Magic Squares
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.
In Class 4, 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.
What is Magic Squares - Class 4 Maths (Patterns)?
A magic square is a square grid of numbers where:
- Each row adds up to the same sum.
- Each column adds up to the same sum.
- Both diagonals add up to the same sum.
This common sum is called the magic constant (or magic sum).
For a 3×3 magic square using 1 to 9: Magic Constant = 15
Magic Squares Formula
Finding the magic constant:
For a 3×3 magic square using numbers 1 to 9:
Step 1: Sum of all numbers = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
Step 2: There are 3 rows, and each row has the same sum.
Step 3: Magic constant = 45 ÷ 3 = 15
Magic Constant = Sum of all numbers ÷ Number of rows
Solved Examples
Example 1: Example 1: Verifying a 3×3 magic square
Problem: Is this a magic square?
| 2 | 7 | 6 |
| 9 | 5 | 1 |
| 4 | 3 | 8 |
Solution:
Rows: 2+7+6=15, 9+5+1=15, 4+3+8=15 ✓
Columns: 2+9+4=15, 7+5+3=15, 6+1+8=15 ✓
Diagonals: 2+5+8=15, 6+5+4=15 ✓
Answer: Yes, it is a magic square with magic constant 15.
Example 2: Example 2: Finding a missing number
Problem: Find the missing number (?).
| 2 | 7 | 6 |
| 9 | ? | 1 |
| 4 | 3 | 8 |
Solution:
Step 1: Magic constant = 15 (since numbers are 1 to 9).
Step 2: Middle row: 9 + ? + 1 = 15
Step 3: ? = 15 − 9 − 1 = 5
Answer: The missing number is 5.
Example 3: Example 3: Completing a magic square
Problem: Complete the magic square (magic constant = 15):
| 8 | ? | ? |
| ? | 5 | ? |
| ? | ? | 2 |
Solution:
Step 1: Diagonal (top-left to bottom-right): 8 + 5 + 2 = 15 ✓ (already complete).
Step 2: Row 3: ? + ? + 2 = 15. Diagonal (bottom-left to top-right): ? + 5 + ? = 15.
Step 3: Try numbers 1–9 not yet used: 1, 3, 4, 6, 7, 9.
Step 4: Row 1: 8 + ? + ? = 15 → need pair summing to 7 from {1,3,4,6,7,9}. Options: (1,6) or (3,4).
Step 5: Column 1: 8 + ? + ? = 15 → need pair summing to 7. Options: (1,6) or (3,4).
Step 6: Using systematic trial: Row 1 = 8, 1, 6; Row 2 = 3, 5, 7; Row 3 = 4, 9, 2.
Answer:
| 8 | 1 | 6 |
| 3 | 5 | 7 |
| 4 | 9 | 2 |
Example 4: Example 4: Magic constant for different numbers
Problem: A 3×3 magic square uses the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18. What is the magic constant?
Solution:
Step 1: Sum = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 = 90
Step 2: Magic constant = 90 ÷ 3 = 30
Answer: The magic constant is 30.
Example 5: Example 5: Word problem
Problem: Aditi arranges the numbers 1 to 9 in a 3×3 grid so that each row, column, and diagonal adds to 15. Which number must go in the centre?
Solution:
Step 1: The centre number appears in 1 row, 1 column, and 2 diagonals = 4 lines.
Step 2: In the standard 3×3 magic square, only the middle value (5) satisfies all constraints.
Step 3: Verify: 5 is the median of 1–9.
Answer: The centre must be 5.
Example 6: Example 6: Finding two missing numbers
Problem: Complete (magic constant = 15):
| 6 | 1 | ? |
| 7 | 5 | 3 |
| ? | 9 | 4 |
Solution:
Step 1: Row 1: 6 + 1 + ? = 15 → ? = 8
Step 2: Row 3: ? + 9 + 4 = 15 → ? = 2
Step 3: Verify columns and diagonals: 6+7+2=15, 1+5+9=15, 8+3+4=15, 6+5+4=15, 8+5+2=15 ✓
Answer: The missing numbers are 8 (top-right) and 2 (bottom-left).
Example 7: Example 7: Not a magic square
Problem: Is this a magic square?
| 1 | 2 | 3 |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
Solution:
Row sums: 6, 15, 24 — all different ✗
Answer: No, this is not a magic square. The rows have different sums.
Example 8: Example 8: Magic square with multiples of 3
Problem: Create a 3×3 magic square using 3, 6, 9, 12, 15, 18, 21, 24, 27.
Solution:
Step 1: Sum = 3+6+9+12+15+18+21+24+27 = 135
Step 2: Magic constant = 135 ÷ 3 = 45
Step 3: These are 3 × {1,2,3,4,5,6,7,8,9}. Multiply the standard magic square by 3:
| 6 | 21 | 18 |
| 27 | 15 | 3 |
| 12 | 9 | 24 |
Answer: Each row, column, and diagonal sums to 45.
Example 9: Example 9: Strategy — start with the centre
Problem: Kavi wants to make a 3×3 magic square with numbers 1 to 9. What strategy should he use?
Solution:
Step 1: Place 5 in the centre (always the middle value).
Step 2: Place numbers in opposite positions that add up to 10 (since 5+5=10 and each pair through centre must sum to the same).
Step 3: Pairs: (1,9), (2,8), (3,7), (4,6). Place these in opposite corners and opposite edges.
Answer: Start with 5 in the centre, then place pairs summing to 10 on opposite sides.
Example 10: Example 10: 4×4 magic square introduction
Problem: A 4×4 magic square uses the numbers 1 to 16. What is the magic constant?
Solution:
Step 1: Sum of 1 to 16 = 16 × 17 ÷ 2 = 136
Step 2: Magic constant = 136 ÷ 4 = 34
Answer: The magic constant for a 4×4 magic square (1 to 16) is 34.
Key Points to Remember
- A magic square has equal sums for all rows, columns, and diagonals.
- The sum is called the magic constant.
- For a 3×3 magic square with 1 to 9, the magic constant is 15.
- The centre number of a 3×3 magic square (1–9) is always 5.
- Magic constant = Total sum of all numbers ÷ Number of rows.
- Opposite numbers through the centre always sum to the same value (10 for 1–9).
- Use known sums to find missing numbers: Missing = Magic constant − sum of other numbers in that row/column.
Practice Problems
- Verify that this is a magic square: Row 1: 4, 9, 2; Row 2: 3, 5, 7; Row 3: 8, 1, 6.
- Find the missing number: Row 1: 2, 9, 4; Row 2: 7, ?, 3; Row 3: 6, 1, 8. (Magic constant = 15)
- A 3×3 magic square uses numbers 5, 10, 15, 20, 25, 30, 35, 40, 45. Find the magic constant.
- Complete: Row 1: ?, 3, 8; Row 2: 9, 5, 1; Row 3: 4, 7, ?. (Magic constant = 15)
- Why must 5 always be in the centre of a 3×3 magic square using 1 to 9?
- A 4×4 magic square uses numbers 1 to 16. What is its magic constant?
- Is this a magic square? Row 1: 3, 8, 1; Row 2: 2, 4, 6; Row 3: 7, 0, 5.
Frequently Asked Questions
Q1. What is a magic square?
A magic square is a grid of numbers where every row, column, and diagonal adds up to the same total. That total is called the magic constant.
Q2. What is the magic constant for a 3×3 magic square with numbers 1 to 9?
The magic constant is 15. The sum of numbers 1 to 9 is 45, and 45 divided by 3 rows gives 15.
Q3. Why is 5 always in the centre of a 3×3 magic square (1–9)?
The centre cell is part of one row, one column, and both diagonals — 4 lines in total. Only the number 5, being the median of 1 to 9, can satisfy all four sum conditions of 15.
Q4. How do you find a missing number in a magic square?
Find a row, column, or diagonal that has only one missing number. Subtract the sum of the known numbers from the magic constant to get the missing value.
Q5. Can a magic square use numbers other than 1 to 9?
Yes. Any set of 9 numbers can form a magic square if they can be arranged to give equal row, column, and diagonal sums. Common sets include multiples of 2, 3, or consecutive even numbers.
Q6. How many different 3×3 magic squares exist using 1 to 9?
There is essentially one basic 3×3 magic square using 1 to 9 (with 5 always in the centre). However, it can be rotated and reflected to create 8 different-looking versions.
Q7. What is the magic constant formula?
Magic constant = (Sum of all numbers in the square) ÷ (Number of rows). For the standard 3×3 square: 45 ÷ 3 = 15.
Q8. Are magic squares used in real life?
Magic squares appear in mathematical puzzles, art (like Albrecht Durer's famous 4×4 magic square), and ancient Indian and Chinese mathematics. They are also used to develop logical thinking and problem-solving skills.
