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:

Area Perimeter

30 22

30 34

42 34

60 34

70 34

72 34

42 46

60 46

102 46

120 46

126 46

132 46

72 54

126 54

180 54

120 58

180 58

210 58

70 74

102 74

132 74

210 74

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.