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?

Paul knows the perimeter and since all possible combinations of that perimeter did not contain any unique areas, he also knew in the second clue that Peter would have no idea what the perimeter was. That narrowed the choices down to the follwoing 14 combinations

x y P A

2 9 22 18

3 8 22 24

4 7 22 28

5 6 22 30

2 15 34 30

3 14 34 42

4 13 34 52

5 12 34 60

6 11 34 66

7 10 34 70

8 9 34 72

2 21 46 42

3 20 46 60

4 19 46 76

5 18 46 90

6 17 46 102

7 16 46 112

8 15 46 120

9 14 46 126

10 13 46 130

11 12 46 132

knowing this, Peter still could not find the perimeter. So, there must have been more than one combination for his known area. That narrows it down to the following 6 combinations

x y area perimeter

5 6 30 22

2 15 30 34

3 14 42 34

2 21 42 46

5 12 60 34

3 20 60 46

Since Paul now knows the area, that means it has to be the only unique value left for his known perimeter, which must be 22.

Therefore the answer is a 5x6 rectangle.