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magic

Magic square

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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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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)
ans = 3×1

    15
    15
    15
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

Figure contains 16 axes objects. Hidden axes object 1 with title 9 contains an object of type image. Hidden axes object 2 with title 10 contains an object of type image. Hidden axes object 3 with title 11 contains an object of type image. Hidden axes object 4 with title 12 contains an object of type image. Hidden axes object 5 with title 13 contains an object of type image. Hidden axes object 6 with title 14 contains an object of type image. Hidden axes object 7 with title 15 contains an object of type image. Hidden axes object 8 with title 16 contains an object of type image. Hidden axes object 9 with title 17 contains an object of type image. Hidden axes object 10 with title 18 contains an object of type image. Hidden axes object 11 with title 19 contains an object of type image. Hidden axes object 12 with title 20 contains an object of type image. Hidden axes object 13 with title 21 contains an object of type image. Hidden axes object 14 with title 22 contains an object of type image. Hidden axes object 15 with title 23 contains an object of type image. Hidden axes object 16 with title 24 contains an object of type image.

Input Arguments

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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 [].

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | char

Extended Capabilities

Version History

Introduced before R2006a

See Also

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