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

What rectangular boxes, with integer sides, have their surface
areas equal to their volumes? One example is $4$ by $6$ by $12$.
There are $10$ solutions. Can you find them all?

Suppose the dimensions of the box are $a$, $b$ and $c$ units
where $a \le b \le c$, then the volume of the box is $abc$ and the
surface area is $2(ab+ bc + ca)$. If these are equal to each other
you can divide the expression you get by abc to give:

\[ 1 = \frac{2}{a} + \frac{2}{b} + \frac{2}{c} \]

Now you have $3$ positive numbers (fractions) adding up to $1$
and there are only a few possible ways this can happen. None of the
fractions can be very small or very big. You need to show that

\[ 3 \le a \le 6 , 3 \le b \le 12 , 3 \le c \le 144. \]

This limits the number of possibilities. 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.

According to Plutarch, the Greeks found all the rectangles with
integer sides, whose areas are equal to their perimeters. Can you
find them? You can use the technique described above in this
simpler case.