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?
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.
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
\[ AB = 2A + 2B \]
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.