how to create odd numbered magic squares

And now for something completely different.

WordPress’s prompt today is: “Take a complicated subject you know more about than most people, and explain it to a friend who knows nothing about it at all.” I contemplated explaining how to make curry powder, become a US citizen, insert an invisible zipper, address an automated flat mail piece, or French braid.

But my inner geek won out.

You might not know this about me, but I’m a little bit of a math(s) nerd (a very little bit). I discovered recreational mathematics at an early age (five? six?) when my dad introduced me to magic squares.

A magic square is a grid in which all the columns, rows and both diagonals add up to the same number. You may have encountered them in primary/elementary school, where you had to fill in the blanks on incomplete magic squares.

In the 3-square grid above, for example, each column, row, and diagonal adds up to 15.

I think they’re cool, mainly because Dad taught me where to put each number, no trial and error required! This method works for any size magic square with an odd number of rows/columns.

There are three basic rules.
1. Start by putting “1” at the center of the top row.
2. For each successive number, try to place it in the square that is diagonally up and to the right (northeast) of the last number. If there is no square there, imagine that the entire grid is duplicated beside or above your actual grid, and place your next number in the corresponding square. (Stay with me, I’ll show you in a minute.)
3. If your target square is already occupied, place the number below the current number.

Here’s the above grid again, step by step.

“1” goes in the top row, center position. (Rule #1)
The square northeast does not exist, so we imagine that our grid is duplicated above, and see that “2” belongs in the bottom right corner. (Rule #2) We write it there on our grid.

With me so far?

Again, there is no square to the northeast of “2”, so we imagine a duplicated grid and see that “3” belongs at the far left. (Rule #2)

The square northeast of “3” is already occupied, so “4” goes below “3”. (Rule #3).

“5” and “6” require no shenanigans.

“7” wants to go in the bottom left corner, but that spot’s taken, so it goes under “6”.

And I’m sure you can, by now, figure out where “8” and “9” go, and why. Voila! A completed magic square, no addition required!

Here’s a 5-square version, where each column, row, and both diagonals adds up to 65. As a kid, I knocked myself out making larger and larger magic squares.

Fun times, right?

You’re welcome.

Leave a comment


  1. Lola

     /  October 25, 2012

    Is this how to do Sudoku?? I’ve always wanted to learn that!

    • Lola, no, Sudoku has different rules and goals. Sorry I can’t help you with that!

    • In some ways, suduko is similar but requires no maths, but follows a pattern. Every row and column should have the numbers one to nine. Also the nine “large boxes” (indicated by the bold line) should also contain the numbers one to nine. I hope that makes sense, it is a bit addictive once you learn it though! :-)

      • Lola

         /  October 26, 2012

        That is exciting! (The no math part) How do you decide which number goes where?

    • Lola I think solving Sudoku is just a matter of trial and error

  2. Etol Bagam

     /  October 25, 2012

    Very cool!


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