MATLAB Function Reference

magic - Magic square

Syntax

M = magic(n)

Description

M = magic(n) returns an n-by-n matrix constructed from the integers 1 through n^2 with equal row and column sums. The order n must be a scalar greater than or equal to 3.

Remarks

A magic square, scaled by its magic sum, is doubly stochastic.

Examples

The magic square of order 3 is

M = magic(3)

M = 

    8    1    6
    3    5    7
    4    9    2

This is called a magic square because the sum of the elements in each column is the same.

 sum(M) =

     15    15    15

And the sum of the elements in each row, obtained by transposing twice, is the same.

 sum(M')' = 

     15
     15
     15

This is also a special magic square because the diagonal elements have the same sum.

sum(diag(M)) =

     15

The value of the characteristic sum for a magic square of order n is

sum(1:n^2)/n

which, when n = 3, is 15.

Algorithm

There are three different algorithms:

n odd
n even but not divisible by four
n divisible by four

To make this apparent, type

for n = 3:20
    A = magic(n);
    r(n) = rank(A);
end

For n odd, the rank of the magic square is n. For n divisible by 4, the rank is 3. For n even but not divisible by 4, the rank is n/2 + 2.

[(3:20)',r(3:20)']
ans =
     3     3
     4     3
     5     5
     6     5
     7     7
     8     3
     9     9
    10     7
    11    11
    12     3
    13    13
    14     9
    15    15
    16     3
    17    17
    18    11
    19    19
    20     3

Plotting A for n = 18, 19, 20 shows the characteristic plot for each category.

Limitations

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