Magic Squares Questions

Magic squares questions with solutions are provided here for practice. A magic square is an arrangement of n^2.numbers in an n × n matrix form, where n > 2. These n² numbers are in an arithmetic progression. This n × n arrangement of numbers is in such a way that the sum of the numbers in each row, column and diagonal is always the same.

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The number which is equal to the sum of each row, column and diagonal is called the magic sum constant M.

If the magic square numbers are from 1 to n², then

M = n(n² + 1)/2.

Learn more about magic square puzzles.

The Story of Magic Squares

Practice Magic Squares Questions with Solution

Let us practise some magic squares questions.

Question 1:

Solve the magic square problem, whose sum is 30.

Solution:

Given the sum is 30

∴ for the first row 8 + 18 + __ = 30

⇒ 8 + 18 + 4 = 30

Similarly, 2nd column =18 + 10 + __ = 30

⇒ 18 + 10 + 2 = 30

In this way, by completing the missing numbers, we get the magic square

Question 2:

Complete the following magic square:

Solution:

A property of a 3 × 3 magic square is that when the middle number of the grid is multiplied by 9 and then divided by 3, gives the sum of the magic square

∴ (9 × 12)/3 = 108/3 = 36

Thus, each row, column and diagonal sum up to 36.

So, completing the missing numbers, we get the magic square

Question 3:

Solve the following magic square whose sum is 50:

Solution:

Given the sum is 50, then in the 2nd column,

13 + __ + 10 + 16 = 50, solving this we get the missing number 11. Finding all the numbers in this way, we get the magic square

Question 4:

Complete the magic square whose rows, columns and diagonals sum up to 65.

Solution:

Since the sum of each row, column and diagonal is 65. Hence the missing numbers of the magic square are

Question 5:

Complete the given magic square whose sum of each row, column and diagonal is 72.

Solution:

Given the sum of each row, column and diagonal is 72

∴ 21 + 26 + 25 = 72

21 + 28 + 23 = 72 and so on.

The magic square is:

Also, go through:

• Maths Patterns
• Algebra as Patterns
• Maths Tricks
• Multiplication Tricks

Question 6:

Solve the following magic square:

Solution:

A property of a 3 × 3 magic square is that when the middle number of the grid is multiplied by 9 and then divided by 3, gives the sum of the magic square

∴ (15 × 9)/3 = 45

Then, 24 + 3 + 18 = 45

And 12 + 15 + 21 = 45 and so on.

The magic square is:

Question 7:

Solve the 4 × 4 magic square, which begins with 1 and ends with 16.

Solution:

Given,

a = 1, l = 16, n = 16

Sum of all the numbers in the magic square = sum of n terms of an AP with last term l = n/2(a + l)

= 16/2 (1 + 16) = 8 × 136

Sum of each row, column and diagonal = 136/4 = 34.

Thus, the magic square is:

Question 8:

Solve the following magic square:

Solution:

In a 3 × 3 magic square,

Sum of each row, column and diagonal = middle element × 3 = 1.5 × 3 = 4.5

Thus, calculating each missing number we get,

Question 9:

Find the value of x and complete the magic square:

Solution:

In a 3 × 3 magic square,

Sum of each row, column and diagonal = middle element × 3 = 6 × 3 = 18

∴ (x – 1) + 6 + (x + 1) = 18

⇒ x = 6

Thus, calculating each missing number we get,

Question 10:

Solve the following 4 × 4 magic square, whose sum is 58.

Solution:

First row: 22 + 8 + 9 + 19 = 58

Third column: 9 + 16 + 12 + 21 = 58 solving so on, we get

Practice Questions on Magic Squares

1. Solve the following magic squares:

2. Solve the 4 × 4 magic square whose magic sum is 50.

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