Fascinating Magic Squares

I took up BS Mathematics in college because ever since I was in high school I was a competitor to different math competitions such as Philippine Math Olympiad, SIPNAYAN of Ateneo, and the most famous Mathematics Teachers Association of the Philippines or more commonly known as MTAP.

During my stay in high school, I was one of the chosen students to train in MTAP. Every Saturday, we go to a school in Cubao Quezon City to train. We have different modules to study. As the competition nears, we were pulled out from class to train more for the upcoming battle.

Unfortunately, the highest rank that I got was top 4. And the ranks that were chosen to represent the school in the competition were up to top 3 only. I did not qualify.

One of the things that fascinated me was the so-called Magic Squares. I did a self-study on this. Magic Squares are a set of numbers 1 to n squared, without repetition, that were placed in an n by n grid of square such that the sum of all the columns, rows, and diagonals were all the same and equivalent to a single number.

To illustrate this, please see the example below:
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3x3 Magic Square

In this example, n represents the number 3. The numbers 1 to 9 were placed in a 3-by-3 square grid without repetition. Notice that the sum of all the columns, rows and diagonals equate to only one single sum and that’s 15.

In my study, there’s a trick in creating n-by-n magic squares where n corresponds to an ODD number. This means you can create a magic square of 3-by-3, 5-by-5, 7-by-7 and so on. And the sum will always be equal to n(n squared + 1) divided by 2.

So in the case of the following magic squares:

· 3-by-3 magic square, the sum will always be 15.
· 5-by-5 magic square, the sum will always be 65.
· 7-by-7 magic square, the sum will always be 175.
· 9-by-9 magic square, the sum will always be 369.

And so on…

Take a look at the following few more examples:

1. 5-by-5 magic square: Unique Sum is 65.
2. 7-by-7 magic square: Unique Sum is 175.
3. 9-by-9 magic square: Unique Sum is 369.

Did you get the trick? I hope you are fascinated too.

You can take a look at my other blog for more tricks at: Math Olympian