Magic Constants

'Magic Constants' printed from https://nrich.maths.org/

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Magic Constants

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

Magic square : 15, 10, 3, 6; 4, 5, 16, 9; 14, 11, 2, 7; 1, 8, 13, 12.

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

Here are some questions about this Magic Square.

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.