### Double Digit

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.

### Quick Times

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

# Pyramids

##### Stage: 3 Challenge Level:

When completing a pyramid, each number above the base is made up from the sum of the two supporting it.

(i)

What are the missing numbers?

(ii)

What is the value of a?

(iii)

Find the value of b for which the pyramid can be filled with whole numbers in this pattern. With this value of b find some different ways of doing so.

(iv)

Find eight different pairs of positive values of x and y . How many solutions are there altogether?

Can you find a relationship between x and y ?