"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?