You may also like

problem icon

Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?

problem icon

3388

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

problem icon

Bull's Eye

What fractions of the largest circle are the two shaded regions?

Plutarch's Boxes

Age 11 to 14 Challenge Level:

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.