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 Possible solution by Milind)

Man oh man, I made a big mistake. I still believe my logic is correct, but I made an error in my last steps – "From Peter’s second statement we can rule out any areas that are "unique" in this set of 79 rectangles. There are 28 such rectangles, leaving us with 51." I got that backwards. There were 51 "unique" areas in this set of 79, leaving us with 28 rectangles to consider. For some reason, I deleted the areas that had repeats (and then deleted the corresponding rectangles in my perimeter table) instead of keeping the repeats and deleting the unique ones.

Duh!

So Milind was correct, and the end of my solution should have read:

From Peter’s second statement we can rule out any areas that are "unique" in this set of 79 rectangles. There are 51 such rectangles, leaving us with 28.
Of the 4 ways to make a perimeter of 22, Peter’s statement rules out 3, leaving only 1.
Of the 7 ways to make a perimeter of 34, Peter’s statement only rules out 1.
Of the 10 ways to make a perimeter of 46, Peter’s statement only rules out 4.
Of the 12 ways to make a perimeter of 54, Peter’s statement only rules out 9.
Of the 13 ways to make a perimeter of 58, Peter’s statement only rules out 10.
Of the 16 ways to make a perimeter of 70, Peter’s statement only rules out 13.
Of the 17 ways to make a perimeter of 74, Peter’s statement only rules out 11.
Now Paul says he knows the area. The only way he could know that is if the perimeter was 22. The only rectangle that wasn’t eliminated by Peter’s second statement was 5 by 6. So he knows the area is 30.

And since Peter was keeping track of all this too, and now that Paul said he knew the area, Peter can figure out the perimeter of his rectangle: 22.

So the rectangle is 5 by 6.

Penny, I am still trying to make my brain work… I’ll get back to you if I see anything in your approach.

Posted by nikki on 2005-01-31 15:43:22