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?

Since all of the sides are integers that are greater than one give a pair of opposite sides measures of 2 inches.  Now you have four inches or less to make up the remaining two sides.  Since each side is longer than one inch, the other two sides msut be 2 inches also.  So the dimensions of the rectangle are 2 x 2. 

If we were to let the first pair of sides be three inches long  then we would only have two inches of perimeter to work with, thus leaving the other pair of sides with length one and since all sides must be integers larger than one this is not possible.

Posted by casey on 2005-01-28 20:31:34