If you have been following the opening of the windows in the Mathematical Awareness Month Poster, you might want to go back to window #1 and review Magic Squares. If you haven’t been there yet, please take a look at it. You will learn how to amaze your friends with your magical math abilities.
Magic squares come in many types, shapes, and sizes. Below you will see a magic square, a magic circle, and a magic star. If you would like to see hundreds more, you might want to check out The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions by Clifford Pickover.
Normal Magic Squares
This is a third-order normal magic square where all of the rows, columns, and diagonals add to 15.
Is this the only solution to this magic square? Can you find others?
You could also have a 4 x 4 square or a 5 x 5 square and so on. How big of a square can you solve?
Below you will see a magic circle composed of eight circles of four numbers each and the numbers on each circle all add to 18. The thing that makes this magic circle special is that each number is at the intersection of four circles but no other point is common to the same four circles.
The magic star below is one of the simplest. They can get extremely complicated and also quite beautiful.
So, where’s the math?
Well, you should have noticed already that there are numbers on this page. However, there is more to math than numbers. Let’s add at least one equation.
If we go back to the normal magic square you should know that all these magic squares have the same number of rows and columns, they are n2. The constant that is the same for every column, row, and diagonal is called the magic sum and we will call it M. Now we can figure out what that constant should be. If we use our 3 x 3 square above, we know that n = 3. If we plug our n into the given formula below we will find what our constant has to be.
Since our n = 3, the formula says M = [3 (32 + 1)]/2, which simplifies to 15. For normal magic squares of order n = 4, 5, and 6 the magic constants are, respectively: 34, 65, and 111. What would M be for n = 8? See if you can solve this square. (The figure for the normal square is from Wikipedia.)