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.

Posted by Pat Whitaker on 2005-04-04 18:43:34