Magic Squares

by Jillian Gregory

If you are a puzzle and math lover, then you will want to try your hand at a "Magic Square". It is a game that requires your basic arithmetic, logic, and planning skills to master. You can find several resources online about Magic Squares, but you only need a pencil, paper, and a sharp mind to get started.

What is a Magic Square? A Magic Square is an n x n matrix with each cell containing a number from 1 to n². You need to figure out where to place each number in the cells so that the sum of the vertical columns, horizontal rows, and main diagonal cells is the same. You can start out with a 3 x 3 matrix and build in complexity by working towards a 4 x 4 matrix and so on.

For example, let's take a look at a simple 3 x 3 matrix. On a piece of paper construct a matrix that has 3 columns and 3 rows. Next, we will need to figure out where to place the numbers from 1 to n² or 1 to 3² = 1 to 9 in this case. Trial and error is the common first method to employ when solving this puzzle. Verify that the sum of each vertical column, horizontal row, and main diagonal is the same. The main diagonal means the two diagonals that go through the corners of the matrix.

To understand how the Magic Square can be constructed I have provided an example solution. There is more than one solution to each Magic Square with regards to the arrangement of the numbers. The diagram shows a 3 x 3 matrix and solution.

In the example, the sum of each vertical column, horizontal row, and main diagonal was 15. The arrangement of these numbers can vary. To begin understanding Magic Squares, work on a 3 x 3 matrix and see how many solutions you can find.

Once you have mastered a 3 x 3 matrix, set your sights on a 4 x 4 matrix. For a 4 x 4 matrix, you will place the numbers 1 to 16 in the cells. Test your skills by working on a 5 x 5 matrix and so on.

So you have figured out how to create a Magic Square of a n x n matrix? Good job! However, let's test your math skills even further. Notice that for the 3 x 3 matrix, the sum of each column, row, and main diagonal was 15. Can you find a mathematical relationship that leads to the value of 15? Write your relationship in terms of "n".

Observe the sum of the columns, rows, and main diagonals in 4 x 4 and 5 x 5 matrices to help obtain and justify your answer. You may have found the arrangement of the numbers by trial and error; however the sum of the numbers can be found using a mathematical relationship. Solve a 4 x 4 and 5 x 5 matrix and attempt to construct a mathematical relationship in terms of "n" before you read further.

The mathematical relationship to find the sum in terms of "n" is:

Sum of columns, rows, main diagonals = n[(1 + n²)/2]

Therefore, for the 3 x 3 matrix we have: 3[(1+3²)/2] = 15

Following this relationship you can find the sum for the columns, rows, and main diagonals of any n x n matrix.

Magic Squares are a treat for anyone to solve. Benjamin Franklin is known for his complex 8 x 8 Magic Square. See if you can match Benjamin Franklin's accomplishment by constructing a 8 x 8 Magic Square of your own. What are you waiting for? Get started right now!