### Pyramids

What are the missing numbers in the pyramids?

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.

# AP Rectangles

##### Stage: 3 Challenge Level:

An AP rectangle is one whose area is numerically equal to its perimeter. Michael of Madras College, St Andrews sent the following solution and one nearly the same came in from Prav and Sheli from the North London Collegiate School Maths Puzzle Club.

If the sides of the rectangle have lengths $A$ and $B$ then

$AB = 2A + 2B$

so

$A = \frac{2B}{B-2}$

If you are given the length of one side, say $B$, then for any value of $B$ greater than 2, you can always find an AP rectangle with one side the given length. As the sides of the rectangle cannot have negative length $B$ cannot be less than 2.