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(2): solution is right but logic might be flawed by Penny)

Although 1x4 is disallowed from the statement "I am thinking of a rectangle with integer sides, each of which are greater than one inch", I don't think 4x4 should have been eliminated, for this reason:

Peter and Paul would both have 16.

Paul knew that perimeter 16 corresponds to areas 12, 15 and 16. He would know the area is not 15, because that only corresponds to perimeter 16, and Peter would have spoken up. So Paul had to assume initially that the area was either 12 or 16. In either case he knew that Peter would not know the perimeter initially.

Peter knew that area 16 corresponds to perimeters 16 and 20. Perimeter 16 has been discussed already. Perimeter 20 corresponds to areas 16, 21, 24 and 25.  He would know the perimeter couldn't be 21 or 25, because these only correspond to area 16, and Paul would have spoken up. So Peter knew the perimeter was either 16 or 24.   

 

Edited on February 1, 2005, 8:59 am

Posted by Penny on 2005-02-01 08:17:29