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 Answer + Solution by nikki)

Some thoughts on Nikki's solution. (Possibly, I have misunderstood the solution or missed something :) )

Possible sides for rectangle with area 52: 2 * 26, 4 * 13
Once Paul says he does not know the area, we have only possible perimeters as: 22, 34, 46, 54, 58, 70, and 74 

Now, when Peter says for the second time that he does not know the perimeter, he is wrong; in the sense that the only possible perimeter can be 34 from sides 4, 13 (We can rule out 2,26 because 2,26 gives perimeter as 56 - which is not present in the above list).

So, may be 4, 13 is not the answer....

Posted by Milind on 2005-01-31 09:14:31