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?

Continuing from Nikkis solution...

From Peter’s second statement we can rule out any areas that are "unique" in this set of 79 rectangles. ..

Of the 4 ways to make a perimeter of 22, Peter’s statement  rules out 3 and leaves 1.
Of the 7 ways to make a perimeter of 34, Peter’s statement rules out 1, leaving 6.

so on and so forth.

Now Paul says he knows the area. The only way he could know that is if the perimeter was 22. [sides: 5,6]

Following up on this, Peter will understand that the perimeter is 22. (Perimeter cannot be 34[sides: 2,15] since Paul could figure out the area; If perimeter was 34, Paul could not have figured out the area)

Hence, the sides are 5, 6.

Am I right or am I going bonkers :) ...this late in the night ?!!

Posted by Milind on 2005-01-31 10:02:18