Magic Square

When adding numbers, it does not matter what order you add in.

Materials Needed

A piece of paper with a 4x4 grid drawn on it and a pen.

Demonstrations

Start with a 4x4 grid, with the numbers from one to sixteen in it.

Choose one number, then cross out the other numbers in the same row and column. (e.g. if you choose 7, cross out 5, 6, 8, 3, 11 and 15). Repeat until you only have four numbers left. If you add these numbers together you will get 34.

Further Information

This sort of magic square can be set up for any magic number. For example, let us set up a magic square that has the magic number 42.

Start by choosing eight numbers that add to 42. Let's use 3, 5, 6, 8, 4, 1, 9 and 6.

Now draw up a five by five grid. Leave the top right hand corner empty and write four of the numbers in the remaining squares in the top row. Write the remaining four numbers in the first column, leaving the top right corner empty again.

Next fill in the grid as an addition table, as shown.

Now remove the top row and left hand column.

Now chose the numbers the same way as before and they will total 42.

It works because the method of choosing the numbers guarantees that you will select one number in each row and one in each column. For example, you might choose 10, 4, 17 and 11 (one from each row). This is same as (6+4), (1+3), (9+8) and (6+5). So from the original set of eight numbers, each number occurs once. So when we add them we get:

10+4+17+11=(6+4)+(1+3)+(9+8)+(6+5)=42

To produce the magic square with numbers from 1 to 16, use the numbers 1,2,3 and 4 on the top row and 0,4,8,12 in the left hand column.

Further Activities

Make a magic square with your age, height, belt size etc. as the magic number.

Make a larger or smaller magic square.

Does the magic square work if you use multiplication instead of addition?

Can you make two magic squares with the same magic number?

Can you use two different sets of numbers to make the same magic square?