Magic squares of odd order

From Rosetta Code
Task
Magic squares of odd order
You are encouraged to solve this task according to the task description, using any language you may know.

A magic square is an square matrix whose numbers (usually integers) consist of consecutive numbers arranged so that the sum of each row and column, and both long (main) diagonals are equal to the same sum (which is called the magic number or magic constant).

The numbers are usually (but not always) the 1st positive integers.

A magic square whose rows and columns add up to a magic number but whose main diagonals do not, is known as a semimagic square.

8 1 6
3 5 7
4 9 2
Task

For any odd , generate a magic square with the integers and show the results. Optionally, show the magic number.

You should demonstrate the generator by showing at least a magic square for .

Also see

Ada

<lang Ada>with Ada.Text_IO, Ada.Command_Line;

procedure Magic_Square is

  N: constant Positive := Positive'Value(Ada.Command_Line.Argument(1));
  
  subtype Constants is Natural range 1 .. N*N;
  package CIO is new Ada.Text_IO.Integer_IO(Constants);
  Undef: constant Natural := 0;
  subtype Index is Natural range 0 .. N-1;
  function Inc(I: Index) return Index is (if I = N-1 then 0 else I+1);
  function Dec(I: Index) return Index is (if I = 0 then N-1 else I-1);
   
  A: array(Index, Index) of Natural := (others => (others => Undef));
    -- initially undefined; at the end holding the magic square
 
  X: Index := 0; Y: Index := N/2; -- start position for the algorithm

begin

  for I in Constants loop -- write 1, 2, ..., N*N into the magic array
     A(X, Y) := I; -- write I into the magic array
     if A(Dec(X), Inc(Y)) = Undef then 

X := Dec(X); Y := Inc(Y); -- go right-up

     else

X := Inc(X); -- go down

     end if;
  end loop;
  
  for Row in Index loop -- output the magic array
     for Collumn in Index loop

CIO.Put(A(Row, Collumn), Width => (if N*N < 10 then 2 elsif N*N < 100 then 3 else 4));

     end loop;
     Ada.Text_IO.New_Line;
  end loop;

end Magic_Square;</lang>

Output:
>./magic_square 3
 8 1 6
 3 5 7
 4 9 2
>./magic_square 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

ALGOL W

<lang algolw>begin

   % construct a magic square of odd order - as a procedure can't return an %
   % array, the caller must supply one that is big enough                   %
   logical procedure magicSquare( integer array square ( *, * )
                                ; integer value order
                                ) ;
       if not odd( order ) or order < 1 then begin
           % can't make a magic square of the specified order               %
           false
           end
       else begin
           % order is OK - construct the square using de la Loubère's       %
           % algorithm as in the wikipedia page                             %
           % ensure a row/col position is on the square                     %
           integer procedure inSquare( integer value pos ) ;
               if pos < 1 then order else if pos > order then 1 else pos;
           % move "up" a row in the square                                  %
           integer procedure up   ( integer value row ) ; inSquare( row - 1 );
           % move "accross right" in the square                             %
           integer procedure right( integer value col ) ; inSquare( col + 1 );
           integer  row, col;
           % initialise square                                              %
           for i := 1 until order do for j := 1 until order do square( i, j ) := 0;
           % initial position is the middle of the top row                  %
           col := ( order + 1 ) div 2;
           row := 1;
           % construct square                                               %
           for i := 1 until ( order * order ) do begin
               square( row, col ) := i;
               if square( up( row ), right( col ) ) not = 0 then begin
                   % the up/right position is already taken, move down      %
                   row := row + 1;
                   end
               else begin
                   % can move up/right                                      %
                   row := up(    row );
                   col := right( col );
               end
           end for_i;
           % sucessful result                                               %
           true
       end magicSquare ;
   % prints the magic square                                                %
   procedure printSquare( integer array  square ( *, * )
                        ; integer value  order
                        ) ;
   begin
       integer sum, w;
       % set integer width to accomodate the largest number in the square   %
       w := ( order * order ) div 10;
       i_w := s_w := 1;
       while w > 0 do begin i_w := i_w + 1; w := w div 10 end;
       for i := 1 until order do sum := sum + square( 1, i );
       write( "maqic square of order ", order, ": sum: ", sum );
       for i := 1 until order do begin
           write( square( i, 1 ) );
           for j := 2 until order do writeon( square( i, j ) )
       end for_i
   end printSquare ;
   % test the magic square generation                                       %
   integer array sq ( 1 :: 11, 1 :: 11 );
   for i := 1, 3, 5, 7 do begin
       if magicSquare( sq, i ) then printSquare( sq, i )
                               else write( "can't generate square" );
   end for_i

end.</lang>

Output:
maqic square of order 1 : sum: 1 
1 
maqic square of order 3 : sum: 15 
8 1 6 
3 5 7 
4 9 2 
maqic square of order  5 : sum: 65 
17 24  1  8 15 
23  5  7 14 16 
 4  6 13 20 22 
10 12 19 21  3 
11 18 25  2  9 
maqic square of order  7 : sum: 175 
30 39 48  1 10 19 28 
38 47  7  9 18 27 29 
46  6  8 17 26 35 37 
 5 14 16 25 34 36 45 
13 15 24 33 42 44  4 
21 23 32 41 43  3 12 
22 31 40 49  2 11 20 

AutoHotkey

<lang autohotkey> msgbox % OddMagicSquare(5) msgbox % OddMagicSquare(7) return

OddMagicSquare(oddN){ sq := oddN**2 obj := {} loop % oddN obj[A_Index] := {} ; dis is row mid := Round((oddN+1)/2) sum := Round(sq*(sq+1)/2/oddN) obj[1][mid] := 1 cR := 1 , cC := mid loop % sq-1 { done := 0 , a := A_index+1 while !done { nR := cR-1 , nC := cC+1 if !nR nR := oddN if (nC>oddN) nC := 1 if obj[nR][nC] ;filled cR += 1 else cR := nR , cC := nC if !obj[cR][cC] obj[cR][cC] := a , done := 1 } }

str := "Magic Constant for " oddN "x" oddN " is " sum "`n" for k,v in obj { for k2,v2 in v str .= " " v2 str .= "`n" } return str } </lang>

Output:
Magic Constant for 5x5 is 65
 17 24 1 8 15
 23 5 7 14 16
 4 6 13 20 22
 10 12 19 21 3
 11 18 25 2 9


Magic Constant for 7x7 is 175
 30 39 48 1 10 19 28
 38 47 7 9 18 27 29
 46 6 8 17 26 35 37
 5 14 16 25 34 36 45
 13 15 24 33 42 44 4
 21 23 32 41 43 3 12
 22 31 40 49 2 11 20

AWK

<lang AWK>

  1. syntax: GAWK -f MAGIC_SQUARES_OF_ODD_ORDER.AWK

BEGIN {

   build(5)
   build(3,1) # verify sum
   build(7)
   exit(0)

} function build(n,check, arr,i,width,x,y) {

   if (n !~ /^[0-9]*[13579]$/ || n < 3) {
     printf("error: %s is invalid\n",n)
     return
   }
   printf("\nmagic constant for %dx%d is %d\n",n,n,(n*n+1)*n/2)
   x = 0
   y = int(n/2)
   for (i=1; i<=(n*n); i++) {
     arr[x,y] = i
     if (arr[(x+n-1)%n,(y+n+1)%n]) {
       x = (x+n+1) % n
     }
     else {
       x = (x+n-1) % n
       y = (y+n+1) % n
     }
   }
   width = length(n*n)
   for (x=0; x<n; x++) {
     for (y=0; y<n; y++) {
       printf("%*s ",width,arr[x,y])
     }
     printf("\n")
   }
   if (check) { verify(arr,n) }

} function verify(arr,n, total,x,y) { # verify sum of each row, column and diagonal

   print("\nverify")
  1. horizontal
   for (x=0; x<n; x++) {
     total = 0
     for (y=0; y<n; y++) {
       printf("%d ",arr[x,y])
       total += arr[x,y]
     }
     printf("\t: %d row %d\n",total,x+1)
   }
  1. vertical
   for (y=0; y<n; y++) {
     total = 0
     for (x=0; x<n; x++) {
       printf("%d ",arr[x,y])
       total += arr[x,y]
     }
     printf("\t: %d column %d\n",total,y+1)
   }
  1. left diagonal
   total = 0
   for (x=y=0; x<n; x++ y++) {
     printf("%d ",arr[x,y])
     total += arr[x,y]
   }
   printf("\t: %d diagonal top left to bottom right\n",total)
  1. right diagonal
   x = n - 1
   total = 0
   for (y=0; y<n; y++ x--) {
     printf("%d ",arr[x,y])
     total += arr[x,y]
   }
   printf("\t: %d diagonal bottom left to top right\n",total)

} </lang>

Output:
magic constant for 5x5 is 65
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

magic constant for 3x3 is 15
8 1 6
3 5 7
4 9 2

verify
8 1 6  : 15 row 1
3 5 7  : 15 row 2
4 9 2  : 15 row 3
8 3 4  : 15 column 1
1 5 9  : 15 column 2
6 7 2  : 15 column 3
8 5 2  : 15 diagonal top left to bottom right
4 5 6  : 15 diagonal bottom left to top right

magic constant for 7x7 is 175
30 39 48  1 10 19 28
38 47  7  9 18 27 29
46  6  8 17 26 35 37
 5 14 16 25 34 36 45
13 15 24 33 42 44  4
21 23 32 41 43  3 12
22 31 40 49  2 11 20

bc

Works with: GNU bc

<lang bc>define magic_constant(n) {

   return(((n * n + 1) / 2) * n)

}

define print_magic_square(n) {

   auto i, x, col, row, len, old_scale
   old_scale = scale
   scale = 0
   len = length(n * n)
   print "Magic constant for n=", n, ": ", magic_constant(n), "\n"
   for (row = 1; row <= n; row++) {
       for (col = 1; col <= n; col++) {
           x = n * ((row + col - 1 + (n / 2)) % n) + \
               ((row + 2 * col - 2) % n) + 1
           for (i = 0; i < len - length(x); i++) {
               print " "
           }
           print x
           if (col != n) print " "
       }
       print "\n"
   }
   scale = old_scale

}

temp = print_magic_square(5)</lang>

Output:
Magic constant for n=5: 65
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

C

Generates an associative magic square. If the size is larger than 3, the square is also panmagic. <lang c>#include <stdio.h>

  1. include <stdlib.h>

int f(int n, int x, int y) { return (x + y*2 + 1)%n; }

int main(int argc, char **argv) { int i, j, n = atoi(argv[1]); for (i = 0; i < n; i++) { for (j = 0; j < n; j++) printf("% 4d", f(n, n - j - 1, i)*n + f(n, j, i) + 1); putchar('\n'); }

return 0; }</lang>

Output:
% ./a.out 5
   2  23  19  15   6
  14  10   1  22  18
  21  17  13   9   5
   8   4  25  16  12
  20  11   7   3  24

C++

<lang cpp>

  1. include <iostream>
  2. include <sstream>
  3. include <iomanip>

using namespace std;

class magicSqr { public:

   magicSqr() { sqr = 0; }
   ~magicSqr() { if( sqr ) delete [] sqr; }
   void create( int d )
   {
       if( sqr ) delete [] sqr;
       if( !( d & 1 ) ) d++; sz = d;
       sqr = new int[sz * sz];
       memset( sqr, 0, sz * sz * sizeof( int ) );
       fillSqr();
   }
   void display()
   {
       cout << "Odd Magic Square: " << sz << " x " << sz << "\n";
       cout << "It's Magic Sum is: " << magicNumber() << "\n\n";
       ostringstream cvr; cvr << sz * sz;
       int l = cvr.str().size();

for( int y = 0; y < sz; y++ ) { int yy = y * sz; for( int x = 0; x < sz; x++ ) cout << setw( l + 2 ) << sqr[yy + x];

cout << "\n"; }

       cout << "\n\n";
   }

private:

   void fillSqr()
   {

int sx = sz / 2, sy = 0, c = 0; while( c < sz * sz ) { if( !sqr[sx + sy * sz] ) { sqr[sx + sy * sz]= c + 1; inc( sx ); dec( sy ); c++; } else { dec( sx ); inc( sy ); inc( sy ); } }

   }
   int magicNumber()
   { return ( ( ( sz * sz + 1 ) / 2 ) * sz ); }
   void inc( int& a )
   { if( ++a == sz ) a = 0; }
   void dec( int& a )
   { if( --a < 0 ) a = sz - 1; }
   bool checkPos( int x, int y )
   { return( isInside( x ) && isInside( y ) && !sqr[sz * y + x] ); }
   bool isInside( int s )
   { return ( s < sz && s > -1 ); }
   int* sqr;
   int sz;

};

int main( int argc, char* argv[] ) {

   magicSqr s;
   s.create( 5 );
   s.display();
   return system( "pause" );

} </lang>

Output:
Odd Magic Square: 5 x 5
It's Magic Sum is: 65

  17  24   1   8  15
  23   5   7  14  16
   4   6  13  20  22
  10  12  19  21   3
  11  18  25   2   9

Odd Magic Square: 7 x 7
It's Magic Sum is: 175

  30  39  48   1  10  19  28
  38  47   7   9  18  27  29
  46   6   8  17  26  35  37
   5  14  16  25  34  36  45
  13  15  24  33  42  44   4
  21  23  32  41  43   3  12
  22  31  40  49   2  11  20

Common Lisp

<lang lisp>(defun magic-square (n)

 (loop for i from 1 to n
       collect 
         (loop for j from 1 to n
               collect 
                 (+ (* n (mod (+ i j (floor n 2) -1) 
                              n)) 
                    (mod (+ i (* 2 j) -2) 
                         n) 
                    1))))

(defun magic-constant (n)

 (* n 
    (/ (1+ (* n n)) 
       2)))

(defun output (n)

 (format T "Magic constant for n=~a: ~a~%" n (magic-constant n))
 (let* ((size (length (write-to-string (* n n))))
        (format-str (format NIL "~~{~~{~~~ad~~^ ~~}~~%~~}~~%" size)))
   (format T format-str (magic-square n))))</lang>
Output:
> (output 5)
Magic constant for n=5: 65
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

D

Translation of: Python

<lang d>void main(in string[] args) {

   import std.stdio, std.conv, std.range, std.algorithm,std.exception;
   immutable n = args.length == 2 ? args[1].to!uint : 5;
   enforce(n > 0 && n % 2 == 1, "Only odd n > 1");
   immutable len = text(n ^^ 2).length.text;
   foreach (immutable r; 1 .. n + 1)
       writefln("%-(%" ~ len ~ "d %)",
                iota(1, n + 1)
                .map!(c => n * ((r + c - 1 + n / 2) % n) +
                           ((r + 2 * c - 2) % n)));
   writeln("\nMagic constant: ", (n * n + 1) * n / 2);

}</lang>

Output:
16 23  0  7 14
22  4  6 13 15
 3  5 12 19 21
 9 11 18 20  2
10 17 24  1  8

Magic constant: 65

Alternative Version

Translation of: C

<lang d>import std.stdio, std.conv, std.string, std.range, std.algorithm;

uint[][] magicSquare(immutable uint n) pure nothrow @safe in {

   assert(n > 0 && n % 2 == 1);

} out(mat) {

   // mat is square of the right size.
   assert(mat.length == n);
   assert(mat.all!(row => row.length == n));
   immutable magic = mat[0].sum;
   // The sum of all rows is the same magic number.
   assert(mat.all!(row => row.sum == magic));
   // The sum of all columns is the same magic number.
   //assert(mat.transposed.all!(col => col.sum == magic));
   assert(mat.dup.transposed.all!(col => col.sum == magic));
   // The sum of the main diagonals is the same magic number.
   assert(mat.enumerate.map!(ir => ir[1][ir[0]]).sum == magic);
   //assert(mat.enumerate.map!({i, r} => r[i]).sum == magic);
   assert(mat.enumerate.map!(ir => ir[1][ir[0]]).sum == magic);

} body {

   enum M = (in uint x) pure nothrow @safe @nogc => (x + n - 1) % n;
   auto m = new uint[][](n, n);
   uint i = 0;
   uint j = n / 2;
   foreach (immutable uint k; 1 .. n ^^ 2 + 1) {
       m[i][j] = k;
       if (m[M(i)][M(j)]) {
           i = (i + 1) % n;
       } else {
           i = M(i);
           j = M(j);
       }
   }
   return m;

}

void showSquare(in uint[][] m) in {

   assert(m.all!(row => row.length == m[0].length));

} body {

   immutable maxLen = text(m.length ^^ 2).length.text;
   writefln("%(%(%" ~ maxLen ~ "d %)\n%)", m);
   writeln("\nMagic constant: ", m[0].sum);

}

int main(in string[] args) {

   if (args.length == 1) {
       5.magicSquare.showSquare;
       return 0;
   } else if (args.length == 2) {
       immutable n = args[1].to!uint;
       if (n > 0 && n % 2 == 1) {
           n.magicSquare.showSquare;
           return 0;
       }
   }
   stderr.writefln("Requires n odd and larger than 0.");
   return 1;

}</lang>

Output:
15  8  1 24 17
16 14  7  5 23
22 20 13  6  4
 3 21 19 12 10
 9  2 25 18 11

Magic constant: 65

Elixir

Translation of: Ruby

<lang elixir>defmodule RC do

 require Integer
 def odd_magic_square(n) when Integer.is_odd(n) do
   for i <- 0..n-1 do
     for j <- 0..n-1 do
       n * rem(i+j+1+div(n,2),n) + rem(i+2*j+2*n-5,n) + 1
     end
   end
 end

end

Enum.each([3,5,9], fn n ->

 IO.puts "\nSize #{n}, magic sum #{div(n*n+1,2)*n}"
 Enum.each(RC.odd_magic_square(n), fn x -> IO.inspect x end)

end)</lang>

Output:
Size 3, magic sum 15
[8, 1, 6]
[3, 5, 7]
[4, 9, 2]

Size 5, magic sum 65
[16, 23, 5, 7, 14]
[22, 4, 6, 13, 20]
[3, 10, 12, 19, 21]
[9, 11, 18, 25, 2]
[15, 17, 24, 1, 8]

Size 9, magic sum 369
[50, 61, 72, 74, 4, 15, 26, 28, 39]
[60, 71, 73, 3, 14, 25, 36, 38, 49]
[70, 81, 2, 13, 24, 35, 37, 48, 59]
[80, 1, 12, 23, 34, 45, 47, 58, 69]
[9, 11, 22, 33, 44, 46, 57, 68, 79]
[10, 21, 32, 43, 54, 56, 67, 78, 8]
[20, 31, 42, 53, 55, 66, 77, 7, 18]
[30, 41, 52, 63, 65, 76, 6, 17, 19]
[40, 51, 62, 64, 75, 5, 16, 27, 29]

Fortran

Works with: Fortran version 95 and later

<lang fortran>program Magic_Square

 implicit none
 integer, parameter :: order = 15
 integer :: i, j

 write(*, "(a, i0)") "Magic Square Order: ", order
 write(*, "(a)")     "----------------------"
 do i = 1, order
   do j = 1, order
     write(*, "(i4)", advance = "no") f1(order, i, j)
   end do
   write(*,*)
 end do
 write(*, "(a, i0)") "Magic number = ", f2(order)

contains

integer function f1(n, x, y)

 integer, intent(in) :: n, x, y
 f1 = n * mod(x + y - 1 + n/2, n) + mod(x + 2*y - 2, n) + 1

end function

integer function f2(n)

 integer, intent(in) :: n
 f2 = n * (1 + n * n) / 2

end function end program</lang> Output:

Magic Square Order: 15
----------------------
 122 139 156 173 190 207 224   1  18  35  52  69  86 103 120 
 138 155 172 189 206 223  15  17  34  51  68  85 102 119 121 
 154 171 188 205 222  14  16  33  50  67  84 101 118 135 137 
 170 187 204 221  13  30  32  49  66  83 100 117 134 136 153 
 186 203 220  12  29  31  48  65  82  99 116 133 150 152 169 
 202 219  11  28  45  47  64  81  98 115 132 149 151 168 185 
 218  10  27  44  46  63  80  97 114 131 148 165 167 184 201 
   9  26  43  60  62  79  96 113 130 147 164 166 183 200 217 
  25  42  59  61  78  95 112 129 146 163 180 182 199 216   8 
  41  58  75  77  94 111 128 145 162 179 181 198 215   7  24 
  57  74  76  93 110 127 144 161 178 195 197 214   6  23  40 
  73  90  92 109 126 143 160 177 194 196 213   5  22  39  56 
  89  91 108 125 142 159 176 193 210 212   4  21  38  55  72 
 105 107 124 141 158 175 192 209 211   3  20  37  54  71  88 
 106 123 140 157 174 191 208 225   2  19  36  53  70  87 104 
Magic number = 1695

FreeBASIC

<lang FreeBASIC>' version 23-06-2015 ' compile with: fbc -s console

Sub magicsq(size As Integer, filename As String ="")

   If (size And 1) = 0 Or size < 3 Then
       Print : Beep ' alert
       Print "error: size is not odd or size is smaller then 3"
       Sleep 3000,1  'wait 3 seconds, ignore keypress
       Exit Sub
   End If
   ' filename <> "" then save magic square in a file
   ' filename can contain directory name
   ' if filename exist it will be overwriten, no error checking
   Dim As Integer sq(size,size) ' array to hold square
   ' start in the middle of the first row
   Dim As Integer nr = 1, x = size - (size \ 2), y = 1
   Dim As Integer max = size * size
   ' create format string for using
   Dim As String frmt = String(Len(Str(max)) +1, "#")
   ' main loop for creating magic square
   Do
       If sq(x, y) = 0 Then
           sq(x, y) = nr
           If nr Mod size = 0 Then
               y += 1
           Else
               x += 1
               y -= 1
           End If
           nr += 1
       End If
       If x > size Then
           x = 1
           Do While sq(x,y) <> 0
               x += 1
           Loop
       End If
       If y < 1 Then
           y = size
           Do While sq(x,y) <> 0
               y -= 1
           Loop
       EndIf
   Loop Until nr > max
   ' printing square's bigger than 19 result in a wrapping of the line
   Print "Odd magic square size:"; size; " *"; size
   Print "The magic sum ="; ((max +1) \ 2) * size
   Print
   For y = 1 To size
       For x = 1 To size
           Print Using frmt; sq(x,y);
       Next
       Print
   Next
   print
   ' output magic square to a file with the name provided
   If filename <> "" Then
       nr = FreeFile
       Open filename For Output As #nr
       Print #nr, "Odd magic square size:"; size; " *"; size
       Print #nr, "The magic sum ="; ((max +1) \ 2) * size
       Print #nr,
       For y = 1 To size
           For x = 1 To size
               Print #nr, Using frmt; sq(x,y);
           Next
           Print #nr,
       Next
   End If

End Sub

' ------=< MAIN >=------

magicsq(5) magicsq(11) ' the next line will also print the square to a file called: magic_square_19.txt magicsq(19, "magic_square_19.txt")


' empty keyboard buffer While Inkey <> "" : Var _key_ = Inkey : Wend Print : Print "hit any key to end program" Sleep End</lang>

Output:
Odd magic square size: 5 * 5        Odd magic square size: 11 * 11
The magic sum = 65                  The magic sum = 671           
             
 17 24  1  8 15                       68  81  94 107 120   1  14  27  40  53  66
 23  5  7 14 16                       80  93 106 119  11  13  26  39  52  65  67
  4  6 13 20 22                       92 105 118  10  12  25  38  51  64  77  79
 10 12 19 21  3                      104 117   9  22  24  37  50  63  76  78  91
 11 18 25  2  9                      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
Only the first 2 square shown.        56  69  82  95 108 121   2  15  28  41  54

Go

Translation of: C

<lang go>package main

import (

   "fmt"
   "log"

)

func ms(n int) (int, []int) {

   M := func(x int) int { return (x + n - 1) % n }
   if n <= 0 || n&1 == 0 {
       n = 5
       log.Println("forcing size", n)
   }
   m := make([]int, n*n)
   i, j := 0, n/2
   for k := 1; k <= n*n; k++ {
       m[i*n+j] = k
       if m[M(i)*n+M(j)] != 0 {
           i = (i + 1) % n
       } else {
           i, j = M(i), M(j)
       }
   }
   return n, m

}

func main() {

   n, m := ms(5)
   i := 2
   for j := 1; j <= n*n; j *= 10 {
       i++
   }
   f := fmt.Sprintf("%%%dd", i)
   for i := 0; i < n; i++ {
       for j := 0; j < n; j++ {
           fmt.Printf(f, m[i*n+j])
       }
       fmt.Println()
   }

}</lang>

Output:
  15   8   1  24  17
  16  14   7   5  23
  22  20  13   6   4
   3  21  19  12  10
   9   2  25  18  11

Haskell

Translation of: cpp

<lang haskell> -- as a translation from imperative code, this is probably not a "good" implementation import Data.List

type Var = (Int, Int, Int, Int) -- sx sy sz c

magicSum :: Int -> Int magicSum x = ((x * x + 1) `div` 2) * x

wrapInc :: Int -> Int -> Int wrapInc max x

  | x + 1 == max    = 0
  | otherwise       = x + 1

wrapDec :: Int -> Int -> Int wrapDec max x

  | x == 0    = max - 1
  | otherwise = x - 1

isZero :: Int -> Int -> Int -> Bool isZero m x y = m !! x !! y == 0

setAt :: (Int,Int) -> Int -> Int -> Int setAt (x, y) val table

  | (upper, current : lower) <- splitAt x table,
    (left, this : right) <- splitAt y current
        = upper ++ (left ++ val : right) : lower
  | otherwise = error "Outside"

create :: Int -> Int create x = replicate x $ replicate x 0

cells :: Int -> Int cells m = x*x where x = length m

fill :: Var -> Int -> Int fill (sx, sy, sz, c) m

  | c < cells m =
     if isZero m sx sy 
     then fill ((wrapInc sz sx), (wrapDec sz sy), sz, c + 1) (setAt (sx, sy) (c + 1) m)
     else fill ((wrapDec sz sx), (wrapInc sz(wrapInc sz sy)), sz, c) m
  | otherwise = m

magicNumber :: Int -> Int magicNumber d = transpose $ fill (d `div` 2, 0, d, 0) (create d)

display :: Int -> String display (x:xs)

  | null xs = vdisplay x
  | otherwise = vdisplay x ++ ('\n' : display xs)

vdisplay :: [Int] -> String vdisplay (x:xs)

  | null xs = show x
  | otherwise = show x ++ " " ++ vdisplay xs


magicSquare x = do

  putStr "Magic Square of "
  putStr $ show x
  putStr " = "
  putStrLn $ show $ magicSum x
  putStrLn $ display $ magicNumber x

</lang>

Icon and Unicon

This is a Unicon-specific solution because of the use of the [: ... :] construct. <lang unicon>procedure main(A)

   n := integer(!A) | 3
   write("Magic number: ",n*(n*n+1)/2)
   sq := buildSquare(n)
   showSquare(sq)

end

procedure buildSquare(n)

   sq := [: |list(n)\n :]
   r := 0
   c := n/2
   every i := !(n*n) do {
       /sq[r+1,c+1] := i
       nr := (n+r-1)%n
       nc := (c+1)%n
       if /sq[nr+1,nc+1] then (r := nr,c := nc) else r := (r+1)%n
       }
   return sq

end

procedure showSquare(sq)

   n := *sq
   s := *(n*n)+2
   every r := !sq do every writes(right(!r,s)|"\n")

end</lang>

Output:
->ms 5
Magic number: 65
  17  24   1   8  15
  23   5   7  14  16
   4   6  13  20  22
  10  12  19  21   3
  11  18  25   2   9
->

J

Based on http://www.jsoftware.com/papers/eem/magicsq.htm

<lang J>ms=: i:@<.@-: |."0 1&|:^:2 >:@i.@,~</lang>

In other words, generate a square of counting integers, like this: <lang J> >:@i.@,~ 3 1 2 3 4 5 6 7 8 9</lang>

Then generate a list of integers centering on 0 up to half of that value, like this: <lang J> i:@<.@-: 3 _1 0 1</lang>

Finally, rotate each corresponding row and column of the table by the corresponding value in the list. We can use the same instructions to rotate both rows and columns if we transpose the matrix before rotating (and perform this transpose+rotate twice).

Example use:

<lang J> ms 5

9 15 16 22  3

20 21 2 8 14

1  7 13 19 25

12 18 24 5 6 23 4 10 11 17

  ~.+/ms 5

65

  ~.+/ms 101

515201</lang>

Java

<lang java>public class MagicSquare {

   public static void main(String[] args) {
       int n = 5;
       for (int[] row : magicSquareOdd(n)) {
           for (int x : row)
               System.out.format("%2s ", x);
           System.out.println();
       }
       System.out.printf("\nMagic constant: %d ", (n * n + 1) * n / 2);
   }
   public static int[][] magicSquareOdd(final int base) {
       if (base % 2 == 0 || base < 3)
           throw new IllegalArgumentException("base must be odd and > 2");
       int[][] grid = new int[base][base];
       int r = 0, number = 0;
       int size = base * base;
       int c = base / 2;
       while (number++ < size) {
           grid[r][c] = number;
           if (r == 0) {
               if (c == base - 1) {
                   r++;
               } else {
                   r = base - 1;
                   c++;
               }
           } else {
               if (c == base - 1) {
                   r--;
                   c = 0;
               } else {
                   if (grid[r - 1][c + 1] == 0) {
                       r--;
                       c++;
                   } else {
                       r++;
                   }
               }
           }
       }
       return grid;
   }

}</lang>

Output:
17 24  1  8 15 
23  5  7 14 16 
 4  6 13 20 22 
10 12 19 21  3 
11 18 25  2  9

Magic constant: 65 

jq

Adapted from #AWK <lang jq>def odd_magic_square:

 if type != "number" or . % 2 == 0 or . <= 0
 then error("odd_magic_square requires an odd positive integer")
 else
   . as $n
   | reduce range(1; 1 + ($n*$n)) as $i
        ( [0, (($n-1)/2), []];
	  .[0] as $x | .[1] as $y
         | .[2]

| setpath([$x, $y]; $i )

         | if getpath([(($x+$n-1) % $n), (($y+$n+1) % $n)])
           then [(($x+$n+1) % $n), $y, .]
           else [ (($x+$n-1) % $n), (($y+$n+1) % $n), .]
	    end )  | .[2]
 end ;</lang>

Examples <lang jq>def task:

 def pp: if length == 0 then empty
         else "\(.[0])", (.[1:] | pp )
         end;
 "The magic sum for a square of size \(.) is \( (.*. + 1)*./2 ):",
   (odd_magic_square | pp)

(3, 5, 9) | task</lang>

Output:

<lang sh>$ jq -n -r -M -c -f odd_magic_square.jq The magic sum for a square of size 3 is 15: [8,1,6] [3,5,7] [4,9,2] The magic sum for a square of size 5 is 65: [17,24,1,8,15] [23,5,7,14,16] [4,6,13,20,22] [10,12,19,21,3] [11,18,25,2,9] The magic sum for a square of size 9 is 369: [47,58,69,80,1,12,23,34,45] [57,68,79,9,11,22,33,44,46] [67,78,8,10,21,32,43,54,56] [77,7,18,20,31,42,53,55,66] [6,17,19,30,41,52,63,65,76] [16,27,29,40,51,62,64,75,5] [26,28,39,50,61,72,74,4,15] [36,38,49,60,71,73,3,14,25] [37,48,59,70,81,2,13,24,35]</lang>

Liberty BASIC

<lang lb> Dim m(1,1)

Call magicSquare 5 Call magicSquare 17

End

Sub magicSquare n

   ReDim m(n,n)
   inc = 1
   count = 1
   row = 1
   col=(n+1)/2
   While count <= n*n
       m(row,col) = count
       count = count + 1
       If inc < n Then
           inc = inc + 1
           row = row - 1
           col = col + 1
           If row <> 0 Then
               If col > n Then col = 1
           Else
               row = n
           End If
       Else
           inc = 1
           row = row + 1
       End If
   Wend
   Call printSquare n

End Sub

Sub printSquare n

   'Arbitrary limit to fit width of A4 paper
   If n < 23 Then
       Print n;" x ";n;" Magic Square --- ";
       Print "Magic constant is ";Int((n*n+1)/2*n)
       For row = 1 To n
           For col = 1 To n
               Print Using("####",m(row,col));
           Next col
           Print
           Print
       Next row
   Else
       Notice "Magic Square will not fit on one sheet of paper."
   End If

End Sub </lang>

Output:
5 x 5 Magic Square --- Magic constant is 65
  17  24   1   8  15

  23   5   7  14  16

   4   6  13  20  22

  10  12  19  21   3

  11  18  25   2   9

17 x 17 Magic Square --- Magic constant is 2465
 155 174 193 212 231 250 269 288   1  20  39  58  77  96 115 134 153

 173 192 211 230 249 268 287  17  19  38  57  76  95 114 133 152 154

 191 210 229 248 267 286  16  18  37  56  75  94 113 132 151 170 172

 209 228 247 266 285  15  34  36  55  74  93 112 131 150 169 171 190

 227 246 265 284  14  33  35  54  73  92 111 130 149 168 187 189 208

 245 264 283  13  32  51  53  72  91 110 129 148 167 186 188 207 226

 263 282  12  31  50  52  71  90 109 128 147 166 185 204 206 225 244

 281  11  30  49  68  70  89 108 127 146 165 184 203 205 224 243 262

  10  29  48  67  69  88 107 126 145 164 183 202 221 223 242 261 280

  28  47  66  85  87 106 125 144 163 182 201 220 222 241 260 279   9

  46  65  84  86 105 124 143 162 181 200 219 238 240 259 278   8  27

  64  83 102 104 123 142 161 180 199 218 237 239 258 277   7  26  45

  82 101 103 122 141 160 179 198 217 236 255 257 276   6  25  44  63

 100 119 121 140 159 178 197 216 235 254 256 275   5  24  43  62  81

 118 120 139 158 177 196 215 234 253 272 274   4  23  42  61  80  99

 136 138 157 176 195 214 233 252 271 273   3  22  41  60  79  98 117

 137 156 175 194 213 232 251 270 289   2  21  40  59  78  97 116 135

Mathematica

Rotate rows and columns of the initial matrix with rows filled in order 1 2 3 .... N^2

Method from http://www.jsoftware.com/papers/eem/magicsq.htm

<lang Mathematica> rp[v_, pos_] := RotateRight[v, (Length[v] + 1)/2 - pos]; rho[m_] := MapIndexed[rp, m]; magic[n_] :=

 rho[Transpose[rho[Table[i*n + j, {i, 0, n - 1}, {j, 1, n}]]]];

square = magic[11] // Grid Print["Magic number is ", Total[square1, 1]] </lang>

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

Magic number is 671

Output from code that checks the results Rows

{671,671,671,671,671,671,671,671,671,671,671}

Columns

{671,671,671,671,671,671,671,671,671,671,671}

Diagonals

671

671

Maxima

<lang Maxima>wrap1(i):= if i>%n% then 1 else if i<1 then %n% else i; wrap(P):=maplist('wrap1, P);

uprigth(P):= wrap(P + [-1, 1]); down(P):= wrap(P + [1, 0]);

magic(n):=block([%n%: n,

 M: zeromatrix (n, n),
 P: [1, (n + 1)/2],
 m: 1, Pc],
 do (
   M[P[1],P[2]]: m,
   m: m + 1,
   if m>n^2 then return(M),
   Pc: uprigth(P),
   if M[Pc[1],Pc[2]]=0 then P: Pc
   else while(M[P[1],P[2]]#0) do P: down(P)));</lang>

Usage: <lang output>(%i6) magic(3);

                                 [ 8  1  6 ]
                                 [         ]

(%o6) [ 3 5 7 ]

                                 [         ]
                                 [ 4  9  2 ]

(%i7) magic(5);

                           [ 17  24  1   8   15 ]
                           [                    ]
                           [ 23  5   7   14  16 ]
                           [                    ]

(%o7) [ 4 6 13 20 22 ]

                           [                    ]
                           [ 10  12  19  21  3  ]
                           [                    ]
                           [ 11  18  25  2   9  ]

(%i8) magic(7);

                       [ 30  39  48  1   10  19  28 ]
                       [                            ]
                       [ 38  47  7   9   18  27  29 ]
                       [                            ]
                       [ 46  6   8   17  26  35  37 ]
                       [                            ]

(%o8) [ 5 14 16 25 34 36 45 ]

                       [                            ]
                       [ 13  15  24  33  42  44  4  ]
                       [                            ]
                       [ 21  23  32  41  43  3   12 ]
                       [                            ]
                       [ 22  31  40  49  2   11  20 ]

/* magic number for n=7 */ (%i9) lsum(q, q, first(magic(7))); (%o9) 175</lang>

Nim

Translation of: Python

<lang nim>import strutils

proc `^`*(base: int, exp: int): int =

 var (base, exp) = (base, exp)
 result = 1
 while exp != 0:
   if (exp and 1) != 0:
     result *= base
   exp = exp shr 1
   base *= base

proc magic(n) =

 for row in 1 .. n:
   for col in 1 .. n:
     let cell = (n * ((row + col - 1 + n div 2) mod n) +
                 ((row + 2 * col - 2) mod n) + 1)
     stdout.write align($cell, len($(n^2)))," "
   echo ""
 echo "\nAll sum to magic number ", ((n * n + 1) * n div 2)

for n in [5, 3, 7]:

 echo "\nOrder ",n,"\n======="
 magic(n)</lang>
Output:
Order 5
=======
17 24  1  8 15 
23  5  7 14 16 
 4  6 13 20 22 
10 12 19 21  3 
11 18 25  2  9 

All sum to magic number 65

Order 3
=======
8 1 6 
3 5 7 
4 9 2 

All sum to magic number 15

Order 7
=======
30 39 48  1 10 19 28 
38 47  7  9 18 27 29 
46  6  8 17 26 35 37 
 5 14 16 25 34 36 45 
13 15 24 33 42 44  4 
21 23 32 41 43  3 12 
22 31 40 49  2 11 20 

All sum to magic number 175

Oforth

<lang Oforth>func: magicSquare(n) { | i j wd |

  n sq log asInteger 1 + ->wd
  n loop: i [
     n loop: j [
        i j + 1 - n 2 / + n mod n *
        i j + j + 2 - n mod 1 + + 
        System.Out swap <<w(wd) " " << drop 
        ]
     printcr
     ] 
  System.Out "Magic constant is : " << n sq 1 + 2 / n * << cr

}</lang>

Output:
magicSquare(5)
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9
Magic constant is : 65

Perl 6

<lang perl6>sub MAIN (Int $n = 5) {

   note "Sorry, must be a positive odd integer." and exit if $n %% 2 or $n < 0;
   my $x = $n/2;
   my $y = 0;
   my $i = 1;
   my @sq;
   @sq[($i % $n ?? $y-- !! $y++) % $n][($i % $n ?? $x++ !! $x) % $n] = $i++ for ^($n * $n);
   my $f = "%{$i.chars}d";
   say .fmt($f, ' ') for @sq;
   say "\nThe magic number is ", [+] @sq[0].list;

}</lang>

Output:

Default, No parameter:

17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

The magic number is 65

With a parameter of 19

192 213 234 255 276 297 318 339 360   1  22  43  64  85 106 127 148 169 190
212 233 254 275 296 317 338 359  19  21  42  63  84 105 126 147 168 189 191
232 253 274 295 316 337 358  18  20  41  62  83 104 125 146 167 188 209 211
252 273 294 315 336 357  17  38  40  61  82 103 124 145 166 187 208 210 231
272 293 314 335 356  16  37  39  60  81 102 123 144 165 186 207 228 230 251
292 313 334 355  15  36  57  59  80 101 122 143 164 185 206 227 229 250 271
312 333 354  14  35  56  58  79 100 121 142 163 184 205 226 247 249 270 291
332 353  13  34  55  76  78  99 120 141 162 183 204 225 246 248 269 290 311
352  12  33  54  75  77  98 119 140 161 182 203 224 245 266 268 289 310 331
 11  32  53  74  95  97 118 139 160 181 202 223 244 265 267 288 309 330 351
 31  52  73  94  96 117 138 159 180 201 222 243 264 285 287 308 329 350  10
 51  72  93 114 116 137 158 179 200 221 242 263 284 286 307 328 349   9  30
 71  92 113 115 136 157 178 199 220 241 262 283 304 306 327 348   8  29  50
 91 112 133 135 156 177 198 219 240 261 282 303 305 326 347   7  28  49  70
111 132 134 155 176 197 218 239 260 281 302 323 325 346   6  27  48  69  90
131 152 154 175 196 217 238 259 280 301 322 324 345   5  26  47  68  89 110
151 153 174 195 216 237 258 279 300 321 342 344   4  25  46  67  88 109 130
171 173 194 215 236 257 278 299 320 341 343   3  24  45  66  87 108 129 150
172 193 214 235 256 277 298 319 340 361   2  23  44  65  86 107 128 149 170

The magic number is 3439

PL/I

<lang PL/I>magic: procedure options (main); /* 18 April 2014 */

  declare n fixed binary;
  put skip list ('What is the order of the magic square?');
  get list (n);
  if n < 3 | iand(n, 1) = 0 then
     do; put skip list ('The value is out of range'); stop; end;
  put skip list ('The order is ' || trim(n));
  begin;
     declare m(n, n) fixed, (i, j, k) fixed binary;
     on subrg snap put data (i, j, k);
     m = 0;
     i = 1; j = (n+1)/2;
     do k = 1 to n*n;
        if m(i,j) = 0 then
           m(i,j) = k;
        else
           do;
              i = i + 2; j = j + 1;
              if i > n then i = mod(i,n);
              if j > n then j = 1;
              m(i,j) = k;
           end;
        i = i - 1; j = j - 1;
        if i < 1 then i = n;
        if j < 1 then j = n;
     end;
     do i = 1 to n;
        put skip edit (m(i, *)) (f(4));
     end;
     put skip list ('The magic number is' || sum(m(1,*)));
  end;

end magic;</lang>

Output:
What is the order of the magic square? 

The order is 5 
  15   8   1  24  17
  16  14   7   5  23
  22  20  13   6   4
   3  21  19  12  10
   9   2  25  18  11
The magic number is                65 
What is the order of the magic square? 

The order is 7 
  28  19  10   1  48  39  30
  29  27  18   9   7  47  38
  37  35  26  17   8   6  46
  45  36  34  25  16  14   5
   4  44  42  33  24  15  13
  12   3  43  41  32  23  21
  20  11   2  49  40  31  22
The magic number is               175

Python

<lang python>>>> def magic(n):

   for row in range(1, n + 1):
       print(' '.join('%*i' % (len(str(n**2)), cell) for cell in
                      (n * ((row + col - 1 + n // 2) % n) +
                      ((row + 2 * col - 2) % n) + 1
                      for col in range(1, n + 1))))
   print('\nAll sum to magic number %i' % ((n * n + 1) * n // 2))


>>> for n in (5, 3, 7): print('\nOrder %i\n=======' % n) magic(n)


Order 5

=

17 24 1 8 15 23 5 7 14 16

4  6 13 20 22

10 12 19 21 3 11 18 25 2 9

All sum to magic number 65

Order 3

=

8 1 6 3 5 7 4 9 2

All sum to magic number 15

Order 7

=

30 39 48 1 10 19 28 38 47 7 9 18 27 29 46 6 8 17 26 35 37

5 14 16 25 34 36 45

13 15 24 33 42 44 4 21 23 32 41 43 3 12 22 31 40 49 2 11 20

All sum to magic number 175 >>> </lang>

Racket

<lang racket>#lang racket

Using "helpful formulae" in
http://en.wikipedia.org/wiki/Magic_square#Method_for_constructing_a_magic_square_of_odd_order

(define (squares n) n)

(define (last-no n) (sqr n))

(define (middle-no n) (/ (add1 (sqr n)) 2))

(define (M n) (* n (middle-no n)))

(define ((Ith-row-Jth-col n) I J)

 (+ (* (modulo (+ I J -1 (exact-floor (/ n 2))) n) n)
    (modulo (+ I (* 2 J) -2) n)
    1))

(define (magic-square n)

 (define IrJc (Ith-row-Jth-col n))
 (for/list ((I (in-range 1 (add1 n)))) (for/list ((J (in-range 1 (add1 n)))) (IrJc I J))))

(define (fmt-list-of-lists l-o-l width)

 (string-join
  (for/list ((row l-o-l))
    (string-join (map (λ (x) (~a #:align 'right #:width width x)) row) "  "))
  "\n"))

(define (show-magic-square n)

 (format "MAGIC SQUARE ORDER:~a~%~a~%MAGIC NUMBER:~a~%"
         n (fmt-list-of-lists (magic-square n) (+ (order-of-magnitude (last-no n)) 1)) (M n)))

(displayln (show-magic-square 3)) (displayln (show-magic-square 5)) (displayln (show-magic-square 9))</lang>

Output:
MAGIC SQUARE ORDER:3
8  1  6
3  5  7
4  9  2
Magic Number:15

MAGIC SQUARE ORDER:5
17  24   1   8  15
23   5   7  14  16
 4   6  13  20  22
10  12  19  21   3
11  18  25   2   9
Magic Number:65

MAGIC SQUARE ORDER:9
47  58  69  80   1  12  23  34  45
57  68  79   9  11  22  33  44  46
67  78   8  10  21  32  43  54  56
77   7  18  20  31  42  53  55  66
 6  17  19  30  41  52  63  65  76
16  27  29  40  51  62  64  75   5
26  28  39  50  61  72  74   4  15
36  38  49  60  71  73   3  14  25
37  48  59  70  81   2  13  24  35
Magic Number:369

REXX

<lang rexx>/*REXX program generates and displays true magic squares (for odd N). */ parse arg N .; if N== then N=5 /*matrix size ¬given? Use default*/ w=length(N*N); r=2; c=(n+1)%2-1 /*define initial row and column. */ @.=. /* [↓] uses the Siamese method.*/

   do j=1  for n*n;   br=r==N & c==N; r=r-1;  c=c+1   /*BR=bottom right*/
   if r<1 & c>N then do;  r=r+2;  c=c-1;    end       /*R under, C over*/
   if r<1       then r=n; if r>n  then r=1; if c>n then c=1  /*overflow*/
   if @.r.c\==. then do; r=r+2; c=c-1; if br then do; r=N; c=c+1; end;end
   @.r.c=j                            /*assign #───►square matrix cell.*/
   end   /*j*/                        /* [↑]  can handle even N matrix.*/
                                      /* [↓]  displays (aligned) matrix*/
      do   r=1  for N;  _=            /*display 1 matrix row at a time.*/
        do c=1  for N;  _=_ right(@.r.c, w);  end  /*c*/    /*build row*/
      say substr(_,2)                 /*row has an extra leading blank.*/
      end   /*c*/                     /* [↑]   also right-justified #s.*/

say /*might as well show a blank line*/ if N//2 then say 'The magic number (or magic constant is): ' N*(n*n+1)%2

                                      /*stick a fork in it, we're done.*/</lang>
Output:
using the default input of   5
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

The magic number  (or magic constant is):  65
Output:
using the default input of   3
8 1 6
3 5 7
4 9 2

The magic number  (or magic constant is):  15
Output:
using the input of
  19:
192 213 234 255 276 297 318 339 360   1  22  43  64  85 106 127 148 169 190
212 233 254 275 296 317 338 359  19  21  42  63  84 105 126 147 168 189 191
232 253 274 295 316 337 358  18  20  41  62  83 104 125 146 167 188 209 211
252 273 294 315 336 357  17  38  40  61  82 103 124 145 166 187 208 210 231
272 293 314 335 356  16  37  39  60  81 102 123 144 165 186 207 228 230 251
292 313 334 355  15  36  57  59  80 101 122 143 164 185 206 227 229 250 271
312 333 354  14  35  56  58  79 100 121 142 163 184 205 226 247 249 270 291
332 353  13  34  55  76  78  99 120 141 162 183 204 225 246 248 269 290 311
352  12  33  54  75  77  98 119 140 161 182 203 224 245 266 268 289 310 331
 11  32  53  74  95  97 118 139 160 181 202 223 244 265 267 288 309 330 351
 31  52  73  94  96 117 138 159 180 201 222 243 264 285 287 308 329 350  10
 51  72  93 114 116 137 158 179 200 221 242 263 284 286 307 328 349   9  30
 71  92 113 115 136 157 178 199 220 241 262 283 304 306 327 348   8  29  50
 91 112 133 135 156 177 198 219 240 261 282 303 305 326 347   7  28  49  70
111 132 134 155 176 197 218 239 260 281 302 323 325 346   6  27  48  69  90
131 152 154 175 196 217 238 259 280 301 322 324 345   5  26  47  68  89 110
151 153 174 195 216 237 258 279 300 321 342 344   4  25  46  67  88 109 130
171 173 194 215 236 257 278 299 320 341 343   3  24  45  66  87 108 129 150
172 193 214 235 256 277 298 319 340 361   2  23  44  65  86 107 128 149 170

The magic number  (or magic constant is):  3439

Ruby

<lang ruby>def odd_magic_square(n)

 raise ArgumentError "Need odd positive number" if n.even? || n <= 0
 n.times.map{|i| n.times.map{|j| n*((i+j+1+n/2)%n) + ((i+2*j-5)%n) + 1} }

end

[3, 5, 9].each do |n|

 puts "\nSize #{n}, magic sum #{(n*n+1)/2*n}"
 fmt = "%#{(n*n).to_s.size + 1}d" * n
 odd_magic_square(n).each{|row| puts fmt % row}

end </lang>

Output:
Size 3, magic sum 15
 8 1 6
 3 5 7
 4 9 2

Size 5, magic sum 65
 16 23  5  7 14
 22  4  6 13 20
  3 10 12 19 21
  9 11 18 25  2
 15 17 24  1  8

Size 9, magic sum 369
 50 61 72 74  4 15 26 28 39
 60 71 73  3 14 25 36 38 49
 70 81  2 13 24 35 37 48 59
 80  1 12 23 34 45 47 58 69
  9 11 22 33 44 46 57 68 79
 10 21 32 43 54 56 67 78  8
 20 31 42 53 55 66 77  7 18
 30 41 52 63 65 76  6 17 19
 40 51 62 64 75  5 16 27 29

Rust

<lang rust>fn main() {

   let n = 9;
   let mut square = Vec::from_fn(n, |_| Vec::from_fn(n, |_| 0u));
   for (i, row) in square.iter_mut().enumerate() {
       for (j, e) in row.iter_mut().enumerate() {
           *e = n * (((i + 1) + (j + 1) - 1 + (n >> 1)) % n) + (((i + 1) + (2 * (j + 1)) - 2) % n) + 1;
           print!("{:3} ", e);
       }
       println!("");
   }
   let sum = n * (((n * n) + 1) / 2);
   println!("The sum of the square is {}.", sum);

}</lang>

Output:
 47  58  69  80   1  12  23  34  45 
 57  68  79   9  11  22  33  44  46 
 67  78   8  10  21  32  43  54  56 
 77   7  18  20  31  42  53  55  66 
  6  17  19  30  41  52  63  65  76 
 16  27  29  40  51  62  64  75   5 
 26  28  39  50  61  72  74   4  15 
 36  38  49  60  71  73   3  14  25 
 37  48  59  70  81   2  13  24  35 
The sum of the square is 369.

Scala

<lang scala> def magicSquare( n:Int ) : Option[Array[Array[Int]]] = {

   require(n % 2 != 0, "n must be an odd number")
   val a = Array.ofDim[Int](n,n)
   // Make the horizontal by starting in the middle of the row and then taking a step back every n steps
   val ii = Iterator.continually(0 to n-1).flatten.drop(n/2).sliding(n,n-1).take(n*n*2).toList.flatten
   // Make the vertical component by moving up (subtracting 1) but every n-th step, step down (add 1)
   val jj = Iterator.continually(n-1 to 0 by -1).flatten.drop(n-1).sliding(n,n-2).take(n*n*2).toList.flatten
   // Combine the horizontal and vertical components to create the path
   val path = (ii zip jj) take (n*n)
   // Fill the array by following the path
   for( i<-1 to (n*n); p=path(i-1) ) { a(p._1)(p._2) = i }
   Some(a)
 }
 def output() :  Unit = {
   def printMagicSquare(n: Int): Unit = {
     val ms = magicSquare(n)
     val magicsum = (n * n + 1) / 2
     assert(
       if( ms.isDefined ) {
         val a = ms.get
         a.forall(_.sum == magicsum) &&
           a.transpose.forall(_.sum == magicsum) &&
           (for(i<-0 until n) yield { a(i)(i) }).sum == magicsum
       }
       else { false }
     )
     if( ms.isDefined ) {
       val a = ms.get
       for (y <- 0 to n * 2; x <- 0 until n) (x, y) match {
         case (0, 0) => print("╔════╤")
         case (i, 0) if i == n - 1 => print("════╗\n")
         case (i, 0) => print("════╤")
         case (0, j) if j % 2 != 0 => print("║ " + f"${ a(0)((j - 1) / 2) }%2d" + " │")
         case (i, j) if j % 2 != 0 && i == n - 1 => print(" " + f"${ a(i)((j - 1) / 2) }%2d" + " ║\n")
         case (i, j) if j % 2 != 0 => print(" " + f"${ a(i)((j - 1) / 2) }%2d" + " │")
         case (0, j) if j == (n * 2) => print("╚════╧")
         case (i, j) if j == (n * 2) && i == n - 1 => print("════╝\n")
         case (i, j) if j == (n * 2) => print("════╧")
         case (0, _) => print("╟────┼")
         case (i, _) if i == n - 1 => print("────╢\n")
         case (i, _) => print("────┼")
       }
     }
   }
   printMagicSquare(7)
 }</lang>
Output:
╔════╤════╤════╤════╤════╤════╤════╗
║ 30 │ 39 │ 48 │  1 │ 10 │ 19 │ 28 ║
╟────┼────┼────┼────┼────┼────┼────╢
║ 38 │ 47 │  7 │  9 │ 18 │ 27 │ 29 ║
╟────┼────┼────┼────┼────┼────┼────╢
║ 46 │  6 │  8 │ 17 │ 26 │ 35 │ 37 ║
╟────┼────┼────┼────┼────┼────┼────╢
║  5 │ 14 │ 16 │ 25 │ 34 │ 36 │ 45 ║
╟────┼────┼────┼────┼────┼────┼────╢
║ 13 │ 15 │ 24 │ 33 │ 42 │ 44 │  4 ║
╟────┼────┼────┼────┼────┼────┼────╢
║ 21 │ 23 │ 32 │ 41 │ 43 │  3 │ 12 ║
╟────┼────┼────┼────┼────┼────┼────╢
║ 22 │ 31 │ 40 │ 49 │  2 │ 11 │ 20 ║
╚════╧════╧════╧════╧════╧════╧════╝

Seed7

<lang seed7>$ include "seed7_05.s7i";

const func integer: succ (in integer: num, in integer: max) is

 return succ(num mod max);

const func integer: pred (in integer: num, in integer: max) is

 return succ((num - 2) mod max);

const proc: main is func

 local
   var integer: size is 3;
   var array array integer: magic is 0 times 0 times 0;
   var integer: row is 1;
   var integer: column is 1;
   var integer: number is 0;
 begin
   if length(argv(PROGRAM)) >= 1 then
     size := integer parse (argv(PROGRAM)[1]);
   end if;
   magic := size times size times 0;
   column := succ(size div 2);
   for number range 1 to size ** 2 do
     magic[row][column] := number;
     if magic[pred(row, size)][succ(column, size)] = 0 then 
       row := pred(row, size);
       column := succ(column, size);
     else
       row := succ(row, size);
     end if;
   end for;
   for key row range magic do
     for key column range magic[row] do
       write(magic[row][column] lpad 4);
     end for;
     writeln;
   end for;
 end func;</lang>
Output:
> s7 magicSquaresOfOddOrder 7
SEED7 INTERPRETER Version 5.0.5203  Copyright (c) 1990-2014 Thomas Mertes
  30  39  48   1  10  19  28
  38  47   7   9  18  27  29
  46   6   8  17  26  35  37
   5  14  16  25  34  36  45
  13  15  24  33  42  44   4
  21  23  32  41  43   3  12
  22  31  40  49   2  11  20

Sidef

<lang ruby>__USE_INTNUM__

func magic_square(n) {

   (n % 2 == 0) || (n < 0) && (
       warn "Sorry, must be a positive odd integer.";
       return;
   );
   var x = int(n/2);
   var y = 0;
   var i = 1;
   var sq = n.of { n.of(0) };
   range(0, n*n - 1).each {
       sq[(i % n ? y-- : y++) % n][(i % n ? x++ : x) % n] = i++;
   }
   return sq;

}

func print_square(sq) {

   var f = "%#{(sq.len**2).len}d";
   sq.each {|row|
       say row.map{ f % _ }.join(' ');
   };

}

var(n=5) = ARGV.map{.to_i}...; var sq = magic_square(n); print_square(sq);

say "\nThe magic number is: #{sq[0].sum}";</lang>

Output:
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

The magic number is: 65

Tcl

<lang tcl>proc magicSquare {order} {

   if {!($order & 1) || $order < 0} {

error "order must be odd and positive"

   }
   set s [lrepeat $order [lrepeat $order 0]]
   set x [expr {$order / 2}]
   set y 0
   for {set i 1} {$i <= $order**2} {incr i} {

lset s $y $x $i set x [expr {($x + 1) % $order}] set y [expr {($y - 1) % $order}] if {[lindex $s $y $x]} { set x [expr {($x - 1) % $order}] set y [expr {($y + 2) % $order}] }

   }
   return $s

}</lang> Demonstrating:

Works with: Tcl version 8.6

<lang tcl>package require Tcl 8.6

set square [magicSquare 5] puts [join [lmap row $square {join [lmap n $row {format "%2s" $n}]}] "\n"] puts "magic number = [tcl::mathop::+ {*}[lindex $square 0]]"</lang>

Output:
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9
magic number = 65

VBScript

Translation of: Liberty BASIC

<lang vb> Sub magic_square(n) Dim ms() ReDim ms(n,n) inc = 1 count = 1 row = 1 col = (n+1)/2 Do While count <= n*n ms(row,col) = count count = count + 1 If inc < n Then inc = inc + 1 row = row - 1 col = col + 1 If row <> 0 Then If col > n Then col = 1 End If Else row = n End If Else inc = 1 row = row + 1 End If Loop For i = 1 To n For j = 1 To n If j = n Then WScript.StdOut.Write ms(i,j) Else WScript.StdOut.Write ms(i,j) & vbTab End If Next WScript.StdOut.WriteLine Next End Sub

magic_square(5) </lang>

Output:
17	24	1	8	15
23	5	7	14	16
4	6	13	20	22
10	12	19	21	3
11	18	25	2	9

zkl

Translation of: Ruby

<lang zkl>fcn rmod(n,m){ n=n%m; if (n<0) n+=m; n } // Ruby: -5%3-->1 fcn odd_magic_square(n){ //-->list of n*n numbers, row order

  if (n.isEven or n <= 0) throw(Exception.ValueError("Need odd positive number"));
  [[(i,j); n; n; '{ n*((i+j+1+n/2):rmod(_,n)) + ((i+2*j-5):rmod(_,n)) + 1 }]]

}

T(3, 5, 9).pump(Void,fcn(n){

  "\nSize %d, magic sum %d".fmt(n,(n*n+1)/2*n).println();
  fmt := "%%%dd".fmt((n*n).toString().len() + 1) * n;
  odd_magic_square(n).pump(Console.println,T(Void.Read,n-1),fmt.fmt);

});</lang>

Output:
Size 3, magic sum 15
 8 1 6
 3 5 7
 4 9 2

Size 5, magic sum 65
 16 23  5  7 14
 22  4  6 13 20
  3 10 12 19 21
  9 11 18 25  2
 15 17 24  1  8

Size 9, magic sum 369
 50 61 72 74  4 15 26 28 39
 60 71 73  3 14 25 36 38 49
 70 81  2 13 24 35 37 48 59
 80  1 12 23 34 45 47 58 69
  9 11 22 33 44 46 57 68 79
 10 21 32 43 54 56 67 78  8
 20 31 42 53 55 66 77  7 18
 30 41 52 63 65 76  6 17 19
 40 51 62 64 75  5 16 27 29