Copyright © University of Cambridge. All rights reserved.

## 'Plutarch's Boxes' printed from http://nrich.maths.org/

According to Plutarch, the Greeks found all the rectangles with
integer sides, whose areas are equal to their perimeters. Can you
find them?

What rectangular boxes, with integer sides, have their surface
areas equal to their volumes? One example is $4$ by $6$ by
$12$.

How to do this? No doubt different people will suggest different
methods. Suppose the dimensions of the box are $a$, $b$ and $c$
units where $a \leq b \leq c$ . You might like to show that the
problem amounts to solving the equation$1 = 2/a + 2/ b + 2/c$ and
then show $3 \leq a\leq 6 , 3 \leq b \leq 12 , 3 \leq c \leq
144$.

Knowing how far to go in the search, it is then easy to write a
short program to find all possible boxes. You could use a
spreadsheet. You could just go through all possible cases
systematically as people would have done before the days of
computers.