A while ago I pointed out that while I’ve never found creating a simple 3×3 magic square very useful in the classroom there are plenty of spinoff activities that really do work well. I devised Polo Squares when I was giving a session for 300 children at the Royal Institution in the middle of London. The children were from a dozen or more schools and I wanted to offer something that was easily accessible and could be instantly usable no matter how big the groups were or whether they arrived early or with just five minutes to spare.
Every child had a set of ten digit cards, from 0 to 9. I asked them to make a hollow square using eight of the cards, so the cards on each side of the square add up to the same total. I provided lots of blank record sheets; clearly a lot of work was done at home after the session, because the postman was delivering solutions for weeks afterwards.
I’ve no idea how many different Polo Squares there are, but I know it’s several dozen, so no-one’s going to say “I’m finished, what do I do next?” Each solution uses eight of the ten cards, so there’s an element of Trial and Improvement.
And there are plenty of challenges – “What’s the highest / lowest Polo total?”, “Can you make all totals in between?”, “Which totals have the highest number of solutions?”, …..
Polo Squares offer a nice challenge – even wider than I’d realised; George was just five, and the younger brother of someone who’d attended the session.
Using digit cards helps the activity and also makes it feel like fun. The cards invite the use of some problem-solving skills (look at L & A’s) , and give some meaty mental arithmetic practice.
And in just the same way I could use them with children waiting for my session to start, teachers can have some digit cards and answer blanks available on parents’ evenings!
Ladies and Gentlemen: Roll up, Roll up, because I’m going to give you, at positively no charge, my all-time winner of the “Sir’s busy and must not be interrupted for any reason whatsoever” award. You can use it with virtually any group and every child will get lots of lovely arithmetic practice; there’s pattern and symmetry, exploration, hypothesis, searching for completeness, and no doubt more bedsides.
The previous post gives you a 4×4 magic square, so we know that each of the four rows, the four columns, and the two diagonals all add up to the magic number. For example:
But what we often fail to notice is that these aren’t the only sets of four numbers that add up to 34. Try, for example, the numbers in the corners. Then look at some of these configurations (and won’t each one suggest some more arrangements to try?).
How many patterns do you think there are? (I give children a sheet with 35 grids – but that may or may not give you a clue.) This is, incidentally, another wonderful activity to have available for parents to try on an Open Evening. Give them plenty of grids, pens, a large expanse of wall to fill, and let them get on with it.
Rather surprisingly I find 4×4 magic squares offer more opportunity in the classroom than 3×3.
To start with, though I can never remember the details for a 3×3 square, it’s easy to create a 4×4 square in just a few moments.
Start by writing the numbers 1 to 16 into a 4×4 square in the natural way.
Then exchange the numbers in the two yellow positions, and do the same with the numbers in the other two corners.
You now have this arrangement.
Now exchange the two numbers in the two red positions, and do the same with the other two numbers in the central square.
That’s it! You now have a 4×4 magic square in which every row, every column, and the two diagonals add up to 34.
I’ve never found much success asking children to use the numbers 1 to 9 to create a magic square in which each of the rows, columns, and the two diagonals totals the same number.
True, you can give them a nudge by telling them the central number has to be 5 and / or that the magic total is 15, but I’m rather against telling children anything in what’s supposed to be an exploration. In any case, for most children it’s not helpful enough, and failed attempts don’t give much helpful feedback that enables you to do better next time – instead of the “Trial and Improvement” we like to see, it’s Trial and Error and then Trial and Error again.
Consequently, for most children it’s a frustrating exercise in failure – they’ve been given a challenge and failed it, perhaps after putting a lot of time and effort into it. So it’s not a very rewarding task at all.
That’s not the only disadvantage. What does the teacher do when one or two children do succeed? Apart from rotations and reflections there’s only the one solution (green cards), so you can’t ask them to find any more, and if finding the 3×3 magic square presents plenty of problems, you can be sure finding a 4×4 magic square can definitely be crossed off the list of potential extensions.
So how do you get round all these problems? Easy, dead easy. Simply reverse the task, and ask children to find an Anti-Magic Square, where the rows, columns and diagonals all add up to different totals (example: yellow cards). Whatever array you choose is bound to give three different totals or more, so children are getting some success immediately, and the task then becomes to build on success rather than failure and to do better than your last attempt.
There are further advantages. There’s more than one solution (no, I’ve no idea how many), and if anyone really wants to, finding a 4×4 Anti-Magic Square is a worthwhile challenge.
PS: one final benefit: have some digit cards available on Open Evenings, and let parents have a go at creating an Anti-Magic Square!