Magic Square Calculator
Generate magic squares with equal row, column, and diagonal sums using mathematical algorithms
Generate Magic Square
Enter a number from 1 to 10 (Note: 2×2 magic squares are impossible)
3×3 Magic Square
Validation Results
Row Sums:
Column Sums:
Famous Magic Squares
3×3 Magic Square (Lo Shu Square)
Magic Constant: 15
Origin: Ancient China (~190 BC)
Legend: Found on the shell of a turtle
4×4 Magic Square (Dürer's Square)
Magic Constant: 34
Origin: Albrecht Dürer (1514)
Special: Date "1514" in bottom row
Extra: 2×2 corners sum to 34
Magic Square Properties
Magic Constant
M = n(n² + 1)/2
Sum of each row, column, and diagonal
Distinct Numbers
Uses numbers 1 through n²
Each number appears exactly once
Equal Sums
All rows, columns, diagonals equal M
Total of 2n + 2 equal sums
Construction Methods
Odd Order (n = 1,3,5,...)
Siamese/De la Loubère method
Start top-middle, move up-right
Doubly Even (n = 4,8,12,...)
Diagonal method
Fill diagonals, then complement
Singly Even (n = 6,10,14,...)
Conway's LUX method
Combine odd squares with swaps
2×2 Square
Mathematically impossible
No solution with distinct integers
Magic Square Facts
Magic squares exist for all orders except 2
Used in ancient cultures for mystical purposes
A 3×3 square is essentially unique
Dürer included one in his 1514 engraving "Melencolia I"
Higher order squares have astronomical numbers of variations
Understanding Magic Squares
What Makes a Square "Magic"?
A magic square is an arrangement of distinct positive integers in a square grid where every row, column, and diagonal sums to the same number, called the magic constant.
Historical Significance
- •First recorded in China around 190 BC (Lo Shu Square)
- •Used in Islamic world for astrological purposes
- •Featured in Renaissance art (Dürer's Melencolia I)
- •Studied by mathematicians like Euler and Franklin
Mathematical Properties
Magic Constant: M = n(n² + 1)/2
For n=3: M = 3(9+1)/2 = 15
For n=4: M = 4(16+1)/2 = 34
For n=5: M = 5(25+1)/2 = 65
Special Types
Pandiagonal: Broken diagonals also sum to M
Associative: Opposite cells sum to n²+1
Most-Perfect: Both pandiagonal and associative