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Double Digit

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

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What are the missing numbers in the pyramids?

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A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.

Plutarch's Boxes

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.