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: Addendum to Another solution
by Pat Whitaker)
I mistakenly included P = 62 instead of P = 58 in my complete list. The correct list is:
When generating the list of allowable perimeters (not generated by the sum of two primes) it is only necessary to consider the odd numbers (Goldbach's conjecture - verified way beyond 40).
Clearly the only candidates are those odds which are not of the form p + 2, p an odd prime. This leaves only the first prime of the twin primes between 11 and 39, and the numbers 23 and 27.
Sorry for being so prolix.