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?

Because Peter cannot determine the perimeter based on the area, we know that there must be multiple sets of dimensions that produce the area he was given.  In other words, any area with only two prime factors is ruled out.  So I constructed a multiplication table of prime numbers 2-37.  This yielded several areas.

Paul knew that Peter wouldn't be able to guess the perimeter, so his perimeter must not have been a perimeter corresponding to one of the areas found above.  When all the possible dimensions that yield perimeters corresponding to areas ruled out by Peter's statement have been generated, we know that all those dimensions are wrong.

At this point I opened up excel and created a multiplication table from 1-36.  I then colored all the areas eliminated by step one red, and all the additional areas eliminated by step two blue.   (you will see a diagonal pattern reinforcing this if you are doing it correctly). 
The next step is to find the areas remaining.  Because Peter is still unable to figure out the perimeter, highlight in green any non-duplicated areas remaining.  The only way Peter would still not know the area is if there were duplicates remaining, and there are.
Finally, find the perimeters corresponding to any areas remaining.  At this point Paul knows what the area is, so there must be one and only one unique perimeter, and there is, 22.  Thus the dimensions are 5x6.

Posted by sam on 2005-02-02 07:00:52