Magic Square
Compute the magic constant M = n(n²+1)/2 for any magic square order. Free calculator showing row, column and diagonal sums for 3×3, 4×4, 5×5 and beyond.
About This Calculator
This Magic Square Calculator computes the magic constant for any normal magic square of order n. A magic square is a square grid of distinct positive integers where the sum of every row, every column, and both main diagonals is identical. That identical sum is called the magic constant. For a normal magic square that contains the integers 1 through n^2 exactly once, the magic constant is given by the formula M = n(n^2 + 1)/2. Simply enter the order (size) of the square -- 3 for a 3x3 square, 4 for 4x4, and so on -- and instantly see the magic constant.
The formula comes from the fact that the sum of all numbers 1 + 2 + ... + n^2 equals n^2(n^2 + 1)/2. Since every row must sum to the same constant M, and there are n rows, we have n x M = n^2(n^2 + 1)/2. Solving for M gives the formula above. For example, a 3x3 normal magic square contains the numbers 1 through 9 (which sum to 45), so each of the three rows must total 45/3 = 15. A 4x4 square uses numbers 1 through 16 (sum 136), giving a constant of 136/4 = 34. A 5x5 square yields 65, and a 6x6 square yields 111.
Note that a 2x2 magic square is impossible -- the constraints force two cells to hold the same number. For n = 1, the trivial single-cell square has a magic constant of 1. Magic squares have fascinated mathematicians for millennia, appearing in ancient Chinese, Indian, and Islamic mathematics before reaching Europe. They remain a rich area of recreational mathematics with applications in puzzle design, art, and even cryptography.
Our calculator supports any positive integer order up to large values, always returning the exact magic constant. Use the shareable URL feature to bookmark or share specific calculations.
Frequently Asked Questions
What is a magic square?
A magic square is a square grid of distinct positive integers where the sum of every row, column, and main diagonal equals the same value, called the magic constant. For a normal magic square of order n, it contains the numbers 1 through n^2 exactly once each.
What is the magic constant formula?
The magic constant M for a normal magic square of order n is given by M = n(n^2 + 1)/2. For a 3x3 square, M = 15. For a 4x4 square, M = 34. For a 5x5 square, M = 65.
Can I build a 2x2 magic square?
No, a 2x2 magic square is impossible. The system of equations forces two cells to have the same value, which violates the rule that no number repeats. The magic constant formula returns 5 for n = 2, but the square itself cannot be constructed.
How many 3x3 magic squares exist?
There is exactly one unique 3x3 magic square up to rotations and reflections. All 3x3 magic squares are rotations or reflections of the same arrangement, where the center is always 5 and opposite cells sum to 10.
What are the different types of magic squares?
Beyond normal magic squares, variants include pandiagonal magic squares (broken diagonals also sum to the constant), associative magic squares (opposite cells sum to n^2 + 1), and most-perfect magic squares (pandiagonal with additional 2x2 subsquare properties). Semi-magic squares satisfy only rows and columns but not diagonals.
What are the algorithms to construct a magic square?
For odd-order squares, the Siamese method places numbers by moving up and right. For doubly even orders (multiples of 4), fill diagonals first then fill remaining cells in reverse order. For singly even orders (4k + 2), divide into four odd-order sub-squares, fill them, then swap selected columns.
Is this magic square calculator free to use?
Yes, this magic square calculator on Calculy is completely free to use with no registration required. You can calculate magic constants for any order and share results via URL.