#MathArtChallenge 95: Magic Squares

Overhead view of lego towers arranged in a 3×3 grid.

The Challenge: Create a magic square. Bonus points if you make it a physical thing.

Materials Needed: Legos, blocks or coins all work well for making these towers. Could also be pen/pencil & paper, of course.
Math Concepts: Algebra, arithmetic, counting, proportions/ratios, structure, sum of 1-n integers

Magic Squares go WAY back. I’d heard of them, but hadn’t really dug in until I started reading George Gheverghese Joseph’s The Crest of the Peacock. Here’s the gist:

  • Using a square grid, place the counting numbers (equal to the number of spaces in the grid) in a manner such that every row, column and diagonal sum to the same value.
    • So for a 3×3 grid, you want to place the numbers 1 through 9. The sum of each row, column and diagonal will be 15.
    • For a 4×4, place the numbers 1 through 16, the sums will be… ? (It’s more fun if you work it out.)

Part of my attraction to this is that Joseph shares Yang Hui’s description of how to form a 3×3 magic square, and it felt like a story that begged to be made with legos. So I did that:

Yang Hui’s work on magic squares was published in 1275, and even he said he was “passing on the works of earlier scholars and would make no claim to originality,” (Peacock, 209).

Spoiler alert…. there’s only one 3×3 square. I would love to lego up a 4×4, but I don’t have enough legos currently, but you’re all far more creative than I and I’m sure you can sort out a way to explore a bit with the 4×4 and the 5×5.

Magic squares have been around for a….while. There are examples of them in China back to 190 BCE and in India as early as 587 CE. The Chinese examples were connected to an emperor Yu (Joseph calls him “semimythical”) and are just beautiful.

Yu magic squares

Hui went so far as to create magic circles. Joseph notes that some of his explanations for how he derived the higher-number magic squares and circles were “either absent or cryptic to the point of obscurity”. I would love to see some of you illustrate his magic circle… but I’ll also be thrilled just to see some magic squares.


Things to ponder:

  • How many 4×4 squares are possible? What about 5×5? Can you create an algorithm for deriving them?
  • What impact does it have on the creation if there is a center square (like in a 3×3 or a 5×5 grid) and when there is not (like a 4×4 grid)?
  • How could you use my video (and, please, definitely legos or blocks that the students can play with themselves) to help students think about means/averages?

Depending on how you use this activity, you may engage with different mathematical standards. I’ve listed possible connected math content above. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

1.) Make sense of problems and persevere in solving them. Is there an algorithm you can create for where to place the 4×4 towers?

2.) Reason abstractly and quantitatively. How many pieces are needed for a 4×4, 5×5 etc? What about an nxn?

4.) Model with mathematics How can we connect the 3×3 model with the larger ones? With a circular grid?

Author: Ms. P

Math Teacher in Minneapolis, MN.

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