### Counting Counters

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

### Cuisenaire Rods

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

### Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

# Magic Constants

## Magic Constants

This is a 4 x 4 Magic Square made from the numbers 1 to 16.

In a Magic Square all the rows, columns and diagonals add to the same number. This number is called the 'Magic Constant'.

1/What is the Magic Constant of this Magic Square?

This particular square is especially 'magic' as some 2 x 2 squares within it also add to that number.

2/How many of these squares can you find?

3/What happens to the Magic Constant if you add 2 to each number in the square?

4/What happens if you double each number?

5/Can you make a square in which the Magic Constant is 17?

How did you do it?

6/Can you make a square in which the Magic Constant is 38?

How did you do it?

7/What other numbers under 100 can you make into the Magic Constant by changing all the numbers in the square in the same way?

8/Can some be made in more than one way?

9/Are there some numbers you really cannot make?

### Why do this problem?

This investigation gives a lot of opportunities for a wide range of learners to increase both their spatial and number awareness.

### Possible approach

It might be neccesary for the pupils to be introduced to simpler, more common, magic squares.

If there is a problem in identifying the 2 x 2 little squares within the 4 x 4 square the first one in the bottom left hand corner could be selected.

### Key questions

Can you tell me about the way you are doing this?
What have you decided to do to the first set of numbers?

### Possible extension

Questions 7, 8 & 9 act as a good set of extension activities, further ones could be suggested by the pupils. Then magic squares of a different size could be explored.

### Possible support

Some pupils will find it useful to have small square cards with the numbers on a a prepared grid to place them on.