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 Rectangular Logic (Posted on 2005-01-28)
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?

 No Solution Yet Submitted by Sam Rating: 3.7500 (12 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re(2): Correction to Addendum to Another solution | Comment 33 of 44 |
(In reply to re: Addendum to Another solution by Pat Whitaker)

I mistakenly included P = 62 instead of P = 58 in my complete list. The correct list is:

Area    Perimeter

30      22
30      34
42      34
60      34
70      34
72      34
42      46
60      46
102      46
120      46
126      46
132      46
72      54
126      54
180      54
120      58
180      58
210      58
70      74
102      74
132      74
210      74

When generating the list of allowable perimeters (not generated by the sum of two primes) it is only necessary to consider the odd numbers (Goldbach's conjecture - verified way beyond 40).

Clearly the only candidates are those odds which are not of the form p + 2, p an odd prime. This leaves only the first prime of the twin primes  between 11 and 39, and the numbers 23 and 27.

Sorry for being so prolix.

 Posted by Pat Whitaker on 2005-04-04 18:43:34
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