What is a Magic Square?

A magic square is a square grid (normally 4×4) with numbers in each cell. The numbers in each row, column, and diagonal all add up to the same number. Normally in magic squares, the numbers in the cells are all different, and are the lowest numbers other than 0. Example: In a 4×4 grid, the numbers would be 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16.

How to Construct a Magic Square

To start, create a grid that has the letters a, b, c, and d in every row, column, and diagonal. An example is shown below.

a c d b
b d c a
c a b d
d b a c

Now, you need to make another version of the magic square, only on a 90 degree angle.

a c d b  D  C  B  A
b d c a  B  A  D  C
c a b d  A  B  C  D
d b a c  C  D  A  B

Next, you need to combine them into one grid that has each cell as an addition sentence

a+D c+C d+B b+A
b+B d+A c+D a+C
c+A a+B b+C d+D
d+C b+D a+A c+B

You notice that each addition sentence only appears once. Example: there is only one addition sentence that is a+A. This is what makes every number different.

Our next step is to make each variable equal a value.

Assign the variables a, b, c, and d, the values 1, 2, 3, and 4. (It doesn’t matter which is which.)

Assign the variables A, B, C, and D, the values 0, 4, 8, and 12. (It doesn’t matter which is which.)

Finally, add up your equations.

13 10 8 2
6 4 15 9
3 5 10 16
12 14 1 7

There is your magic square. In this magic square, (or any magic square you make using these instructions,) the rows, columns, and diagonals all add up to 34. This number is called the magic sum.

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