Magic Square Calculator

Generate magic squares of odd order where every row, column, and diagonal sums to the same magic constant.

Configure Magic Square

Magic squares of odd order are generated using the Siamese (De la Loubere) method.

About Magic Squares

A magic square is an n x n grid filled with distinct integers from 1 to n2, where every row, column, and both main diagonals sum to the same value.

This value is called the magic constant and equals n(n2 + 1) / 2.

Result

Magic Constant
15
sum of each row, column & diagonal
Order (n) 3
Total Cells 9
Numbers Used 1 to 9
Sum of All Entries 45

Construction Method (Siamese)

M = n(n^2 + 1)/2 = 3(9 + 1)/2 = 15

Understanding Magic Squares

A magic square is one of the oldest mathematical puzzles, dating back to ancient China (the Lo Shu square, circa 2200 BCE). It consists of an n x n grid filled with distinct positive integers such that every row, column, and both main diagonals all sum to the same number, known as the magic constant.

The Magic Constant Formula

For a normal magic square of order n (using integers 1 through n2), the magic constant M is given by:

M = n(n2 + 1) / 2

This formula comes from the fact that the total sum of all integers from 1 to n2 is n2(n2 + 1)/2, and since there are n rows that must each sum to the same value, the constant per row is this total divided by n.

Construction Methods

Siamese Method (Odd Order)

Place 1 in the center of the top row. Move diagonally up-right for each successive number. Wrap around edges; if blocked, move down instead.

Works for all odd n: 3, 5, 7, ...

Doubly Even Order

For orders divisible by 4 (4, 8, 12...), use a diagonal marking method to determine which entries to complement.

n = 4k (k = 1, 2, 3, ...)

Singly Even Order

For orders like 6, 10, 14, use the LUX method or strachey method combining smaller odd-order squares.

n = 4k + 2 (k = 1, 2, 3, ...)

Lo Shu Square (3x3)

The oldest known magic square. There is essentially only one 3x3 normal magic square (up to rotations and reflections).

2 7 6 | 9 5 1 | 4 3 8

Properties of Magic Squares

  • Every row, column, and both main diagonals sum to the magic constant M.
  • For odd-order squares, the center cell always contains (n2 + 1) / 2.
  • The number of distinct normal magic squares grows rapidly with order: 1 for 3x3, 880 for 4x4, and over 275 million for 5x5.
  • Magic squares can be rotated 90, 180, and 270 degrees, and reflected, producing equivalent squares.

Historical Significance

Magic squares have appeared in art, architecture, and literature throughout history. The famous Durer magic square (1514) appears in the engraving "Melencolia I" and features the year 1514 in the bottom row. In many cultures, magic squares were believed to have mystical or protective properties.