Magic Squares
In a Magic Square the rows, columns and diagonals all add up to the same number.
Image credit: User:Phidauex, Public domain, via Wikimedia Commons
People have been interested in Magic Squares for thousands of years. The ancient Chinese Lo-Shu Square is the world’s oldest example of a Magic Square and it is around 4000 years old.
Lo-Shu Square
Image credit: Wikipedia - Magic square
Things to do with Magic Squares
Learning about Magic Squares is like opening a Pandora’s Box. Once you discover one thing, you realise that there are five more things to find out. This is why we have collected together a list of 8 ideas to help get you started.
1. Make your own Odd Order Magic Square
Siamese (or De la Loubère) Method
The easiest way to make your own Odd Order Magic Square is to use the Siamese Method. Follow these simple steps.
Order 3 Magic Square, Siamese Method
Put a number 1 in the central box in the first row
Then move one up and one to the right for your next number.
If you leave the square, imagine that the square ‘wraps round’.
If you hit a box which already has a number in it, then move down one box instead.
Image credit: Wikipedia - Siamese method
Image Credit:
Victuallers,
CC BY-SA 3.0, via Wikimedia Commons
You might find it easier to watch this Video showing the construction of the Order 3 Magic Square using the Siamese Method.
Siamese Method, Arithmetic Sequences
Did you know? You can start with a different starting number, other than 1. You can use any arithmetic sequence to construct a Magic Square using the Siamese method. For example, you could use 3,6,9,12,15,18,21,24,27,30,33,36.
Challenge
Can your pupils construct an Order 5 Magic Square using the Siamese Method?
2. Make a Magic Birthday Age Square
You can construct a Magic Square with any total greater than 21 by following the steps in this Numberphile Video. Perfect as a Magic Square Birthday Gift for anyone older than 21:
3. Make a Magic Birthday Date Square
It is also possible to construct a Magic Square which has a specific date dd/mm/yy on the top row. This is fairly complex, so would be more suitable for older pupils. Here are two different explanations of how to do this:
Nrich Article: Magic Squares for Special Occasions
Video: How to make a Birthday Magic Square (with a specific date in mind).
4. Transform Your Magic Squares
How can you transform a Magic Square so that it remains magic? There are a number of transformations which will preserve the Magic Property such as:
- Adding the same number to every entry in the Magic Square
- Exchanging two rows which are equidistant from the centre
- In Even Order Squares you can exchange the two diagonally opposite quadrants.
Read about other Transformations which preserve the Magic Property in this Nrich article on Magic Squares.
5. Make a Magic Square using a Knights Tour
Photo by Piotr Makowski on Unsplash
This video from Maths YouTuber Ayliean shows how some Magic Squares can be constructed using the moves a Knight makes in the game of chess. If pupils want to try this out for themselves then they should first try it out on an Order 5 grid.
6. Magic Square of Squares
In 1770 mathematician Euler announced a 4 x 4 Magic Square consisting just of square numbers.
Image credit: Math Garden CC BY-NC-SA 3.0
It is an unsolved problem whether it is possible to construct a 3 X 3 Magic Square of Squares. There is a website offering up to 8000 Euros and 12 bottles of champagne for solving this problem and other similar problems. Read more about the Prize Problem here. Introduce your pupils to the idea that Mathematics is not finished, new things are being discovered all the time. Spoiler Alert: Mathematicians think it’s not possible to make a 3 X 3 Magic Square of Squares, but they can’t Prove it and without a Proof we can’t be fully sure!
Magic Squares of Squares (are PROBABLY impossible) – Numberphile
7. Magic Squares Investigation
Photo by K8 on Unsplash
If you want to set an open-ended homework project or investigation, choose Magic Squares. They are simple to understand but offer depth and complexity for any pupils who want to go further. Pupils could be asked to prepare a poster, presentation or PowerPoint to summarise their findings. They could build their own Magic Square and explain what they like about it.
8. Using Magic Squares as Puzzles
Magic Squares can also be used as fun puzzles where pupils work to fill in the missing values. Magic Squares can be used to practise other skills such as adding negative numbers or collecting like terms.
Algebraic Magic Squares, Collecting Like Terms
Negative Numbers Magic Squares
Further Magic Squares Resources
Numberphile Magic Squares Video Collection
Plus Maths Article: Anything but square: from magic squares to Sudoku
Royal Institution Masterclass on Magic Squares
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