Magic Squares and Gauss’s counting formula

In the past couple weeks, Katherine and I have launched no less than 5 math circles, with at least two more on the way–plus a physics circle starting in February or March! Some of these we co-teach; some we handle solo. The age range of the circles is K-1 at the youngest to 8-11 grade at the highest. Needless to say, we’ve been busy.

At the same time, it’s been exciting, and so much happens in each, with so many breakthroughs on behalf of the children, that I’m already kicking myself for not recording every moment here. So here’s a fun lesson that I just did with a group of 2-3 graders. I would say that all but one or two of them loved it. For me, it created a nice connection between magic squares and motivating a classic question (and a great one in its own right): how do you add all the numbers from 1 to n?

(Legend has it that Gauss solved this as an 8 year old, quickly summing the numbers from 1 to 100 after a bullying teacher assigned the question to keep his class busy. How did he do it?)

But back to magic squares.

Magic Squares

A magic square has all rows and columns (and the diagonals, depending on how you want to play it) adding up to the same number. In the squares above, that number is 15, a fact I related to the kids. But I also asked if they could add up all nine numbers in the square.

“We can just do it, right?” one student asked me. “Well, you can choose how you want to add.” One student’s eyes lit up with an idea. “We can just add the rows!” he said. “15+15+15!” And thus, every number is accounted for, and 1+2+…+9=45.

But it gets better. On the back was the next level up.

Magic Squares

As kids went to work, they ran into a problem: they didn’t know what number the rows and columns were supposed to add up to! As I circulated, I told some kids the magic number. But for those seemed to be jones-ing for extra challenge, I made the following suggestion: if you knew what all 16 numbers added up to, and took into account that that each row adds to the same amount, you have a recipe for how to find the magic number.

But how do you add up the numbers from 1 to 16? Well what if you add the 16 and the 1? It’s 17. 15+2? 17. 14+3? 17. Once you notice you get all 17s, you just need to figure out how many equations you have (there are 8). So the numbers add up to 17×8. But the four rows must add up to 17×8, then each row must be a quarter of that amount, or 17×2… =34 (notice it’s easier not to bother doing the multiplication when you’re about to divide).

The kids who undertook this were very proud of their result. Some went on to the 5 by 5 magic squares, and figured out the magic number for those too, in the same way. Pretty impressive for 2nd-3rd graders. It’s a cool scene when a kid who hasn’t mastered the algorithms for multiplying two digit numbers is able to find the product 12.5 times 26 (which he needed because 1+25=26, 2+24=26, etc., and there are 12.5 groups adding to 26 [a weird point, but actually true]; put in the fact about the rows, and he needed to find 12.5 times 26 divided by 5. The following week he did this calculation for a 16 by 16 magic square, and successfully figured out that each row must add to 2056.)

In the last five minutes I pulled the kids together for a game, but they didn’t want to leave the squares. Still, I like to send them home with something to play. Here it is: 15 tic-tac-toe. You play on a tic-tac-toe board, except you fill the squares in with the numbers from one to nine, and no number can be used more than once. First person to write three numbers in a row that sum to fifteen wins.

Here’s what I liked about this lesson:

  1. It was fun. Almost all the kids were pulled in and engaged.
  2. It was challenging, and there were different points of entry. A kid could work on filling in a 3 by 3 magic square, or deal with some tricky computations, or consider much bigger squares, or develop a system for adding lots of numbers together. One student asked me this very deep question: how many magic squares can you write in an n by n square. Beautiful question.
  3. It connected different mathematical ideas and techniques, and in this way it was an organic problem. From the question “how do we make magic squares,” it turns out we need to know a fair amount about square numbers, large sums (and how to group them as products), and logic. In other words, there’s more going on here than meets the eye. I’ve always loved the question “how do you add up the numbers from 1 to 100,” but having to reckon with that question because you need to figure out what the rows in a 10 by 10 magic square must add up to is pretty cool. The motivation spreads from one idea to the next. This is part of what makes math teaching interesting to me: you’re not always “ready” for a type of problem when you get to it, and the fact that you don’t already know how to do it isn’t an impediment, necessarily… it can be an incentive to learn.

Next time, we’ll dip into some topology. If I’m really slick, maybe I can combine the two, and do magic squares on donuts. We’ll see where it goes.

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3 Responses to Magic Squares and Gauss’s counting formula

  1. Stephanie O says:

    I love these activities – I think I’ll try it with my (homeschooling) 2nd grade age son tomorrow.

    Do you have suggestions for where to find activities like these? I keep thinking I should start a math circle for my son, but haven’t gotten to it yet. Do your circles meet every week?

    If you put enough activities for 10-12 weeks of a math circle into an e-book, I bet it would sell. (I’d buy one.)

    • Dan says:

      Thanks, Stephanie–an e-book is a cool idea. I’d love to get some of these ideas and lessons out there in an organized form.

      My circles do meet every week. I generally have ideas about what would make an interesting lesson for a certain group of kids, but I also spend a lot of time flipping through old books of games and puzzles, Martin Gardner books, and various online resources. I cut and pasted some of the magic squares from http://www.dr-mikes-math-games-for-kids.com/index.html and then added my own questions; a lot of great activities on that site.

  2. Breedeen says:

    “One student asked me this very deep question: how many magic squares can you write in an n by n square.”

    *Heartfelt sigh*

    I love it when kids ask amazing questions. I enjoy it even more when I don’t know the answer.

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