Promising them an increase in their allowance if they get the answer, I offer my two sons, Peter and Paul, the following puzzler:

"I am thinking of a rectangle with integer sides, each of which are greater than one inch. The total perimeter of the rectangle is no greater than eighty inches."

I then whisper the total area to Peter and the total perimeter to Paul. Neither of them are allowed to tell the other what they heard: their job is to work out the rectangle's dimensions.

Their subsequent conversation goes like this:

Peter: Hmmm... I have no idea what the perimeter is.
Paul: I knew you were going to say that. However, I don't know what the area is.
Peter: Still no clue as to the perimeter...
Paul: But now I know what the area is!
Peter: And I know what the perimeter is!

What are the dimensions of the rectangle?

(In reply to Another solution by Pat Whitaker)

Paul knew that the perimeter was not of the form 2 * (p + q) and thus was to reply to Peter that he already knew Peter couldn't determine the perimeter immediately.

I incorrectly stated that the submitted duplicate are list  was complete. Here is the complete list of duplicated areas from the allowable perimeters. 22 is the only perimeter with a unique duplicated area:

Area  Perimeter
  30      22
  30      34
  42      34
  60      34
  70      34
  72      34
  42      46
  60      46
102      46
126      46
130      46
132      46
  72      54
126      54
130      62
210      62
  70      74
102      74
132      74
210      74

Posted by Pat Whitaker on 2005-04-04 18:12:15