The Impossible Puzzle

Dr. Shankar’s upcoming talk in the Math Club reminded us of the wonderful problem which Martin Gardner called the Impossible Puzzle.  It seems to have been invented by the mathematician Hans Freudenthal in the 1960’s.   There are now several different versions out there.  We give the version by David Sprows (this and other variations can be found at this website) :

David Sprows,
Mathematics Magazine,
Volume 49, Number 2, March 1976, page 96:

Let x and y be two integers with 1 < x < y and x+y <= 100. Suppose Ms S is given the value of x+y and Mr P is given the value of x*y.

  1. Mr P says: “I don’t know the values of x and y.”
  2. Ms S replies: “I knew that you didn’t know the values.”
  3. Mr P responds: “Oh, then I do know the values of x and y.”
  4. Ms S exclaims: “Oh, then so do I.”

What are the values of x and y?