### 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:

The problem was essentially a simple one using addition and algebra.

The solutions are:

(i)

(ii)

 This gave a = 19 + 33 = 52

(iii)

 All the solutions to this came from the relationship x + y = 2. The value of b was 10.

N.B. Allowing negative integers here, there are infinitely many possible values for x and y, but only one possible value of b .

(iv)

 The relationship for x and y is given as: x + 3 y = 27.

By choosing whole number values for y , this relationship will always give whole number values for x . Again here, allowing for negative integers , there are infinitely many possible values for x and y .