magic

Magic square

Syntax

M = magic(n)

Description

M = magic(n) returns an n-by-n matrix constructed from the integers 1 through n² with equal row and column sums. The order n must be a scalar greater than or equal to 3 in order to create a valid magic square.

example

Examples

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Third-Order Magic Square

Open Live Script

Compute the third-order magic square M.

M = magic(3)

M = 3×3

     8     1     6
     3     5     7
     4     9     2

The sum of the elements in each column and the sum of the elements in each row are the same.

sum(M)

ans = 1×3

    15    15    15

sum(M,2)

Magic Square Visualization

Open Live Script

Visually examine the patterns in magic square matrices with orders between 9 and 24 using imagesc. The patterns show that magic uses three different algorithms, depending on whether the value of mod(n,4) is 0, 2, or odd.

for n = 1:16
    subplot(4,4,n)
    ord = n+8;
    m = magic(ord);
    imagesc(m)
    title(num2str(ord))
    axis equal
    axis off
end

Input Arguments

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`n` — Matrix order
scalar integer

Matrix order, specified as a scalar integer greater than or equal to 3. If n is complex, not an integer, or not scalar, then magic converts it into a usable integer with floor(real(double(n(1)))).

If you supply n less than 3, then magic returns either a nonmagic square, or the degenerate magic squares 1 and [].

Extended Capabilities

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C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

The input argument n must be a constant at code generation time.

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a