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# Plutarch's Boxes

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Age 11 to 14

Challenge Level

- Problem
- Student Solutions

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.