Magic Square – Even Number of Squares

4 x 4 Magic Square

Did you enjoy my previous post about solving a 3 X 3 Magic Square. Now, you might be left wondering what about a 4 X 4 Magic square ? Can it be solved the same way ? Well no right, because unlike 3 X 3 Magic square, there is no middle column in the 4 X 4 Magic Square. Therefore, we need to take a different approach to fill the cells in clockwise and anti-clockwise direction. The cells are filled in such a way that clockwise numbers and anti-clockwise numbers balance each other.

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How to solve Magic Square with Even Number of Cells

Let’s solve a 4 X 4 Magic square which has 4 rows and 4 columns with 4×4= 16 cells to fill in. Let’s fill the squares from numbers 1-16.

Since there are 4 rows, divide the numbers from 1-16 into four different groups.

The Math Behind It

Do you notice a pattern in the numbers from each group ?

Add the numbers in Group 1 and Group 4, starting from opposite direction and what do you see, they all add up to 17.

1+16 = 173 + 14 = 17
2 + 15 = 174 + 13 = 17

Similarly, the numbers in Group 2 and Group 3, when added, first and last and so on, add up to 17.

5 + 12 = 177 + 10 = 17
6 + 11 = 178 + 9 = 17

Using the above knowledge, we can fill out the first and last row of the square with numbers from Group 1 and Group 4 and second and third row with numbers from group 2 and Group 3.

Magic Square 4 X 4

Starting from the left side of the square, fill the cells in row 1 and 4 with numbers from Group 1 -> 1, 2, 3, 4 in anti-clockwise direction.

Similarly, starting from the left side of the square, fill the cells in rows 1 and 4 with numbers from group 4 – > 13, 14, 15, 16 in clockwise direction.

Starting with the right side of the square, fill the cells in rows 2 and 3 with numbers from group 2 – > 5, 6, 7, 8 in anti-clockwise direction.

Similarly, starting from the right side of the square, fill the cells in rows 2 and 3 with numbers from group 3 -> 9, 10, 11, 12 in clockwise direction.

This is a Magic Square with a magic constant of 34, which is twice the sum of the first number and last number in the sequence ie 2 x ( 1 + 16 ) = 34. Sum of the numbers in each rows, columns and diagonally equals 34.

What is a Magic Square

A Magic Square is a square with numbers from a sequence arranged in such a way that the sum of all the numbers in a row, column and diagonal are same. This sum is called the magic constant. They are grouped into two categories – Odd order of Magic Squares and Even order of Magic Squares. In an Odd Magic square, there are odd number of cells on each side of the square. We use the concept of transpose and apply to solve them. Similarly, in an even magic square, there are even number of cells on each side of the square. And we use the concept of filling the cells in clockwise and anti-clockwise manner to solve them.

We can fill the cells of a magic square with any arithmetic progression sequence. It means, a sequence of numbers such that the difference between the consecutive terms is constant. For example, 2,4,6, 8, …. and so on is an arithmetic progression series with a difference of 2 between each number.

How about trying it with different sequence to see the magic yourself.

Do you love learning math tricks, then check out this amazing book from Arthur T. Benjamin who is passionate about two things math and magic. Some of his books have been New York best seller and winner of American Mathematical Society AMA.

And while you are here, check out some of my latest posts.

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