.

    What else can we conclude from their words?

    ReplyDelete
  2. Magnus DaugroutesSeptember 30, 2009 at 8:17 AM

    A partial solution to this problem was posted at Mathematics Resolution on September 30, 2009.

    ReplyDelete

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