Magic Numbers of Magic Squares
How To Work Out What Each Row Of A Magic Square Should Add To

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A magic square is a grid containing the numbers 1, 2, 3, and so on, where each row, column and diagonal add up to the same number. An example is shown below, you will see that each row, column and diagonal add up to 34. This number 34 is the "magic number" of the magic square.

magic square of order 4
Finding magic squares or solving magic square puzzles is much easier if you know the magic number. The good news is, once you know the size of the magic square you want, you can calculate the magic number without too much trouble. This is because of an amazing fact :

Why is this? Suppose you add up each row of a magic square. Each row will add up to the magic number. So if you add all the cells together, you have added up this magic number as many times as there are rows in the grid. Let's note this fact:

But there's another way to add all the cells together, since each number 1, 2, 3 and so on appears exactly once. The biggest number in the grid is the number of rows times the number of columns. Therefore,

A magic square has the same number of rows as columns, so the magic number depends only on the number of rows in the magic square. The following steps help you work it out :

For example, for the four by four square above (or for any four by four magic square), the magic number can be worked out as :

To save you some calculations, I've given below the magic numbers of a few different sizes of magic square :

Yours, Dr Mike!