Magic squares of doubly even order: Difference between revisions

From Rosetta Code
Content added Content deleted
(→‎JS ES6: Simplified diagonals function used in sum checking)
Line 2,197: Line 2,197:


fn main() {
fn main() {
let n: usize = env::args().nth(1).and_then(|arg| arg.parse().ok()).expect(
let n: usize = match env::args()
"
.nth(1)
.and_then(|arg| arg.parse().ok())
Please specify the size of the magic square, as a positive multiple of 4.
.ok_or("Please specify the size of the magic square, as a positive multiple of 4.")
",
);
{
Ok(arg) if arg >= 4 && arg % 4 == 0 => arg,
assert!(
n >= 4 && n % 4 == 0,
Err(e) => panic!(e),
"Argument must be a positive multiple of 4."
_ => panic!("Argument must be a positive multiple of 4."),
);
};


let mc = (n * n + 1) * n / 2;
let mc = (n * n + 1) * n / 2;
Line 2,228: Line 2,228:
i += 1;
i += 1;
}
}
println!("");
println!();
}
}
}</lang>
}</lang>

Revision as of 22:32, 5 October 2019

Task
Magic squares of doubly even order
You are encouraged to solve this task according to the task description, using any language you may know.

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

A magic square of doubly even order has a size that is a multiple of four (e.g. 4, 8, 12). This means that the subsquares also have an even size, which plays a role in the construction.

1 2 62 61 60 59 7 8
9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58 6 5 4 3 63 64


Task

Create a magic square of 8 x 8.

Related tasks
See also



360 Assembly

Translation of: Java

<lang 360asm>* Magic squares of doubly even order 01/03/2017 MAGICSDB CSECT 

        USING  MAGICSDB,R13
        B      72(R15)            skip save area
        DC     17F'0'             save area
        STM    R14,R12,12(R13)
        ST     R13,4(R15)
        ST     R15,8(R13)
        LR     R13,R15            end of prolog
        SR     R8,R8              k=0
        LA     R6,0               i=0
      DO WHILE=(C,R6,LE,=A(N-1))  do i=0 to n-1
        MVC    PG,=CL80' '          clear buffer
        LA     R9,PG                pgi=0
        LA     R7,0                 j=0
      DO WHILE=(C,R7,LE,=A(N-1))    do j=0 to n-1
        LR     R4,R7                  j
        SRDA   R4,32                  >>r5
        D      R4,MULT                /mult
        LR     R2,R5                  r2=j/mult
        LR     R4,R6                  i
        SRDA   R4,32                  >>r5
        D      R4,MULT                /mult
        SLA    R5,2                   r5=(i/mult)*4
        AR     R2,R5                  bitpos=j/mult+(i/mult)*4
        STC    R2,XSLL+3              number_of_shift=bitpos
        L      R5,=F'1'               1
        EX     0,XSLL                 r5=1<<bitpos  (SLL R5,bitpos)
        L      R4,BITS                bits
        NR     R4,R5                  bits and (1<<bitpos)
      IF LTR,R4,NZ,R4 THEN            if (bits and (1<<bitpos))<>0
        LA     R10,1(R8)                x=k+1
      ELSE     ,                      else
        L      R10,SIZE                 size
        SR     R10,R8                   x=size-k
      ENDIF    ,                      endif
        XDECO  R10,XDEC               edit x
        MVC    0(4,R9),XDEC+8         output x
        LA     R9,4(R9)               pgi+=4
        LA     R8,1(R8)               k++
        LA     R7,1(R7)               j++
      ENDDO    ,                    enddo j
        XPRNT  PG,L'PG              print buffer
        LA     R6,1(R6)             i++
      ENDDO    ,                  enddo i
        MVC    PG,=CL80'magic constant='
        L      R1,=A((N*N+1)*N/2) magicnum=(n*n+1)*n/2
        XDECO  R1,XDEC            edit magicnum
        MVC    PG+15(4),XDEC+8    output magicnum
        XPRNT  PG,L'PG            print buffer
        L      R13,4(0,R13)       epilog
        LM     R14,R12,12(R13)
        XR     R15,R15            rc=0
        BR     R14                exit

XSLL SLL R5,0 shift left logical N EQU 8 <= input value n SIZE DC A(N*N) size=n*n MULT DC A(N/4) mult=n/4 (multiple of 4) BITS DC XL2'0000',BL2'1001011001101001' pattern PG DS CL80 buffer XDEC DS CL12 temp for xdeco 

        YREGS
        END    MAGICSDB</lang>
Output:
   1   2  62  61  60  59   7   8
   9  10  54  53  52  51  15  16
  48  47  19  20  21  22  42  41
  40  39  27  28  29  30  34  33
  32  31  35  36  37  38  26  25
  24  23  43  44  45  46  18  17
  49  50  14  13  12  11  55  56
  57  58   6   5   4   3  63  64
magic constant= 260

AppleScript

Translation of: JavaScript

<lang AppleScript>-- MAGIC SQUARE OF DOUBLY EVEN ORDER -----------------------------------------

-- magicSquare :: Int -> Int on magicSquare(n) 

   if n mod 4 > 0 then
       {}
   else
       set sqr to n * n
       
       set maybePowerOfTwo to asPowerOfTwo(sqr)
       if maybePowerOfTwo is not missing value then
           
           -- For powers of 2, the (append not) 'magic' series directly
           -- yields the truth table that we need
           set truthSeries to magicSeries(maybePowerOfTwo)
       else
           -- where n is not a power of 2, we can replicate a
           -- minimum truth table, horizontally and vertically
           
           script scale
               on |λ|(x)
                   replicate(n / 4, x)
               end |λ|
           end script
           
           set truthSeries to ¬
               flatten(scale's |λ|(map(scale, splitEvery(4, magicSeries(4)))))
       end if
       
       set limit to sqr + 1
       script inOrderOrReversed
           on |λ|(x, i)
               cond(x, i, limit - i)
           end |λ|
       end script
       
       -- Taken directly from an integer series  [1..sqr] where True
       -- and from the reverse of that series where False
       splitEvery(n, map(inOrderOrReversed, truthSeries))
   end if

end magicSquare

-- magicSeries :: Int -> [Bool] on magicSeries(n) 

   script boolToggle
       on |λ|(x)
           not x
       end |λ|
   end script
   
   if n ≤ 0 then
       {true}
   else
       set xs to magicSeries(n - 1)
       xs & map(boolToggle, xs)
   end if

end magicSeries


-- TEST ---------------------------------------------------------------------- on run 

   formattedTable(magicSquare(8))

end run

-- formattedTable :: Int -> String on formattedTable(lstTable) 

   set n to length of lstTable
   set w to 2.5 * n
   "magic(" & n & ")" & linefeed & wikiTable(lstTable, ¬
       false, "text-align:center;width:" & ¬
       w & "em;height:" & w & "em;table-layout:fixed;")

end formattedTable

-- wikiTable :: [Text] -> Bool -> Text -> Text on wikiTable(lstRows, blnHdr, strStyle) 

   script fWikiRows
       on |λ|(lstRow, iRow)
           set strDelim to cond(blnHdr and (iRow = 0), "!", "|")
           set strDbl to strDelim & strDelim
           linefeed & "|-" & linefeed & strDelim & space & ¬
               intercalate(space & strDbl & space, lstRow)
       end |λ|
   end script
   
   linefeed & "{| class=\"wikitable\" " & ¬
       cond(strStyle ≠ "", "style=\"" & strStyle & "\"", "") & ¬
       intercalate("", ¬
           map(fWikiRows, lstRows)) & linefeed & "|}" & linefeed

end wikiTable


-- GENERIC FUNCTIONS ---------------------------------------------------------

-- asPowerOfTwo :: Int -> maybe Int on asPowerOfTwo(n) 

   if not isPowerOf(2, n) then
       missing value
   else
       set strCMD to ("echo 'l(" & n as string) & ")/l(2)' | bc -l"
       (do shell script strCMD) as integer
   end if

end asPowerOfTwo

-- concatMap :: (a -> [b]) -> [a] -> [b] on concatMap(f, xs) 

   script append
       on |λ|(a, b)
           a & b
       end |λ|
   end script
   
   foldl(append, {}, map(f, xs))

end concatMap

-- cond :: Bool -> a -> a -> a on cond(bool, f, g) 

   if bool then
       f
   else
       g
   end if

end cond

-- flatten :: Tree a -> [a] on flatten(t) 

   if class of t is list then
       concatMap(my flatten, t)
   else
       t
   end if

end flatten

-- foldl :: (a -> b -> a) -> a -> [b] -> a on foldl(f, startValue, xs) 

   tell mReturn(f)
       set v to startValue
       set lng to length of xs
       repeat with i from 1 to lng
           set v to |λ|(v, item i of xs, i, xs)
       end repeat
       return v
   end tell

end foldl

-- intercalate :: Text -> [Text] -> Text on intercalate(strText, lstText) 

   set {dlm, my text item delimiters} to {my text item delimiters, strText}
   set strJoined to lstText as text
   set my text item delimiters to dlm
   return strJoined

end intercalate

-- isPowerOf :: Int -> Int -> Bool on isPowerOf(k, n) 

   set v to k
   script remLeft
       on |λ|(x)
           x mod v is not 0
       end |λ|
   end script
   
   script integerDiv
       on |λ|(x)
           x div v
       end |λ|
   end script
   
   |until|(remLeft, integerDiv, n) = 1

end isPowerOf

-- map :: (a -> b) -> [a] -> [b] on map(f, xs) 

   tell mReturn(f)
       set lng to length of xs
       set lst to {}
       repeat with i from 1 to lng
           set end of lst to |λ|(item i of xs, i, xs)
       end repeat
       return lst
   end tell

end map


-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f) 

   if class of f is script then
       f
   else
       script
           property |λ| : f
       end script
   end if

end mReturn

-- Egyptian multiplication - progressively doubling a list, appending -- stages of doubling to an accumulator where needed for binary -- assembly of a target length

-- replicate :: Int -> a -> [a] on replicate(n, a) 

   set out to {}
   if n < 1 then return out
   set dbl to {a}
   
   repeat while (n > 1)
       if (n mod 2) > 0 then set out to out & dbl
       set n to (n div 2)
       set dbl to (dbl & dbl)
   end repeat
   return out & dbl

end replicate

-- splitAt :: Int -> [a] -> ([a],[a]) on splitAt(n, xs) 

   if n > 0 and n < length of xs then
       if class of xs is text then
           {items 1 thru n of xs as text, items (n + 1) thru -1 of xs as text}
       else
           {items 1 thru n of xs, items (n + 1) thru -1 of xs}
       end if
   else
       if n < 1 then
           {{}, xs}
       else
           {xs, {}}
       end if
   end if

end splitAt

-- splitEvery :: Int -> [a] -> a on splitEvery(n, xs) 

   if length of xs ≤ n then
       {xs}
   else
       set {gp, t} to splitAt(n, xs)
       {gp} & splitEvery(n, t)
   end if

end splitEvery

-- until :: (a -> Bool) -> (a -> a) -> a -> a on |until|(p, f, x) 

   set mp to mReturn(p)
   set v to x
   
   tell mReturn(f)
       repeat until mp's |λ|(v)
           set v to |λ|(v)
       end repeat
   end tell
   return v

end |until|</lang>

Output:

magic(8)

1 63 62 4 60 6 7 57
56 10 11 53 13 51 50 16
48 18 19 45 21 43 42 24
25 39 38 28 36 30 31 33
32 34 35 29 37 27 26 40
41 23 22 44 20 46 47 17
49 15 14 52 12 54 55 9
8 58 59 5 61 3 2 64

AWK

Translation of: C#

Since standard awk does not support bitwise operators, we provide the function countup to do the magic. <lang AWK># Usage: GAWK -f MAGIC_SQUARES_OF_DOUBLY_EVEN_ORDER.AWK BEGIN { 

   n = 8
   msquare[0, 0] = 0
   if (magicsquaredoublyeven(msquare, n)) {
       for (i = 0; i < n; i++) {
           for (j = 0; j < n; j++) {
               printf("%2d ", msquare[i, j])
           }
           printf("\n")
       }
       printf("\nMagic constant: %d\n", (n * n + 1) * n / 2)
       exit 1
   } else {
       exit 0
   }

} function magicsquaredoublyeven(msquare, n, size, mult, r, c, i) { 

   if (n < 4 || n % 4 != 0) {
       printf("Base must be a positive multiple of 4.\n")
       return 0
   }
   size = n * n
   mult = n / 4 # how many multiples of 4

   i = 0
   for (r = 0; r < n; r++) {
       for (c = 0; c < n; c++) {
           msquare[r, c] = countup(r, c, mult) ? i + 1 : size - i
           i++
       }
   }
   return 1

} function countup(r, c, mult, pattern, bitpos) { 

   # Returns 1, if we are in a count-up zone (0 otherwise)
   pattern = "1001011001101001"
   bitpos =  int(c / mult) + int(r / mult) * 4 + 1
   return substr(pattern, bitpos, 1) + 0

}</lang>

Output:
 1  2 62 61 60 59  7  8
 9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58  6  5  4  3 63 64

Magic constant: 260

Befunge

The size, N, is specified by the first value on the stack. The algorithm is loosely based on the Java implementation.

<lang befunge>8>>>v>10p00g:*1-*\110g2*-*+1+.:00g%!9+,:#v_@ p00:<^:!!-!%3//4g00%g00\!!%3/*:g00*4:::-1<*:</lang>

Output:
1       2       62      61      60      59      7       8
9       10      54      53      52      51      15      16
48      47      19      20      21      22      42      41
40      39      27      28      29      30      34      33
32      31      35      36      37      38      26      25
24      23      43      44      45      46      18      17
49      50      14      13      12      11      55      56
57      58      6       5       4       3       63      64

C

Takes number of rows from command line, prints out usage on incorrect invocation. <lang C>

  1. include<stdlib.h>
  2. include<ctype.h>
  3. include<stdio.h>

int** doublyEvenMagicSquare(int n) { if (n < 4 || n % 4 != 0) return NULL;

int bits = 38505; int size = n * n; int mult = n / 4,i,r,c,bitPos;

int** result = (int**)malloc(n*sizeof(int*));

for(i=0;i<n;i++) result[i] = (int*)malloc(n*sizeof(int));

for (r = 0, i = 0; r < n; r++) { for (c = 0; c < n; c++, i++) { bitPos = c / mult + (r / mult) * 4; result[r][c] = (bits & (1 << bitPos)) != 0 ? i + 1 : size - i; } } return result; }

int numDigits(int n){ int count = 1;

while(n>=10){ n /= 10; count++; }

return count; }

void printMagicSquare(int** square,int rows){ int i,j,baseWidth = numDigits(rows*rows) + 3;

printf("Doubly Magic Square of Order : %d and Magic Constant : %d\n\n",rows,(rows * rows + 1) * rows / 2);

for(i=0;i<rows;i++){ for(j=0;j<rows;j++){ printf("%*s%d",baseWidth - numDigits(square[i][j]),"",square[i][j]); } printf("\n"); } }

int main(int argC,char* argV[]) { int n;

if(argC!=2||isdigit(argV[1][0])==0) printf("Usage : %s <integer specifying rows in magic square>",argV[0]); else{ n = atoi(argV[1]); printMagicSquare(doublyEvenMagicSquare(n),n); } return 0; } </lang> Invocation and Output : 

C:\rosettaCode>doublyEvenMagicSquare 12
Doubly Magic Square of Order : 12 and Magic Constant : 870

     1     2     3   141   140   139   138   137   136    10    11    12
    13    14    15   129   128   127   126   125   124    22    23    24
    25    26    27   117   116   115   114   113   112    34    35    36
   108   107   106    40    41    42    43    44    45    99    98    97
    96    95    94    52    53    54    55    56    57    87    86    85
    84    83    82    64    65    66    67    68    69    75    74    73
    72    71    70    76    77    78    79    80    81    63    62    61
    60    59    58    88    89    90    91    92    93    51    50    49
    48    47    46   100   101   102   103   104   105    39    38    37
   109   110   111    33    32    31    30    29    28   118   119   120
   121   122   123    21    20    19    18    17    16   130   131   132
   133   134   135     9     8     7     6     5     4   142   143   144

C++

<lang cpp>#include <iostream>

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

using namespace std;

class magicSqr { public: 

   magicSqr( int d ) {
       while( d % 4 > 0 ) { d++; }
       sz = d;
       sqr = new int[sz * sz];
       fillSqr();
   }
   ~magicSqr() { delete [] sqr; }

   void display() const {
       cout << "Doubly Even 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() {
       static const bool tempAll[4][4] = {{ 1, 0, 0, 1 }, { 0, 1, 1, 0 }, { 0, 1, 1, 0 }, { 1, 0, 0, 1 } };
       int i = 0;
       for( int curRow = 0; curRow < sz; curRow++ ) {
           for( int curCol = 0; curCol < sz; curCol++ ) {
               sqr[curCol + sz * curRow] = tempAll[curRow % 4][curCol % 4] ? i + 1 : sz * sz - i;
               i++;
           }
       }
   }
   int magicNumber() const { return sz * ( ( sz * sz ) + 1 ) / 2; }

   int* sqr;
   int sz;

};

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

   magicSqr s( 8 );
   s.display();
   return 0;

}</lang>

Output:
Doubly Even Magic Square: 8 x 8
It's Magic Sum is: 260

   1  63  62   4   5  59  58   8
  56  10  11  53  52  14  15  49
  48  18  19  45  44  22  23  41
  25  39  38  28  29  35  34  32
  33  31  30  36  37  27  26  40
  24  42  43  21  20  46  47  17
  16  50  51  13  12  54  55   9
  57   7   6  60  61   3   2  64

C#

Translation of: Java

<lang csharp>using System;

namespace MagicSquareDoublyEven { 

   class Program
   {
       static void Main(string[] args)
       {
           int n = 8;
           var result = MagicSquareDoublyEven(n);
           for (int i = 0; i < result.GetLength(0); i++)
           {
               for (int j = 0; j < result.GetLength(1); j++)
                   Console.Write("{0,2} ", result[i, j]);
               Console.WriteLine();
           }
           Console.WriteLine("\nMagic constant: {0} ", (n * n + 1) * n / 2);
           Console.ReadLine();
       }

       private static int[,] MagicSquareDoublyEven(int n)
       {
           if (n < 4 || n % 4 != 0)
               throw new ArgumentException("base must be a positive "
                       + "multiple of 4");

           // pattern of count-up vs count-down zones
           int bits = 0b1001_0110_0110_1001;
           int size = n * n;
           int mult = n / 4;  // how many multiples of 4

           int[,] result = new int[n, n];

           for (int r = 0, i = 0; r < n; r++)
           {
               for (int c = 0; c < n; c++, i++)
               {
                   int bitPos = c / mult + (r / mult) * 4;
                   result[r, c] = (bits & (1 << bitPos)) != 0 ? i + 1 : size - i;
               }
           }
           return result;
       }
   }

}</lang> 

 1  2 62 61 60 59  7  8
 9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58  6  5  4  3 63 64

Magic constant: 260

D

Translation of: Java

<lang D>import std.stdio;

void main() { 

   int n=8;
   foreach(row; magicSquareDoublyEven(n)) {
       foreach(col; row) {
           writef("%2s ", col);
       }
       writeln;
   }
   writeln("\nMagic constant: ", (n*n+1)*n/2);

}

int[][] magicSquareDoublyEven(int n) { 

   import std.exception;
   enforce(n>=4 && n%4 == 0, "Base must be a positive multiple of 4");

   int bits = 0b1001_0110_0110_1001;
   int size = n * n;
   int mult = n / 4;  // how many multiples of 4

   int[][] result;
   result.length = n;
   foreach(i; 0..n) {
       result[i].length = n;
   }

   for (int r=0, i=0; r<n; r++) {
       for (int c=0; c<n; c++, i++) {
           int bitPos = c / mult + (r / mult) * 4;
           result[r][c] = (bits & (1 << bitPos)) != 0 ? i + 1 : size - i;
       }
   }

   return result;

}</lang>

Output:
 1  2 62 61 60 59  7  8
 9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58  6  5  4  3 63 64

Magic constant: 260

Elena

Translation of: C#

ELENA 4.x : <lang elena>import system'routines; import extensions; import extensions'routines;

MagicSquareDoublyEven(int n) { 

   if(n < 4 || n.mod(4) != 0)
       { InvalidArgumentException.new:"base must be a positive multiple of 4".raise() };

   int bits := 09669h;
   int size := n * n;
   int mult := n / 4;

   var result := new IntMatrix(n,n);
   int i := 0;
   for (int r := 0, r < n, r += 1)
   {
       for(int c := 0, c < n, c += 1, i += 1)
       {
           int bitPos := c / mult + (r / mult) * 4;

           result[r][c] := ((bits && (1 $shl bitPos)) != 0).iif(i+1,size - i)
       }
   };

   ^ result

}

public program() { 

   int n := 8;

   console.printLine(MagicSquareDoublyEven(n));

   console.printLine().printLine("Magic constant: ",(n * n + 1) * n / 2)

}</lang>

Output:
1,2,62,61,60,59,7,8
9,10,54,53,52,51,15,16
48,47,19,20,21,22,42,41
40,39,27,28,29,30,34,33
32,31,35,36,37,38,26,25
24,23,43,44,45,46,18,17
49,50,14,13,12,11,55,56
57,58,6,5,4,3,63,64

Magic constant: 260

Elixir

<lang elixir>defmodule Magic_square do 

 def doubly_even(n) when rem(n,4)!=0, do: raise ArgumentError, "must be even, but not divisible by 4."
 def doubly_even(n) do
   n2 = n * n
   Enum.zip(1..n2, make_pattern(n))
   |> Enum.map(fn {i,p} -> if p, do: i, else: n2 - i + 1 end)
   |> Enum.chunk(n)
   |> to_string(n)
   |> IO.puts
 end
 
 defp make_pattern(n) do
   pattern = Enum.reduce(1..4, [true], fn _,acc ->
               acc ++ Enum.map(acc, &(!&1))
             end) |> Enum.chunk(4)
   for i <- 0..n-1, j <- 0..n-1, do: Enum.at(pattern, rem(i,4)) |> Enum.at(rem(j,4))
 end

 defp to_string(square, n) do
   format = String.duplicate("~#{length(to_char_list(n*n))}w ", n) <> "\n"
   Enum.map_join(square, fn row ->
     :io_lib.format(format, row)
   end)
 end

end

Magic_square.doubly_even(8)</lang>

Output:
 1 63 62  4  5 59 58  8
56 10 11 53 52 14 15 49
48 18 19 45 44 22 23 41
25 39 38 28 29 35 34 32
33 31 30 36 37 27 26 40
24 42 43 21 20 46 47 17
16 50 51 13 12 54 55  9
57  7  6 60 61  3  2 64

Factor

<lang factor>USING: arrays combinators.short-circuit formatting fry generalizations kernel math math.matrices prettyprint sequences

IN: rosetta-code.doubly-even-magic-squares

top? ( loc n -- ? ) [ second ] dip 1/4 * <  ;
bottom? ( loc n -- ? ) [ second ] dip 3/4 * >= ;
left? ( loc n -- ? ) [ first ] dip 1/4 * <  ;
right? ( loc n -- ? ) [ first ] dip 3/4 * >= ;
corner? ( loc n -- ? )
   {
       [ { [ top?    ] [ left?  ] } ]
       [ { [ top?    ] [ right? ] } ]
       [ { [ bottom? ] [ left?  ] } ]
       [ { [ bottom? ] [ right? ] } ]
   } [ 2&& ] map-compose 2|| ;
center? ( loc n -- ? )
   { [ top? ] [ bottom? ] [ left? ] [ right? ] } [ not ]
   map-compose 2&& ;
backward? ( loc n -- ? ) { [ corner? ] [ center? ] } 2|| ;
forward ( loc n -- m ) [ first2 ] dip * 1 + + ;
backward ( loc n -- m ) tuck forward [ sq ] dip - 1 + ;
(doubly-even-magic-square) ( n -- matrix )
   [ dup 2array matrix-coordinates flip ] [ 3 dupn ] bi
   '[ dup _ backward? [ _ backward ] [ _ forward ] if ]
   matrix-map ;

ERROR: invalid-order order ;

check-order ( n -- )
   dup { [ zero? not ] [ 4 mod zero? ] } 1&& [ drop ]
   [ invalid-order ] if ;
doubly-even-magic-square ( n -- matrix )
   dup check-order (doubly-even-magic-square) ;
main ( -- )
   { 4 8 12 } [
       dup doubly-even-magic-square dup
       [ "Order: %d\n" printf ]
       [ simple-table. ]
       [ first sum "Magic constant: %d\n\n" printf ] tri*
   ] each ;

MAIN: main</lang>

Output:
Order: 4
16 2  3  13
5  11 10 8
9  7  6  12
4  14 15 1
Magic constant: 34

Order: 8
64 63 3  4  5  6  58 57
56 55 11 12 13 14 50 49
17 18 46 45 44 43 23 24
25 26 38 37 36 35 31 32
33 34 30 29 28 27 39 40
41 42 22 21 20 19 47 48
16 15 51 52 53 54 10 9
8  7  59 60 61 62 2  1
Magic constant: 260

Order: 12
144 143 142 4   5   6   7   8   9   135 134 133
132 131 130 16  17  18  19  20  21  123 122 121
120 119 118 28  29  30  31  32  33  111 110 109
37  38  39  105 104 103 102 101 100 46  47  48
49  50  51  93  92  91  90  89  88  58  59  60
61  62  63  81  80  79  78  77  76  70  71  72
73  74  75  69  68  67  66  65  64  82  83  84
85  86  87  57  56  55  54  53  52  94  95  96
97  98  99  45  44  43  42  41  40  106 107 108
36  35  34  112 113 114 115 116 117 27  26  25
24  23  22  124 125 126 127 128 129 15  14  13
12  11  10  136 137 138 139 140 141 3   2   1
Magic constant: 870

FreeBASIC

<lang freebasic>' version 18-03-2016 ' compile with: fbc -s console ' doubly even magic square 4, 8, 12, 16...

Sub Err_msg(msg As String) 

   Print msg
   Beep : Sleep 5000, 1 : Exit Sub

End Sub

Sub de_magicsq(n As UInteger, filename As String = "") 

   ' filename <> "" then save square in a file
   ' filename can contain directory name
   ' if filename exist it will be overwriten, no error checking

   If n < 4 Then
       Err_msg( "Error: n is to small")
       Exit Sub
   End If

   If (n Mod 4) <> 0 Then
       Err_msg "Error: not possible to make doubly" + _
       " even magic square size " + Str(n)
       Exit Sub
   End If

   Dim As UInteger sq(1 To n, 1 To n)
   Dim As UInteger magic_sum = n * (n ^ 2 +1) \ 2
   Dim As UInteger q = n \ 4
   Dim As UInteger x, y, nr = 1
   Dim As String frmt = String(Len(Str(n * n)) +1, "#")

   ' set up the square
   For y = 1 To n
       For x = q +1 To n - q
           sq(x,y) = 1
       Next
   Next
   For x = 1 To n
       For y = q +1 To n - q
           sq(x, y) Xor= 1
       Next
   Next

   ' fill the square
   q = n * n +1
   For y = 1 To n
       For x = 1 To n
           If sq(x,y) = 0 Then
               sq(x,y) = q - nr
           Else
               sq(x,y) = nr
           End If
           nr += 1
       Next
   Next

   ' check columms and rows
   For y = 1 To n
       nr = 0 : q = 0
       For x = 1 To n
           nr += sq(x,y)
           q += sq(y,x)
       Next
       If nr <> magic_sum Or q <> magic_sum Then
           Err_msg "Error: value <> magic_sum"
           Exit Sub
       End If
   Next

   ' check diagonals
   nr = 0 : q = 0
   For x = 1 To n
       nr += sq(x, x)
       q += sq(n - x +1, n - x +1)
   Next
   If nr <> magic_sum Or q <> magic_sum Then
       Err_msg "Error: value <> magic_sum"
       Exit Sub
   End If

   ' printing square's on screen bigger when
   ' n > 19 results in a wrapping of the line
   Print "Single even magic square size: "; n; "*"; n
   Print "The magic sum = "; magic_sum
   Print
   For y = 1 To n
       For x = 1 To n
           Print Using frmt; sq(x, y);
       Next
       Print
   Next

   ' output magic square to a file with the name provided
   If filename <> "" Then
       nr = FreeFile
       Open filename For Output As #nr
       Print #nr, "Single even magic square size: "; n; "*"; n
       Print #nr, "The magic sum = "; magic_sum
       Print #nr,
       For y = 1 To n
           For x = 1 To n
               Print #nr, Using frmt; sq(x,y);
           Next
           Print #nr,
       Next
       Close #nr
   End If

End Sub

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

de_magicsq(8, "magic8de.txt") : Print

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

Output:
Single even magic square size: 8*8
The magic sum = 260

 64 63  3  4  5  6 58 57
 56 55 11 12 13 14 50 49
 17 18 46 45 44 43 23 24
 25 26 38 37 36 35 31 32
 33 34 30 29 28 27 39 40
 41 42 22 21 20 19 47 48
 16 15 51 52 53 54 10  9
  8  7 59 60 61 62  2  1

Go

<lang Go>package main

import ( "fmt" "log" "strings" )

const dimensions int = 8

func setupMagicSquareData(d int) ([][]int, error) { var output [][]int if d < 4 || d%4 != 0 { return [][]int{}, fmt.Errorf("Square dimension must be a positive number which is divisible by 4") } var bits uint = 0x9669 // 0b1001011001101001 size := d * d mult := d / 4 for i, r := 0, 0; r < d; r++ { output = append(output, []int{}) for c := 0; c < d; i, c = i+1, c+1 { bitPos := c/mult + (r/mult)*4 if (bits & (1 << uint(bitPos))) != 0 { output[r] = append(output[r], i+1) } else { output[r] = append(output[r], size-i) } } } return output, nil }

func arrayItoa(input []int) []string { var output []string for _, i := range input { output = append(output, fmt.Sprintf("%4d", i)) } return output }

func main() { data, err := setupMagicSquareData(dimensions) if err != nil { log.Fatal(err) } magicConstant := (dimensions * (dimensions*dimensions + 1)) / 2 for _, row := range data { fmt.Println(strings.Join(arrayItoa(row), " ")) } fmt.Printf("\nMagic Constant: %d\n", magicConstant) } </lang>

Output:
   1    2   62   61   60   59    7    8
   9   10   54   53   52   51   15   16
  48   47   19   20   21   22   42   41
  40   39   27   28   29   30   34   33
  32   31   35   36   37   38   26   25
  24   23   43   44   45   46   18   17
  49   50   14   13   12   11   55   56
  57   58    6    5    4    3   63   64

Magic Constant: 260

Haskell

<lang Haskell>import Data.List (transpose, unfoldr, intercalate) import Data.List.Split (chunksOf) import Data.Bool (bool) import Control.Monad (forM_)

magicSquare :: Int -> Int magicSquare n 

 | rem n 4 > 0 = []
 | otherwise =
   chunksOf n $ zipWith (flip (bool =<< (-) limit)) series [1 .. sqr]
 where
   sqr = n * n
   limit = sqr + 1
   series
     | isPowerOf 2 n = magicSeries $ floor (logBase 2 (fromIntegral sqr))
     | otherwise =
       concat . concat . concat . scale $ scale <$> chunksOf 4 (magicSeries 4)
     where
       scale = replicate $ quot n 4

magicSeries :: Int -> [Bool] magicSeries = (iterate ((++) <*> fmap not) [True] !!)

isPowerOf :: Int -> Int -> Bool isPowerOf k n = until ((0 /=) . flip rem k) (`quot` k) n == 1

-- TEST AND DISPLAY FUNCTIONS -------------------------------------------------- checked :: Int -> (Int, Bool) checked square = 

 let diagonals =
       fmap (flip (zipWith (!!)) [0 ..]) . ((:) <*> (return . reverse))
     h:t =
       sum <$>
       square ++ -- rows
       transpose square ++ -- cols
       diagonals square -- diagonals
 in (h, all (h ==) t)

table :: String -> String -> [String] table delim rows = 

 let justifyRight c n s = drop (length s) (replicate n c ++ s)
 in intercalate delim <$>
    transpose
      ((fmap =<< justifyRight ' ' . maximum . fmap length) <$> transpose rows)

main :: IO () main = 

 forM_ [4, 8, 16] $
 \n -> do
   let test = magicSquare n
   putStrLn $ unlines (table " " (fmap show <$> test))
   print $ checked test
   putStrLn []</lang>
Output:
 1 15 14  4
12  6  7  9
 8 10 11  5
13  3  2 16

(34,True)

 1 63 62  4 60  6  7 57
56 10 11 53 13 51 50 16
48 18 19 45 21 43 42 24
25 39 38 28 36 30 31 33
32 34 35 29 37 27 26 40
41 23 22 44 20 46 47 17
49 15 14 52 12 54 55  9
 8 58 59  5 61  3  2 64

(260,True)

  1 255 254   4 252   6   7 249 248  10  11 245  13 243 242  16
240  18  19 237  21 235 234  24  25 231 230  28 228  30  31 225
224  34  35 221  37 219 218  40  41 215 214  44 212  46  47 209
 49 207 206  52 204  54  55 201 200  58  59 197  61 195 194  64
192  66  67 189  69 187 186  72  73 183 182  76 180  78  79 177
 81 175 174  84 172  86  87 169 168  90  91 165  93 163 162  96
 97 159 158 100 156 102 103 153 152 106 107 149 109 147 146 112
144 114 115 141 117 139 138 120 121 135 134 124 132 126 127 129
128 130 131 125 133 123 122 136 137 119 118 140 116 142 143 113
145 111 110 148 108 150 151 105 104 154 155 101 157  99  98 160
161  95  94 164  92 166 167  89  88 170 171  85 173  83  82 176
 80 178 179  77 181  75  74 184 185  71  70 188  68 190 191  65
193  63  62 196  60 198 199  57  56 202 203  53 205  51  50 208
 48 210 211  45 213  43  42 216 217  39  38 220  36 222 223  33
 32 226 227  29 229  27  26 232 233  23  22 236  20 238 239  17
241  15  14 244  12 246 247   9   8 250 251   5 253   3   2 256

(2056,True)

J

<lang J> mask =: ,#: 8 $ 153 102 102 153 t =: (, 65&-) each >:i.64 8 8 $ mask{each t </lang>

Output:
┌──┬──┬──┬──┬──┬──┬──┬──┐
│64│2 │3 │61│60│6 │7 │57│
├──┼──┼──┼──┼──┼──┼──┼──┤
│9 │55│54│12│13│51│50│16│
├──┼──┼──┼──┼──┼──┼──┼──┤
│17│47│46│20│21│43│42│24│
├──┼──┼──┼──┼──┼──┼──┼──┤
│40│26│27│37│36│30│31│33│
├──┼──┼──┼──┼──┼──┼──┼──┤
│32│34│35│29│28│38│39│25│
├──┼──┼──┼──┼──┼──┼──┼──┤
│41│23│22│44│45│19│18│48│
├──┼──┼──┼──┼──┼──┼──┼──┤
│49│15│14│52│53│11│10│56│
├──┼──┼──┼──┼──┼──┼──┼──┤
│8 │58│59│5 │4 │62│63│1 │
└──┴──┴──┴──┴──┴──┴──┴──┘

Java

<lang java>public class MagicSquareDoublyEven { 

   public static void main(String[] args) {
       int n = 8;
       for (int[] row : magicSquareDoublyEven(n)) {
           for (int x : row)
               System.out.printf("%2s ", x);
           System.out.println();
       }
       System.out.printf("\nMagic constant: %d ", (n * n + 1) * n / 2);
   }

   static int[][] magicSquareDoublyEven(final int n) {
       if (n < 4 || n % 4 != 0)
           throw new IllegalArgumentException("base must be a positive "
                   + "multiple of 4");

       // pattern of count-up vs count-down zones
       int bits = 0b1001_0110_0110_1001;
       int size = n * n;
       int mult = n / 4;  // how many multiples of 4

       int[][] result = new int[n][n];

       for (int r = 0, i = 0; r < n; r++) {
           for (int c = 0; c < n; c++, i++) {
               int bitPos = c / mult + (r / mult) * 4;
               result[r][c] = (bits & (1 << bitPos)) != 0 ? i + 1 : size - i;
           }
       }
       return result;
   }

}</lang> 

 1  2 62 61 60 59  7  8 
 9 10 54 53 52 51 15 16 
48 47 19 20 21 22 42 41 
40 39 27 28 29 30 34 33 
32 31 35 36 37 38 26 25 
24 23 43 44 45 46 18 17 
49 50 14 13 12 11 55 56 
57 58  6  5  4  3 63 64 

Magic constant: 260


JavaScript

ES6

Translation of: Haskell

<lang JavaScript>(() => { 

   'use strict';

   // doublyEvenMagicSquare :: Int -> Int
   const doublyEvenMagicSquare = n =>
       0 === n % 4 ? (() => {
           const
               sqr = n * n,
               power = Math.log2(sqr),
               scale = replicate(n / 4);
           return chunksOf(n)(
               map((x, i) => x ? 1 + i : sqr - i)(
                   isInt(power) ? truthSeries(power) : (
                       compose(
                           flatten,
                           scale,
                           map(scale),
                           chunksOf(4)
                       )(truthSeries(4))
                   )
               )
           );
       })() : undefined;

   // truthSeries :: Int -> [Bool]
   const truthSeries = n =>
       0 >= n ? (
           [true]
       ) : (() => {
           const xs = truthSeries(n - 1);
           return xs.concat(xs.map(x => !x));
       })();



   // TEST -----------------------------------------------
   const main = () =>
       // Magic squares of orders 4, 8 and 12, with
       // checks of row, column and diagonal sums.
       intercalate('\n\n')(
           map(n => {
               const
                   lines = doublyEvenMagicSquare(n),
                   sums = map(sum)(
                       lines.concat(
                           transpose(lines)
                           .concat(diagonals(lines))
                       )
                   ),
                   total = sums[0];
               return unlines([
                   "Order: " + str(n),
                   "Summing to: " + str(total),
                   "Row, column and diagonal sums checked: " +
                   str(all(eq(total))(sums)) + '\n',
                   unlines(map(compose(
                       intercalate('  '),
                       map(compose(justifyRight(3)(' '), str))
                   ))(lines))
               ]);
           })([4, 8, 12])
       );



   // GENERIC FUNCTIONS ----------------------------------

   // all :: (a -> Bool) -> [a] -> Bool
   const all = p =>
       // True if p(x) holds for every x in xs.
       xs => xs.every(p);

   // chunksOf :: Int -> [a] -> a
   const chunksOf = n => xs =>
       enumFromThenTo(0)(n)(
           xs.length - 1
       ).reduce(
           (a, i) => a.concat([xs.slice(i, (n + i))]),
           []
       );

   // compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
   const compose = (...fs) =>
       x => fs.reduceRight((a, f) => f(a), x);

   // diagonals :: a -> [[a], [a]]
   const diagonals = rows =>
       // Two diagonal sequences,
       // from top left and bottom left
       // respectively, of a given matrix.
       map(flip(zipWith(index))(
           enumFromTo(0)(pred(
               0 < rows.length ? (
                   rows[0].length
               ) : 0
           ))
       ))([rows, reverse(rows)]);

   // enumFromThenTo :: Int -> Int -> Int -> [Int]
   const enumFromThenTo = x1 => x2 => y => {
       const d = x2 - x1;
       return Array.from({
           length: Math.floor(y - x2) / d + 2
       }, (_, i) => x1 + (d * i));
   };

   // enumFromTo :: Int -> Int -> [Int]
   const enumFromTo = m => n =>
       Array.from({
           length: 1 + n - m
       }, (_, i) => m + i);

   // eq (==) :: Eq a => a -> a -> Bool
   const eq = a => b => a === b;

   // flatten :: NestedList a -> [a]
   const flatten = nest => nest.flat(Infinity);

   // flip :: (a -> b -> c) -> b -> a -> c
   const flip = f =>
       x => y => f(y)(x);

   // index (!!) :: [a] -> Int -> a
   const index = xs => i => xs[i];

   // intercalate :: String -> [String] -> String
   const intercalate = s =>
       xs => xs.join(s);

   // isInt :: Int -> Bool
   const isInt = x => x === Math.floor(x);

   // justifyRight :: Int -> Char -> String -> String
   const justifyRight = n => cFiller => s =>
       n > s.length ? (
           s.padStart(n, cFiller)
       ) : s;

   // map :: (a -> b) -> [a] -> [b]
   const map = f => xs =>
       (Array.isArray(xs) ? (
           xs
       ) : xs.split()).map(f);

   // pred :: Enum a => a -> a
   const pred = x => x - 1;

   // replicate :: Int -> a -> [a]
   const replicate = n => x =>
       Array.from({
           length: n
       }, () => x);

   // reverse :: [a] -> [a]
   const reverse = xs =>
       'string' !== typeof xs ? (
           xs.slice(0).reverse()
       ) : xs.split().reverse().join();

   // show :: a -> String
   const show = x => JSON.stringify(x);

   // str :: a -> String
   const str = x => x.toString();

   // sum :: [Num] -> Num
   const sum = xs => xs.reduce((a, x) => a + x, 0);

   // transpose :: a -> a
   const transpose = xs =>
       xs[0].map((_, iCol) => xs.map((row) => row[iCol]));

   // unlines :: [String] -> String
   const unlines = xs => xs.join('\n');

   // zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
   const zipWith = f => xs => ys =>
       xs.slice(
           0, Math.min(xs.length, ys.length)
       ).map((x, i) => f(x)(ys[i]));

   // MAIN ------------------------------------------------
   return main();

})();</lang>

Output:
Order: 4
Summing to: 34
Row, column and diagonal sums checked: true

  1   15   14    4
 12    6    7    9
  8   10   11    5
 13    3    2   16

Order: 8
Summing to: 260
Row, column and diagonal sums checked: true

  1   63   62    4   60    6    7   57
 56   10   11   53   13   51   50   16
 48   18   19   45   21   43   42   24
 25   39   38   28   36   30   31   33
 32   34   35   29   37   27   26   40
 41   23   22   44   20   46   47   17
 49   15   14   52   12   54   55    9
  8   58   59    5   61    3    2   64

Order: 12
Summing to: 870
Row, column and diagonal sums checked: true

  1  143  142    4    5  139  138    8    9  135  134   12
132   14   15  129  128   18   19  125  124   22   23  121
120   26   27  117  116   30   31  113  112   34   35  109
 37  107  106   40   41  103  102   44   45   99   98   48
 49   95   94   52   53   91   90   56   57   87   86   60
 84   62   63   81   80   66   67   77   76   70   71   73
 72   74   75   69   68   78   79   65   64   82   83   61
 85   59   58   88   89   55   54   92   93   51   50   96
 97   47   46  100  101   43   42  104  105   39   38  108
 36  110  111   33   32  114  115   29   28  118  119   25
 24  122  123   21   20  126  127   17   16  130  131   13
133   11   10  136  137    7    6  140  141    3    2  144

Julia

<lang julia># v0.6

function magicsquaredoubleeven(order::Int) 

   if order % 4 != 0; error("the order must be divisible by 4") end

   sqr = Matrix{Int}(order, order)
   mul = div(order, 4)
   ext = vcat(1:mul, order-mul+1:order)
   isext(i::Int, j::Int) = (i in ext) == (j in ext)
   boolsqr = collect(isext(i, j) for i in 1:order, j in 1:order)
   for i in linearindices(sqr)
       if boolsqr[i]; sqr[i] = i end
       if !boolsqr[end+1-i]; sqr[end+1-i] = i end
   end
   return sqr

end

for n in (4, 8, 12) 

   magicconst = div(n ^ 3 + n, 2)
   sq = magicsquaredoubleeven(n)

   println("Order: $n; magic constant: $magicconst.\nSquare:")
   for r in 1:n, c in 1:n
       @printf("%4i", sq[r, c])
       if c == n; println() end
   end
   println()

end</lang>

Output:
Order: 4; magic constant: 34.
Square:
   1  12   8  13
  15   6  10   3
  14   7  11   2
   4   9   5  16

Order: 8; magic constant: 260.
Square:
   1   9  48  40  32  24  49  57
   2  10  47  39  31  23  50  58
  62  54  19  27  35  43  14   6
  61  53  20  28  36  44  13   5
  60  52  21  29  37  45  12   4
  59  51  22  30  38  46  11   3
   7  15  42  34  26  18  55  63
   8  16  41  33  25  17  56  64

Order: 12; magic constant: 870.
Square:
   1  13  25 108  96  84  72  60  48 109 121 133
   2  14  26 107  95  83  71  59  47 110 122 134
   3  15  27 106  94  82  70  58  46 111 123 135
 141 129 117  40  52  64  76  88 100  33  21   9
 140 128 116  41  53  65  77  89 101  32  20   8
 139 127 115  42  54  66  78  90 102  31  19   7
 138 126 114  43  55  67  79  91 103  30  18   6
 137 125 113  44  56  68  80  92 104  29  17   5
 136 124 112  45  57  69  81  93 105  28  16   4
  10  22  34  99  87  75  63  51  39 118 130 142
  11  23  35  98  86  74  62  50  38 119 131 143
  12  24  36  97  85  73  61  49  37 120 132 144

Kotlin

Translation of: Java

<lang scala>// version 1.1.0

fun magicSquareDoublyEven(n: Int): Array<IntArray> { 

   if ( n < 4 || n % 4 != 0) 
       throw IllegalArgumentException("Base must be a positive multiple of 4")

   // pattern of count-up vs count-down zones
   val bits = 0b1001_0110_0110_1001
   val size = n * n
   val mult = n / 4  // how many multiples of 4 
   val result = Array(n) { IntArray(n) }
   var i = 0
   for (r in 0 until n)
       for (c in 0 until n) {
           val bitPos = c / mult + r / mult * 4
           result[r][c] =  if (bits and (1 shl bitPos) != 0) i + 1 else size - i
           i++
       }
   return result

}

fun main(args: Array<String>) { 

   val n = 8
   for (ia in magicSquareDoublyEven(n)) { 
       for (i in ia) print("%2d  ".format(i))
       println()
   }
   println("\nMagic constant ${(n * n + 1) * n / 2}")

}</lang>

Output:
 1   2  62  61  60  59   7   8
 9  10  54  53  52  51  15  16
48  47  19  20  21  22  42  41
40  39  27  28  29  30  34  33
32  31  35  36  37  38  26  25
24  23  43  44  45  46  18  17
49  50  14  13  12  11  55  56
57  58   6   5   4   3  63  64

Magic constant 260

Lua

For all three kinds of Magic Squares(Odd, singly and doubly even)
See Magic_squares/Lua.

PARI/GP

A magic one-liner: <lang parigp>magicsquare(n)=matrix(n,n,i,j,k=i+j*n-n;if(bitand(38505,2^((j-1)%4*4+(i-1)%4)),k,n*n+1-k))</lang>

Output:

magicsquare(8)

[ 1 56 48 25 33 24 16 57]

[63 10 18 39 31 42 50  7]

[62 11 19 38 30 43 51  6]

[ 4 53 45 28 36 21 13 60]

[ 5 52 44 29 37 20 12 61]

[59 14 22 35 27 46 54  3]

[58 15 23 34 26 47 55  2]

[ 8 49 41 32 40 17  9 64]

BTW: The bit field number 38505 = 9669h seems to come from hell to do the magic...

Perl

See Magic squares/Perl for a general magic square generator. <lang perl></lang>

Perl 6

See Magic squares/Perl 6 for a general magic square generator.

Output:

With a parameter of 8: 

 1  2 62 61 60 59  7  8
 9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58  6  5  4  3 63 64

The magic number is 260

With a parameter of 12: 

  1   2   3 141 140 139 138 137 136  10  11  12
 13  14  15 129 128 127 126 125 124  22  23  24
 25  26  27 117 116 115 114 113 112  34  35  36
108 107 106  40  41  42  43  44  45  99  98  97
 96  95  94  52  53  54  55  56  57  87  86  85
 84  83  82  64  65  66  67  68  69  75  74  73
 72  71  70  76  77  78  79  80  81  63  62  61
 60  59  58  88  89  90  91  92  93  51  50  49
 48  47  46 100 101 102 103 104 105  39  38  37
109 110 111  33  32  31  30  29  28 118 119 120
121 122 123  21  20  19  18  17  16 130 131 132
133 134 135   9   8   7   6   5   4 142 143 144

The magic number is 870

Phix

Translation of: C++

<lang Phix>constant t = {{1,1,0,0}, 

             {1,1,0,0},
             {0,0,1,1},
             {0,0,1,1}}

function magic_square(integer n) 

   if n<4 or mod(n,4)!=0 then return false end if
   sequence square = repeat(repeat(0,n),n)
   integer i = 0
   for r=1 to n do
       for c=1 to n do
           square[r,c] = iff(t[mod(r,4)+1,mod(c,4)+1]?i+1:n*n-i)
           i += 1
       end for
   end for
   return square

end function

procedure check(sequence sq) 

   integer n = length(sq)
   integer magic = n*(n*n+1)/2
   integer bd = 0, fd = 0
   for i=1 to length(sq) do
       if sum(sq[i])!=magic then ?9/0 end if
       if sum(columnize(sq,i))!=magic then ?9/0 end if
       bd += sq[i,i]
       fd += sq[n-i+1,n-i+1]
   end for
   if bd!=magic or fd!=magic then ?9/0 end if

end procedure

--for i=4 to 16 by 4 do for i=8 to 8 by 4 do 

   sequence square = magic_square(i)
   printf(1,"maqic square of order %d, sum: %d\n", {i,sum(square[i])})
   string fmt = sprintf("%%%dd",length(sprintf("%d",i*i)))
   pp(square,{pp_Nest,1,pp_IntFmt,fmt,pp_StrFmt,1,pp_Pause,0})
   check(square)

end for</lang>

Output:
maqic square of order 8, sum: 260
{{ 1,63,62, 4, 5,59,58, 8},
 {56,10,11,53,52,14,15,49},
 {48,18,19,45,44,22,23,41},
 {25,39,38,28,29,35,34,32},
 {33,31,30,36,37,27,26,40},
 {24,42,43,21,20,46,47,17},
 {16,50,51,13,12,54,55, 9},
 {57, 7, 6,60,61, 3, 2,64}}

PureBasic

Translation of: FreeBasic

<lang PureBasic>Procedure.i MagicN(n.i) 

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

EndProcedure

Procedure.i MaxN(mx.i,n.i) 

 If mx>n : ProcedureReturn mx : Else : ProcedureReturn n : EndIf

EndProcedure

Procedure.i MaxL(mx.i) 

 Define.i i
 While mx
   mx/10 : i+1
 Wend
 ProcedureReturn i

EndProcedure

Procedure.b DblEvenMagicSquare(n.i) 

 Define.i q=n/4, nr=1, x, y, max, spc
 Dim sq.i(n,n)
 
 For y=1 To n
   For x=q+1 To n-q
     sq(x,y)=1
   Next
 Next
 
 For x=1 To n
   For y=q+1 To n-q
     sq(x,y) ! 1
   Next
 Next
 
 q=n*n+1
 For y=1 To n
   For x=1 To n
     If sq(x,y)=0
       sq(x,y)=q-nr
     Else
       sq(x,y)=nr
     EndIf
     nr+1
     max=MaxN(max,sq(x,y))
   Next
 Next
 
 spc=MaxL(max)+1
 For y=n To 1 Step -1
   For x=n To 1 Step -1
     Print(RSet(Str(sq(x,y)),spc," "))
   Next
   PrintN("")
 Next

EndProcedure

OpenConsole("Magic-Square-Doubly-Even") Define.i n

Repeat 

 PrintN("Input [4,8,12..n] (0=Exit)")
 While (n<4) Or (n%4)
   Print(">") : n=Val(Input())
   If n=0 : End : EndIf
 Wend  
 PrintN("The magic sum = "+Str(MagicN(n)))
 DblEvenMagicSquare(n)  
 n=0

ForEver</lang>

Output:
Input [4,8,12..n] (0=Exit)

>8
The magic sum = 260



 1  2 62 61 60 59  7  8
 9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58  6  5  4  3 63 64



Input [4,8,12..n] (0=Exit)


>

Python

Procedural

<lang python> def MagicSquareDoublyEven(order): 

   sq = [range(1+n*order,order + (n*order)+1) for n in range(order) ]
   n1 = order/4
   for r in range(n1):
       r1 = sq[r][n1:-n1]
       r2 = sq[order -r - 1][n1:-n1]
       r1.reverse()
       r2.reverse()
       sq[r][n1:-n1] = r2
       sq[order -r - 1][n1:-n1] = r1
   for r in range(n1, order-n1):
       r1 = sq[r][:n1]
       r2 = sq[order -r - 1][order-n1:]
       r1.reverse()
       r2.reverse()
       sq[r][:n1] = r2
       sq[order -r - 1][order-n1:] = r1
   return sq

def printsq(s): 

   n = len(s)
   bl = len(str(n**2))+1
   for i in range(n):
       print .join( [ ("%"+str(bl)+"s")%(str(x)) for x in s[i]] )
   print "\nMagic constant = %d"%sum(s[0])

printsq(MagicSquareDoublyEven(8)) </lang>

Output:


 1  2 62 61 60 59  7  8
 9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58  6  5  4  3 63 64



Magic constant = 260


>>>


Composition of pure functions

Generating test results and a magic square in the form of a wiki table:

Works with: Python version 3.7

<lang python>Magic squares of doubly even order

from itertools import chain, repeat from functools import reduce from math import log


  1. doublyEvenMagicSquare :: Int -> Int

def doublyEvenMagicSquare(n): 

   Magic square of order n

   # magic :: Int -> [Bool]
   def magic(n):
       Truth-table series
       if 0 < n:
           xs = magic(n - 1)
           return xs + [not x for x in xs]
       else:
           return [True]

   sqr = n * n
   power = log(sqr, 2)
   scale = replicate(n / 4)
   return chunksOf(n)([
       succ(i) if bln else sqr - i for i, bln in
       enumerate(magic(power) if isInteger(power) else (
           flatten(scale(
               map(scale, chunksOf(4)(magic(4)))
           ))
       ))
   ])


  1. TEST ----------------------------------------------------
  2. main :: IO()

def main(): 

   Tests

   order = 8
   magicSquare = doublyEvenMagicSquare(order)

   print(
       'Row sums: ',
       [sum(xs) for xs in magicSquare],
       '\nCol sums:',
       [sum(xs) for xs in transpose(magicSquare)],
       '\n1st diagonal sum:',
       sum(magicSquare[i][i] for i in range(0, order)),
       '\n2nd diagonal sum:',
       sum(magicSquare[i][(order - 1) - i] for i in range(0, order)),
       '\n'
   )
   print(wikiTable({
       'class': 'wikitable',
       'style': cssFromDict({
           'text-align': 'center',
           'color': '#605B4B',
           'border': '2px solid silver'
       }),
       'colwidth': '3em'
   })(magicSquare))


  1. GENERIC -------------------------------------------------


  1. chunksOf :: Int -> [a] -> a

def chunksOf(n): 

   A series of lists of length n,
      subdividing the contents of xs.
      Where the length of xs is not evenly divible
      the final list will be shorter than n.
   return lambda xs: reduce(
       lambda a, i: a + [xs[i:n + i]],
       range(0, len(xs), n), []
   ) if 0 < n else []


  1. concatMap :: (a -> [b]) -> [a] -> [b]

def concatMap(f): 

   Concatenated list over which a function has been mapped.
      The list monad can be derived by using a function f which
      wraps its output a in list
      (using an empty list to represent computational failure).
   return lambda xs: list(
       chain.from_iterable(
           map(f, xs)
       )
   )


  1. cssFromDict :: Dict -> String

def cssFromDict(dct): 

   CSS string serialized from key values in a Dictionary.
   return reduce(
       lambda a, k: a + k + ':' + dct[k] + '; ', dct.keys(), 
   )


  1. flatten :: NestedList a -> [a]

def flatten(x): 

   A list of atoms resulting from fully flattening
      an arbitrarily nested list.
   return concatMap(flatten)(x) if isinstance(x, list) else [x]


  1. isInteger :: Num -> Bool

def isInteger(n): 

   Divisible by one without remainder ?
   return 0 == (n - int(n))


  1. replicate :: Int -> a -> [a]

def replicate(n): 

   A list of length n in which every element
      has the value x.
   return lambda x: list(repeat(x, n))


  1. succ :: Enum a => a -> a

def succ(x): 

   The successor of a value. For numeric types, (1 +).
   return 1 + x if isinstance(x, int) else (
       chr(1 + ord(x))
   )


  1. transpose :: Matrix a -> Matrix a

def transpose(m): 

   The rows and columns of the argument transposed.
      (The matrix containers and rows can be lists or tuples).
   if m:
       inner = type(m[0])
       z = zip(*m)
       return (type(m))(
           map(inner, z) if tuple != inner else z
       )
   else:
       return m


  1. wikiTable :: Dict -> a -> String

def wikiTable(opts): 

   List of lists rendered as a wiki table string.
   def colWidth():
       return 'width:' + opts['colwidth'] + '; ' if (
           'colwidth' in opts
       ) else 

   def cellStyle():
       return opts['cell'] if 'cell' in opts else 

   return lambda rows: '{| ' + reduce(
       lambda a, k: (
           a + k + '="' + opts[k] + '" ' if k in opts else a
       ),
       ['class', 'style'],
       
   ) + '\n' + '\n|-\n'.join(
       '\n'.join(
           ('|' if (0 != i and ('cell' not in opts)) else (
               '|style="' + colWidth() + cellStyle() + '"|'
           )) + (
               str(x) or ' '
           ) for x in row
       ) for i, row in enumerate(rows)
   ) + '\n|}\n\n'


if __name__ == '__main__': 

   main()</lang>
Output:
Row sums:  [260, 260, 260, 260, 260, 260, 260, 260] 
Col sums: [260, 260, 260, 260, 260, 260, 260, 260] 
1st diagonal sum: 260 
2nd diagonal sum: 260
1 63 62 4 60 6 7 57
56 10 11 53 13 51 50 16
48 18 19 45 21 43 42 24
25 39 38 28 36 30 31 33
32 34 35 29 37 27 26 40
41 23 22 44 20 46 47 17
49 15 14 52 12 54 55 9
8 58 59 5 61 3 2 64

REXX

"Marked" numbers   (via the   diag   subroutine)   indicate that those (sequentially generated) numbers don't get
swapped   (and thusly, stay in place in the magic square). <lang rexx>/*REXX program constructs a magic square of doubly even sides (a size divisible by 4).*/ n=8; s=n%4; L=n%2-s+1; w=length(n**2) /*size; small sq; low middle; # width*/ @.=0; H=n%2+s /*array default; high middle. */ call gen /*generate a grid in numerical order. */ call diag /*mark numbers on both diagonals. */ call corn /* " " in small corner boxen. */ call midd /* " " in the middle " */ call swap /*swap positive numbers with highest #.*/ call show /*display the doubly even magic square.*/ call sum /* " " magic number for square. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ o: parse arg ?; return n-?+1 /*calculate the "other" (right) column.*/ @: parse arg x,y; return abs(@.x.y) diag: do r=1 for n; @.r.r=-@(r,r); o=o(r); @.r.o=-@(r,o); end; return midd: do r=L to H; do c=L to H; @.r.c=-@(r,c); end; end; return gen: #=0; do r=1 for n; do c=1 for n; #=#+1; @.r.c=#; end; end; return show: #=0; do r=1 for n; $=; do c=1 for n; $=$ right(@(r,c),w); end; say $; end; return sum: #=0; do r=1 for n; #=#+@(r,1); end; say; say 'The magic number is: ' #; return max#: do a=n for n by -1; do b=n for n by -1; if @.a.b>0 then return; end; end /*──────────────────────────────────────────────────────────────────────────────────────*/ swap: do r=1 for n 

               do c=1  for n;  if @.r.c<0  then iterate;  call max#  /*find max number.*/
               parse value  -@.a.b  (-@.r.c)    with    @.r.c  @.a.b /*swap two values.*/
               end  /*c*/
             end    /*r*/
     return

/*──────────────────────────────────────────────────────────────────────────────────────*/ corn: do r=1 for n; if r>s & r<=n-s then iterate /*"corner boxen", size≡S*/ 

               do c=1  for n;  if c>s & c<=n-s  then iterate;  @.r.c=-@(r,c);  end  /*c*/
             end   /*r*/
     return</lang>

output   when using the default input: 

  1  2 62 61 60 59  7  8
  9 10 54 53 52 51 15 16
 48 47 19 20 21 22 42 41
 40 39 27 28 29 30 34 33
 32 31 35 36 37 38 26 25
 24 23 43 44 45 46 18 17
 49 50 14 13 12 11 55 56
 57 58  6  5  4  3 63 64

The magic number is:  260

Ruby

<lang ruby>def double_even_magic_square(n) 

 raise ArgumentError, "Need multiple of four" if n%4 > 0
 block_size, max = n/4, n*n
 pre_pat = [true, false, false, true,
            false, true, true, false]
 pre_pat += pre_pat.reverse
 pattern = pre_pat.flat_map{|b| [b] * block_size} * block_size
 flat_ar = pattern.each_with_index.map{|yes, num| yes ? num+1 : max-num}
 flat_ar.each_slice(n).to_a

end

def to_string(square) 

 n = square.size
 fmt = "%#{(n*n).to_s.size + 1}d" * n
 square.inject(""){|str,row| str << fmt % row << "\n"}

end

puts to_string(double_even_magic_square(8))</lang>

Output:
  1  2 62 61 60 59  7  8
 56 55 11 12 13 14 50 49
 48 47 19 20 21 22 42 41
 25 26 38 37 36 35 31 32
 33 34 30 29 28 27 39 40
 24 23 43 44 45 46 18 17
 16 15 51 52 53 54 10  9
 57 58  6  5  4  3 63 64

Rust

<lang rust>use std::env;

fn main() { 

   let n: usize = match env::args()
       .nth(1)
       .and_then(|arg| arg.parse().ok())
       .ok_or("Please specify the size of the magic square, as a positive multiple of 4.")
   {
       Ok(arg) if arg >= 4 && arg % 4 == 0 => arg,
       Err(e) => panic!(e),
       _ => panic!("Argument must be a positive multiple of 4."),
   };

   let mc = (n * n + 1) * n / 2;
   println!("Magic constant: {}\n", mc);
   let bits = 0b1001_0110_0110_1001u32;
   let size = n * n;
   let width = size.to_string().len() + 1;
   let mult = n / 4;
   let mut i = 0;
   for r in 0..n {
       for c in 0..n {
           let bit_pos = c / mult + (r / mult) * 4;
           print!(
               "{e:>w$}",
               e = if bits & (1 << bit_pos) != 0 {
                   i + 1
               } else {
                   size - i
               },
               w = width
           );
           i += 1;
       }
       println!();
   }

}</lang>

Output:
Magic constant: 260

  1  2 62 61 60 59  7  8
  9 10 54 53 52 51 15 16
 48 47 19 20 21 22 42 41
 40 39 27 28 29 30 34 33
 32 31 35 36 37 38 26 25
 24 23 43 44 45 46 18 17
 49 50 14 13 12 11 55 56
 57 58  6  5  4  3 63 64

Scala

Output:

Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

<lang Scala>object MagicSquareDoublyEven extends App { 

 private val n = 8

 private def magicSquareDoublyEven(n: Int): Array[Array[Int]] = {
   require(n >= 4 || n % 4 == 0, "Base must be a positive multiple of 4.")

   // pattern of count-up vs count-down zones
   val (bits, mult, result, size) = (38505, n / 4, Array.ofDim[Int](n, n), n * n)
   var i = 0

   for (r <- result.indices; c <- result(0).indices) {
     def bitPos = c / mult + (r / mult) * 4

     result(r)(c) = if ((bits & (1 << bitPos)) != 0) i + 1 else size - i
     i += 1
   }
   result
 }

 magicSquareDoublyEven(n).foreach(row => println(row.map(x => f"$x%2s ").mkString))
 println(f"---%nMagic constant: ${(n * n + 1) * n / 2}%d")

}</lang>

Sidef

Translation of: Ruby

<lang ruby>func double_even_magic_square(n) { 

   assert(n%4 == 0, "Need multiple of four")
   var (bsize, max) = (n/4, n*n)
   var pre_pat = [true, false, false, true,
                  false, true, true, false]
   pre_pat += pre_pat.flip
   var pattern = (pre_pat.map{|b| bsize.of(b)... } * bsize)
   pattern.map_kv{|k,v| v ? k+1 : max-k }.slices(n)

}

func format_matrix(a) { 

   var fmt = "%#{a.len**2 -> len}s"
   a.map { .map { fmt % _ }.join(' ') }.join("\n")

}

say format_matrix(double_even_magic_square(8))</lang>

Output:
 1  2 62 61 60 59  7  8
56 55 11 12 13 14 50 49
48 47 19 20 21 22 42 41
25 26 38 37 36 35 31 32
33 34 30 29 28 27 39 40
24 23 43 44 45 46 18 17
16 15 51 52 53 54 10  9
57 58  6  5  4  3 63 64

VBScript

Translation of: Java

<lang vb>' Magic squares of doubly even order n=8 'multiple of 4 pattern="1001011001101001" size=n*n: w=len(size) mult=n\4 'how many multiples of 4 wscript.echo "Magic square : " & n & " x " & n i=0 For r=0 To n-1 l="" For c=0 To n-1 bit=Mid(pattern, c\mult+(r\mult)*4+1, 1) If bit="1" Then t=i+1 Else t=size-i l=l & Right(Space(w) & t, w) & " " i=i+1 Next 'c wscript.echo l Next 'r wscript.echo "Magic constant=" & (n*n+1)*n/2</lang>

Output:
Magic square : 8 x 8
 1  2 62 61 60 59  7  8 
 9 10 54 53 52 51 15 16 
48 47 19 20 21 22 42 41 
40 39 27 28 29 30 34 33 
32 31 35 36 37 38 26 25 
24 23 43 44 45 46 18 17 
49 50 14 13 12 11 55 56 
57 58  6  5  4  3 63 64 
Magic constant=260

Visual Basic .NET

Translation of: C#

<lang vbnet>Module MagicSquares 

   Function MagicSquareDoublyEven(n As Integer) As Integer(,)
       If n < 4 OrElse n Mod 4 <> 0 Then
           Throw New ArgumentException("base must be a positive multiple of 4")
       End If

       'pattern of count-up vs count-down zones
       Dim bits = Convert.ToInt32("1001011001101001", 2)
       Dim size = n * n
       Dim mult As Integer = n / 4 ' how many multiples of 4

       Dim result(n - 1, n - 1) As Integer

       Dim i = 0
       For r = 0 To n - 1
           For c = 0 To n - 1
               Dim bitPos As Integer = Math.Floor(c / mult) + Math.Floor(r / mult) * 4
               Dim test = (bits And (1 << bitPos)) <> 0
               If test Then
                   result(r, c) = i + 1
               Else
                   result(r, c) = size - i
               End If

               i = i + 1
           Next
           Console.WriteLine()
       Next

       Return result
   End Function

   Sub Main()
       Dim n = 8
       Dim result = MagicSquareDoublyEven(n)
       For i = 0 To result.GetLength(0) - 1
           For j = 0 To result.GetLength(1) - 1
               Console.Write("{0,2} ", result(i, j))
           Next
           Console.WriteLine()
       Next
       Console.WriteLine()
       Console.WriteLine("Magic constant: {0} ", (n * n + 1) * n / 2)

       Console.ReadLine()
   End Sub

End Module</lang>

Output:
 1  2 62 61 60 59  7  8
 9 10 54 53 52 51 15 16
48 47 19 20 21 22 42 41
40 39 27 28 29 30 34 33
32 31 35 36 37 38 26 25
24 23 43 44 45 46 18 17
49 50 14 13 12 11 55 56
57 58  6  5  4  3 63 64

Magic constant: 260

zkl

Translation of: Java

<lang zkl>class MagicSquareDoublyEven{ 

  fcn init(n){ var result=magicSquareDoublyEven(n) }
  fcn toString{
     sink,n:=Sink(String),result.len();  // num collumns
     fmt:="%2s ";
     foreach row in (result)
        { sink.write(row.apply('wrap(n){ fmt.fmt(n) }).concat(),"\n") }
     sink.write("\nMagic constant: %d".fmt((n*n + 1)*n/2));
     sink.close();
  }
  fcn magicSquareDoublyEven(n){
     if (n<4 or n%4!=0 or n>16)

throw(Exception.ValueError("base must be a positive multiple of 4")); 

     bits,size,mult:=0b1001011001101001, n*n, n/4;
     result:=n.pump(List(),n.pump(List(),0).copy);  // array[n,n] of zero

     foreach i in (size){

bitsPos:=(i%n)/mult + (i/(n*mult)*4); value:=(bits.bitAnd((2).pow(bitsPos))) and i+1 or size-i; result[i/n][i%n]=value; 

     }
     result;
  }

} MagicSquareDoublyEven(8).println();</lang>

Output:
 1  2 62 61 60 59  7  8 
 9 10 54 53 52 51 15 16 
48 47 19 20 21 22 42 41 
40 39 27 28 29 30 34 33 
32 31 35 36 37 38 26 25 
24 23 43 44 45 46 18 17 
49 50 14 13 12 11 55 56 
57 58  6  5  4  3 63 64 

Magic constant: 260
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