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How To Solve a 3 x 3 Magic Square?

How-To-Solve-a-3-x-3-Magic-Square-

This post is also available in: हिन्दी (Hindi) العربية (Arabic)

There are many amazing numbers, series, concepts, and facts about numbers in mathematics. One such is a magic square.

Magic Square 3×3

Magic Squares are square grids with a special arrangement of numbers in them. These numbers are special because every row, column and diagonal adds up to the same number. So for the example below, 15 is the magic number. Could you work this out just from knowing that the square uses the numbers from 1 to 9?

3x3 magic square solver

Sum of rows = 4 + 9 + 2 = 3 + 5 + 7 = 8 + 1 + 6 = 15

Sum of columns = 4 + 3 + 8 = 9 + 5 + 1 = 2 + 7 + 6 = 15

Sum of diagonals = 4 + 5 + 6 = 2 + 5 + 8 = 15

Also, the two numbers that are opposite each other across the centre number will add up to the same number. So in the square above, 8 + 2 = 10 , 6 + 4 = 10, 1 + 9 = 10 and 3 + 7 = 10.

Why Are They  Called Magic?

So the numbers in the Magic Square are special, but why are they called magic? It seems that from ancient times they were connected with the supernatural and magical world. The earliest record of magic squares is from China in about 2200 BC. and is called “Lo-Shu“. 

Legends dating from as early as 650 BCE tell the story of the Lo Shu (洛書) or “scroll of the river Lo”. According to the legend, there was at one time in ancient China a huge flood. While the great king Yu was trying to channel the water out to sea, a turtle emerged from it with a curious pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, such that the sum of the numbers in each row, column and diagonal was the same: 15. 

Thereafter people were able to use this pattern in a certain way to control the river and protect themselves from floods. The Lo Shu Square, as the magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection.

magic square 3x3

The black knots show even numbers and the white knots show odd numbers. Look closely and you’ll see that this ancient magic square is the same as our example above. Magic squares were first mentioned in the Western world in the work of Theon of Smyrna. They were also used by Arab astrologers in the 9th century to help work out horoscopes. The work of the Greek mathematician Moschopoulos in 1300 A.D. help to spread knowledge about magic squares. So here we are now, more than 700 years later, and teachers are using them in class for problem solving and practicing addition.

3×3 Magic Square Solver – Tricks To Solve Magic Squares

To solve these types of the magic square you might use trial and error method,  that takes a  lot of time. We bring you a 3×3 magic square solver to solve such magic squares quickly and easily. Let’s look at the two very interesting ways of solving such problems.

Trick 1:

To solve any 3 by 3 magic square just remember “HEB CIG FAD”.

Here each alphabet represents a number.

HEBCIGFAD
123456789

Now, fill the square with alphabets in a clockwise direction starting from the top left.

magic square 3x3

Next, replace the alphabets with their corresponding numbers.

magic square 3x3

Let’s now consider one more example of filling numbers from 12 to 20 in a similar magic square.

Designate each number with alphabets from A to I.

HEBCIGFAD
121314151617181920

Now, fill the square with alphabets in a clockwise direction starting from the top left.

magic square 3x3

Next, replace the alphabets with their corresponding numbers.

magic square 3x3

Let’s check the sum of numbers:

Rows: 19 + 14 + 15 = 12 + 16 + 20 = 17 + 18 + 13 = 48

Columns: 19 + 12 + 17 = 14 + 16 + 18 = 15 + 20 + 13 = 48

Diagonals: 19 + 16 + 13 = 15 + 16 + 17 = 48

Trick 2:

To solve any 3 by 3 magic square remember the following pattern:

magic square 3x3

And now fill the numbers from lowest to largest following the lines in the above pattern.

magic square 3x3

Let’s now fill numbers from 24 to 32 in a 3 x 3 magic square.

magic square 3x3

Sum of numbers in rows = 27 + 32 + 25 = 26 + 28 + 30 = 31 + 24 + 29 = 84

Sum of numbers in columns = 27 + 26 + 31 = 32 + 28 + 24 = 25 + 30 + 29 = 84

Sum of numbers in diagonals = 27 + 28 + 29 = 25 + 28 + 31 = 84

Using these tricks you can solve any 3 x 3 magic square with any sequence of 9 consecutive numbers.

Solving these types of problems and playing chess help kids improve their analytical and logical thinking. Chess online classes will be helpful for kids to learn basic chess techniques and strategies.

Do you know any other trick for solving such magic squares? Don’t forget to share it with us.

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