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 re: solution is right but logic might be flawed by Charlie)

Charlie wrote: "How did you eliminate 2x3 in the first step?  The same perimeter of ten can be achieved by this (with area 6) or 1x4 (with area 4). Both area 6 and area 4 are non-unique (6=3x2=6x1; 4=2x2=4x1)."

1x4 is disallowed from the statement "I am thinking of a rectangle with integer sides, each of which are greater than one inch.".

Therefore 2x3 would correspond to a unique area and perimeter, and both Peter and Paul would have known the other's dimension immediately. 

Posted by Penny on 2005-02-01 07:40:36