1 What is a magic square?

A n × n table with numbers on it. It’s called magic when the numbers in all the columns, all the rows and both diagonals add up to the same value. Obviously, all the numbers should be different from each other for this to be non-trivial.

2 Can I see an example?

In this case, all columns, rows and diagonals add up to 15, making this an example of a magic square of order 3

3 Get to the point!

Fine. This method will allow you to construct a magic square of any size as long as it’s odd.

1. Draw up your square

   ·	·	·	·	·
   ·	·	·	·	·
   ·	·	·	·	·
   ·	·	·	·	·
   ·	·	·	·	·

2. Put the number 1 in the middle square of the top row

   ·	·	1	·	·
   ·	·	·	·	·
   ·	·	·	·	·
   ·	·	·	·	·
   ·	·	·	·	·

3. Fill up the square by going up and to the right. If you ever go out of bounds, imagine that the square wraps on itself like the maps in old video games. For instance, the number 2 would go in this square

   ·	·	1	·	·
   ·	·	·	·	·
   ·	·	·	·	·
   ·	·	·	·	·
   ·	·	·	2	·

4. Keep going! If the “next” square is already filled, go down a square before continuing. For instance, after 5 steps the square looks like this:

   ·	·	1	·	·
   ·	5	·	·	·
   4	·	·	·	·
   ·	·	·	·	3
   ·	·	·	2	·

   Then you should write the number 6 like so:

   ·	·	1	·	·
   ·	5	·	·	·
   4	6	·	·	·
   ·	·	·	·	3
   ·	·	·	2	·

5. Keep going!

   17	24	1	8	15
   23	5	7	14	16
   4	6	13	20	22
   10	12	19	21	3
   11	18	25	2	9

6. ???

7. Profit!

4 So what?

The prospective recreative mathematician will notice that this procedure tends to create very bland magic squares. While true, this is the kind of interesting bits of mathematics that I encountered in my youth and as such I believe it’s a good diversion for the young’uns.

Moreover, this invites a short introduction to symmetry. After you—or a kid—have played with this for a bit, you could ask yourself:

• Why does this work?
• What would happen if I swapped columns around? Or rows?
• What if I start with another number?
• Can I use another direction?
• Can I use this to create an even-numbered square?
• Can I use only odd numbers? Or only even? Why?

Even though I was good at mathematics all throughout my education, I’ve never been a good mathematician in the traditional sense. My liking mathematics was of course aided by the fact that my brain is wired in a certain way, but the real appeal of it came in the form of games and curiosities like this one. It’s amazing just how much one can learn when removed from the dry academic view of learning how to compute.