Magic Square

Players: 1
Ages: 8 and up
Cost: Free!
Math Ideas: Addition, deductive reasoning
Questions to Ask
    What do all the rows have to add to?

I really, really dislike state testing. The kids get stressed, the teachers get practically crazed, and in the end I have a great deal of skepticism about the validity of our particular tests.

The silver lining of state tests is that my principal asks that we decrease the workload on that week so that kids are fresh and ready on each testing day. I take that as an excuse to pull out some of my favorite math puzzles and problems, in order to keep my students' brains working without overwhelming them with new math material. 

One of my favorite puzzles is the Magic Square.

How to Play


Like most of my favorite math games and activities, the rules can be summed up in a sentence or two. Take a 3x3 box like the one at right and fill it with the digits 1-9, using each digit only once. The Magic Square is complete when all rows, all columns, and both diagonals add up to the same number.

That's it! My 8th graders love this puzzle because they can easily start solving it anywhere, as long as they have paper and a pencil. Every time I've brought out the Magi Square, I've had a student return from lunch or pop in my room the next morning to show me the scrap of paper where they finally solved it.

For that reason, this is a great activity to give your elementary-aged kids when you are sitting at a restaurant waiting for your food, or on a long car trip. They can grapple with the problem as long as they want, easily stepping away and coming back to the square later.

By the way, if your kids manage to solve the 3x3 Magic Square, they can try the 4x4 Magic Square. The guidelines are the same, except that you must use the numbers from 1-16 to make all the rows, columns, and both diagonals add to the same number.

Where's the Math?

This game is clearly a great way to practice addition, but that's not the most significant mathematical structure of the puzzle.

More importantly, the Magic Square gets kids thinking about how to effectively narrow down their options. After all, there are a ton of ways you can arrange these numbers, and only a few of them complete the Magic Square.


So your child might get two of the rows to add to 18, but then the last row only adds to 9. So what number should all the rows add to, anyway? Is there some way to determine that number for sure? 

Once your child has that number in hand, they can start playing with arrangements of numbers that add to the magic sum (and no, I'm not telling you what it is! Play around with it yourself!)

Questions to Ask

As always with solitaire games, giving your child space to explore on her own is important. She might not initially engage with the problem for very long. That's fine! Just set it aside and bring it up the next time you're bored while waiting for something: "Hey, did you ever figure out that Magic Square?"

If your child is frustrated, you can guide them toward finding that magic sum that all the rows and columns must add to. "Have you figured out which number everything has to add to? How could you figure that out?" Even if your child isn't able to solve the full Magic Square, simply making that breakthrough could give her a sense of accomplishment.


Once she's solved the 3x3 Magic Square, you can ask my favorite question for this activity: "Can you make another Magic Square? Is so, how many can you make?" There are a bunch of other magic squares that you can build by rotating and reflecting the original Magic Square solution. This could lead to a great discussion of whether these are all distinct answers, or whether they are all just different versions of the "same" solution. Which is correct? I honestly don't know, but that's not the point. The point is the discussion itself.

And of course, if your kid is a whiz at the magic square, then maybe they could try the magic hexagon...

Coming up from Games for Young Minds

In my next newsletter, I'll share a classic game of deduction and logic that I can't wait to unpack in my own math class next year.