Class 8 - Magic Squares

A magic square is a square grid with numbers in them. It is a special arrangement of numbers as every row, column or diagonal adds up to the same number, which is called the magic number. A magic square is a special arrangement of numbers in a grid where every row, every column, and both diagonals add up to the same sum. This sum is called the magic constant. you will learn to complete 3×3 magic squares using the numbers 1 to 9, and understand the strategies to fill in missing numbers. Magic squares build strong addition skills and logical thinking.

Table of Contents

• What is a Magic Square Puzzle?
• Magic Square Formula
• How to Solve Magic Square
• Magic Square Trick for Order 3
• Magic Square 3×3
• Magic Square 4×4

What is a Magic Square Puzzle?

A Magic Square is a grid filled with different numbers where the sum of every row, every column, and both diagonals is always the same. That special total is called the Magic Sum or Magic Constant.

Think of it like a number puzzle you must place numbers so that no matter which direction you add them, you always get the same answer.

The Magic Square Formula

The Magic Constant (M) is the number that every row, column, and diagonal must add up to. You can calculate it with a simple formula: M=n(n2+1)2

Where n is the size of the grid (number of rows or columns).

How to Solve Magic square

As mentioned above, the formula of the magic square sum is n(n2 + 1)/2.

For a magic square of order 3, we need to substitute n = 3 to know the magic sum so that we can easily form the magic square 3×3.

When n=3,the sum=3(32+1)2=3(9+1)2=3×102=15

Now, we have to place the numbers in the respective places so that the sum of numbers in each row, column and diagonal is equal to 15.

Magic Square Trick for order 3

Let “x” be the order of the magic square.

In this case, x = 3.

Consider another number, “y” such that the product of x and y is equal to the magic sum, i.e. 15.

So, y=5xy=(3)(5)=15

The value of y should always be at the center of the square and x be on its left cell.

The cell above x is taken as y–1 as given below:

Based on this trick, we can build the magic square of order 3.

Magic Square 3×3

A 3×3 magic square uses numbers 1 to 9. The magic constant is 15. Consider the Magic square 3×3 sum 15 given below:

Let us make the complementary magic square of the above square.

(n2+1)=32+1=9+1=10

Now, subtract each number from (n2+1) , i.e. from 10.

First row numbers:

10 − 4 = 6

10 − 3 = 7

10 − 8 = 2

Second row numbers:

10 − 9 = 1

10 − 5 = 5

10 − 1 = 9

Third row numbers:

10 − 2 = 8

10 − 7 = 3

10 − 6 = 4

Therefore, the new magic square formed is:

Every row, every column, and both diagonals add up to 15. That is the magic.

Magic Square 4×4

A 4×4 magic square uses numbers 1 to 16. The magic constant is 34.For a magic square of order 4, we need to substitute n = 4 to know the magic sum so that we can easily form the magic square 3×3.

When n = 4, the sum = 4(42+1)2=4(16+1)2=4×172=34

Thus, the magic square 3×3 sum 34 is given as: