General algorithm for a magic square

Vineeth T

Is there any algorithm to form a magic square of any size with a desired magic sum ?
Also can we make a magic square not only with the numbers from 1 to n²
but using any random numbers ?

RamaWolf

For a quick overview look at: http://en.wikipedia.org/wiki/Magic_square

Vineeth T

Quote by RamaWolf

For a quick overview look at: http://en.wikipedia.org/wiki/Magic_square

But in this webpage,no general methods of construction of a magic square of any numbers are given.
Can you give me a better one ?

RamaWolf

There is not one algorithm for all n (a natural number); I think, there are at least 3:
for odd numbers, for double even ( ie for numbers with 4 as factor) and for even numbers.

Below you find my little algorithm (written in ARIBAS) to generate an odd magic square;
example for n = 11; for simplicity of the algorithm, a 'vector' is used to store the 'square' .

==> MagicSquareOdd(11).
-: (68, 81, 94, 107, 120, 1, 14, 27, 40, 53, 66, 80, 93, 106, 119, 11, 13, 26,
39, 52, 65, 67, 92, 105, 118, 10, 12, 25, 38, 51, 64, 77, 79, 104, 117, 9, 22,
24, 37, 50, 63, 76, 78, 91, 116, 8, 21, 23, 36, 49, 62, 75, 88, 90, 103, 7, 20,
33, 35, 48, 61, 74, 87, 89, 102, 115, 19, 32, 34, 47, 60, 73, 86, 99, 101, 114,
6, 31, 44, 46, 59, 72, 85, 98, 100, 113, 5, 18, 43, 45, 58, 71, 84, 97, 110,
112, 4, 17, 30, 55, 57, 70, 83, 96, 109, 111, 3, 16, 29, 42, 56, 69, 82, 95,
108, 121, 2, 15, 28, 41, 54)

Code:

   n1:=n - 1; n2:=n*n; n3:=n1 + n1 + 1; 
   realloc(a, n2);
   i:=1;
   j:= n1 div 2;
   a[j]:=i;
   while i < n2 do
      i:=i + 1;
      if j mod n = n1 then
         k:=j - n3;   
      else
         k:=j - n1;
      end;
      if k < 0 then
         k:=k + n2;
      end;
      if a[k] = 0 then
         j:=k;
      else
         j:=j + n;       
      end;
      a[j]:=i;
   end;

ramsey2879

Quote by Vineeth T

But in this webpage,no general methods of construction of a magic square of any numbers are given.
Can you give me a better one ?

What do you mean by "general methods"? I consider "A method of constructing majic squares of even number of rows" and "Method for constructing a majic square of odd order" to be "general methods".

RamaWolf

Below you find my little algorithm (written in ARIBAS) to generate an double-even magic square of side length n (ie 4 divides n);

example for n = 4; for simplicity of the algorithm, a 'vector' is used to store the 'square'

MagicSquareDoubleEven(4).
-: (16, 2, 3, 13, 5, 11, 10, 8, 9, 7, 6, 12, 4, 14, 15, 1)

Code:

   d:= n div 4; d2:=d + d; n2:=n * n;
   realloc(a,n2);
   k:=1 + n2; k2:=0;
   for i:=1 to d do
      for j:=1 to d do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k;
      end;
      for j:=1 to d2 do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k2;
      end;
      for j:=1 to d do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k;
      end;      
   end;
   for i:=1 to d2 do
      for j:=1 to d do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k2;
      end;
      for j:=1 to d2 do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k;
      end;
      for j:=1 to d do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k2;
      end;      
   end;  
   for i:=1 to d do
      for j:=1 to d do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k;
      end;
      for j:=1 to d2 do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k2;
      end;
      for j:=1 to d do
         k:=k - 1; k2:=k2 + 1; a[k2 - 1]:=k;
      end;      
   end;

RamaWolf

Below you find my little algorithm (written in ARIBAS) to generate an simple-even magic square of side length n (ie n div 2 must be odd);

example for n = 6; the algorithm uses my 'MagicSquareOdd' (see above)
to form a matrix a and an auxiliary matrix 'aux' for the needed inter change of
some elements of matix a

MagicSquareSimpleEven(6)

((35, 1, 6, 26, 19, 24),
(3, 32, 7, 21, 23, 25),
(31, 9, 2, 22, 27, 20),
(8, 28, 33, 17, 10, 15),
(30, 5, 34, 12, 14, 16),
(4, 36, 29, 13, 18, 11))

A good companion for this algorithm was: http://www.hp-gramatke.de/magic_sq/index.htm

Code:

   k:= n div 2; k2:= k * k; d:= (k - 1) div 2;
   a:=alloc(array,n,alloc(array,n));
   aux:=alloc(array,k,alloc(array,n));
   for i:=0 to d - 1 do
      for j:=0 to d - 1 do
         aux[i][j]:=1;
      end;
   end;
   for j:=0 to d - 1 do
      aux[d][1+j]:=2;
   end;
   for i:=d + 1 to k - 1 do
      for j:=0 to d - 1 do
         aux[i][j]:=3;      
      end;
   end;
   for i:=0 to k - 1 do
      for j:=0 to d - 2 do
         aux[i][n-j-1]:=4;
      end;
   end;
   realloc(b, k2);
   b:=MagicSquareOdd(k);
   for i:=0 to k - 1 do
      for j:=0 to k - 1 do   
         a[i][j]:=b[i*k+j]; a[i][j+k]:=2*k2+b[i*k+j];
         a[i+k][j]:=3*k2+b[i*k+j]; a[i+k][j+k]:=k2+b[i*k+j];
      end;
   end;
   for i:=0 to k - 1 do
      for j:=0 to n - 1 do
         if aux[i][j] > 0 then
            d:=a[i][j]; a[i][j]:=a[i+k][j]; a[i+k][j]:=d;
         end;
      end;
   end;

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Vineeth T	General algorithm for a magic square Tweet Is there any algorithm to form a magic square of any size with a desired magic sum ? Also can we make a magic square not only with the numbers from 1 to n² but using any random numbers ?

RamaWolf Recognitions: Gold Member	For a quick overview look at: http://en.wikipedia.org/wiki/Magic_square