Magic squares of singly even order

From Rosetta Code
Task
Magic squares of singly 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 singly even order has a size that is a multiple of 4, plus 2 (e.g. 6, 10, 14). This means that the subsquares have an odd size, which plays a role in the construction.

Task

Create a magic square of 6 x 6.

Related tasks
See also



C++[edit]

 
#include <iostream>
#include <sstream>
#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++;
while( d % 4 == 0 ) { d += 2; }
sz = d;
sqr = new int[sz * sz];
memset( sqr, 0, sz * sz * sizeof( int ) );
fillSqr();
}
void display() {
cout << "Singly 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 siamese( int from, int to ) {
int oneSide = to - from, curCol = oneSide / 2, curRow = 0, count = oneSide * oneSide, s = 1;
 
while( count > 0 ) {
bool done = false;
while ( false == done ) {
if( curCol >= oneSide ) curCol = 0;
if( curRow < 0 ) curRow = oneSide - 1;
done = true;
if( sqr[curCol + sz * curRow] != 0 ) {
curCol -= 1; curRow += 2;
if( curCol < 0 ) curCol = oneSide - 1;
if( curRow >= oneSide ) curRow -= oneSide;
 
done = false;
}
}
sqr[curCol + sz * curRow] = s;
s++; count--; curCol++; curRow--;
}
}
void fillSqr() {
int n = sz / 2, ns = n * sz, size = sz * sz, add1 = size / 2, add3 = size / 4, add2 = 3 * add3;
 
siamese( 0, n );
 
for( int r = 0; r < n; r++ ) {
int row = r * sz;
for( int c = n; c < sz; c++ ) {
int m = sqr[c - n + row];
 
sqr[c + row] = m + add1;
sqr[c + row + ns] = m + add3;
sqr[c - n + row + ns] = m + add2;
}
}
 
int lc = ( sz - 2 ) / 4, co = sz - ( lc - 1 );
for( int r = 0; r < n; r++ ) {
int row = r * sz;
for( int c = co; c < sz; c++ ) {
sqr[c + row] -= add3;
sqr[c + row + ns] += add3;
}
}
for( int r = 0; r < n; r++ ) {
int row = r * sz;
for( int c = 0; c < lc; c++ ) {
int cc = c;
if( r == lc ) cc++;
sqr[cc + row] += add2;
sqr[cc + row + ns] -= add2;
}
}
}
int magicNumber() { return sz * ( ( sz * sz ) + 1 ) / 2; }
 
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( 6 );
s.display();
return 0;
}
 
Output:
Singly Even Magic Square: 6 x 6
It's Magic Sum is: 111

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

D[edit]

Translation of: Java
 
import std.exception;
import std.stdio;
 
void main() {
int n = 6;
foreach (row; magicSquareSinglyEven(n)) {
foreach (x; row) {
writef("%2s ", x);
}
writeln();
}
writeln("\nMagic constant: ", (n * n + 1) * n / 2);
}
 
int[][] magicSquareOdd(const int n) {
enforce(n >= 3 && n % 2 != 0, "Base must be odd and >2");
 
int value = 0;
int gridSize = n * n;
int c = n / 2;
int r = 0;
 
int[][] result = new int[][](n, n);
 
while(++value <= gridSize) {
result[r][c] = value;
if (r == 0) {
if (c == n - 1) {
r++;
} else {
r = n - 1;
c++;
}
} else if (c == n - 1) {
r--;
c = 0;
} else if (result[r - 1][c + 1] == 0) {
r--;
c++;
} else {
r++;
}
}
 
return result;
}
 
int[][] magicSquareSinglyEven(const int n) {
enforce(n >= 6 && (n - 2) % 4 == 0, "Base must be a positive multiple of four plus 2");
 
int size = n * n;
int halfN = n / 2;
int subSquareSize = size / 4;
 
int[][] subSquare = magicSquareOdd(halfN);
int[] quadrantFactors = [0, 2, 3, 1];
int[][] result = new int[][](n, n);
 
for (int r = 0; r < n; r++) {
for (int c = 0; c < n; c++) {
int quadrant = (r / halfN) * 2 + (c / halfN);
result[r][c] = subSquare[r % halfN][c % halfN];
result[r][c] += quadrantFactors[quadrant] * subSquareSize;
}
}
 
int nColsLeft = halfN / 2;
int nColsRight = nColsLeft - 1;
 
for (int r = 0; r < halfN; r++) {
for (int c = 0; c < n; c++) {
if (c < nColsLeft || c >= n - nColsRight
|| (c == nColsLeft && r == nColsLeft)) {
if (c == 0 && r == nColsLeft) {
continue;
}
 
int tmp = result[r][c];
result[r][c] = result[r + halfN][c];
result[r + halfN][c] = tmp;
}
}
}
 
return result;
}
 

Elixir[edit]

wp:Conway's LUX method for magic squares:

defmodule Magic_square do
@lux  %{ L: [4, 1, 2, 3], U: [1, 4, 2, 3], X: [1, 4, 3, 2] }
 
def singly_even(n) when rem(n-2,4)!=0, do: raise ArgumentError, "must be even, but not divisible by 4."
def singly_even(2), do: raise ArgumentError, "2x2 magic square not possible."
def singly_even(n) do
n2 = div(n, 2)
oms = odd_magic_square(n2)
mat = make_lux_matrix(n2)
square = synthesis(n2, oms, mat)
IO.puts to_string(n, square)
square
end
 
defp odd_magic_square(m) do # zero beginning, it is 4 multiples.
for i <- 0..m-1, j <- 0..m-1, into: %{},
do: {{i,j}, (m*(rem(i+j+1+div(m,2),m)) + rem(i+2*j-5+2*m, m)) * 4}
end
 
defp make_lux_matrix(m) do
center = div(m, 2)
lux = List.duplicate(:L, center+1) ++ [:U] ++ List.duplicate(:X, m-center-2)
(for {x,i} <- Enum.with_index(lux), j <- 0..m-1, into: %{}, do: {{i,j}, x})
|> Map.put({center, center}, :U)
|> Map.put({center+1, center}, :L)
end
 
defp synthesis(m, oms, mat) do
range = 0..m-1
Enum.reduce(range, [], fn i,acc ->
{row0, row1} = Enum.reduce(range, {[],[]}, fn j,{r0,r1} ->
x = oms[{i,j}]
[lux0, lux1, lux2, lux3] = @lux[mat[{i,j}]]
{[x+lux0, x+lux1 | r0], [x+lux2, x+lux3 | r1]}
end)
[row0, row1 | acc]
end)
end
 
defp to_string(n, square) 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.singly_even(6)
Output:
 5  8 36 33 13 16
 6  7 34 35 14 15
28 25 17 20 12  9
26 27 18 19 10 11
24 21  4  1 32 29
22 23  2  3 30 31

FreeBASIC[edit]

' version 18-03-2016
' compile with: fbc -s console
' singly even magic square 6, 10, 14, 18...
 
Sub Err_msg(msg As String)
Print msg
Beep : Sleep 5000, 1 : Exit Sub
End Sub
 
Sub se_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 < 6 Then
Err_msg( "Error: n is to small")
Exit Sub
End If
 
If ((n -2) Mod 4) <> 0 Then
Err_msg "Error: not possible to make singly" + _
" 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 sq_d_2 = n \ 2, q2 = sq_d_2 ^ 2
Dim As UInteger l = (n -2) \ 4
Dim As UInteger x = sq_d_2 \ 2 + 1, y = 1, nr = 1
Dim As String frmt = String(Len(Str(n * n)) +1, "#")
 
' fill pattern A C
' D B
' main loop for creating magic square in section A
' the value for B,C and D is derived from A
' uses the FreeBASIC odd order magic square routine
Do
If sq(x, y) = 0 Then
sq(x , y ) = nr ' A
sq(x + sq_d_2, y + sq_d_2) = nr + q2 ' B
sq(x + sq_d_2, y ) = nr + q2 * 2 ' C
sq(x , y + sq_d_2) = nr + q2 * 3 ' D
If nr Mod sq_d_2 = 0 Then
y += 1
Else
x += 1 : y -= 1
End If
nr += 1
End If
If x > sq_d_2 Then
x = 1
Do While sq(x,y) <> 0
x += 1
Loop
End If
If y < 1 Then
y = sq_d_2
Do While sq(x,y) <> 0
y -= 1
Loop
End If
Loop Until nr > q2
 
 
' swap left side
For y = 1 To sq_d_2
For x = 1 To l
Swap sq(x, y), sq(x,y + sq_d_2)
Next
Next
' make indent
y = (sq_d_2 \ 2) +1
Swap sq(1, y), sq(1, y + sq_d_2) ' was swapped, restore to orignal value
Swap sq(l +1, y), sq(l +1, y + sq_d_2)
 
' swap right side
For y = 1 To sq_d_2
For x = n - l +2 To n
Swap sq(x, y), sq(x,y + sq_d_2)
Next
Next
 
' check columms and rows
For y = 1 To n
nr = 0 : l = 0
For x = 1 To n
nr += sq(x,y)
l += sq(y,x)
Next
If nr <> magic_sum Or l <> magic_sum Then
Err_msg "Error: value <> magic_sum"
Exit Sub
End If
Next
 
' check diagonals
nr = 0 : l = 0
For x = 1 To n
nr += sq(x, x)
l += sq(n - x +1, n - x +1)
Next
If nr <> magic_sum Or l <> 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 >=------
 
se_magicsq(6, "magicse6.txt") : Print
 
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Output:
Single even magic square size: 6*6
The magic sum = 111

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

Haskell[edit]

Using Conway's LUX method for magic squares

import qualified Data.Map.Strict as M
import Data.List (transpose, intercalate)
import Data.Maybe (fromJust, isJust)
import Control.Monad (forM_)
import Data.Monoid ((<>))
 
magic :: Int -> [[Int]]
magic n = mapAsTable ((4 * n) + 2) (hiResMap n)
 
-- Order of square -> sequence numbers keyed by cartesian coordinates
hiResMap :: Int -> M.Map (Int, Int) Int
hiResMap n =
let mapLux = luxMap n
mapSiam = siamMap n
in M.fromList $
foldMap
(\(xy, n) ->
luxNums xy (fromJust (M.lookup xy mapLux)) ((4 * (n - 1)) + 1))
(M.toList mapSiam)
 
-- LUX table coordinate -> L|U|X -> initial number -> 4 numbered coordinates
luxNums :: (Int, Int) -> Char -> Int -> [((Int, Int), Int)]
luxNums xy lux n =
zipWith (\x d -> (x, n + d)) (hiRes xy) $
case lux of
'L' -> [3, 0, 1, 2]
'U' -> [0, 3, 1, 2]
'X' -> [0, 3, 2, 1]
_ -> [0, 0, 0, 0]
 
-- Size of square -> integers keyed by coordinates -> rows of integers
mapAsTable :: Int -> M.Map (Int, Int) Int -> [[Int]]
mapAsTable nCols xyMap =
let axis = [0 .. nCols - 1]
in fmap (fromJust . flip M.lookup xyMap) <$>
(axis >>= \y -> [axis >>= \x -> [(x, y)]])
 
-- Dimension of LUX table -> LUX symbols keyed by coordinates
luxMap :: Int -> M.Map (Int, Int) Char
luxMap n =
(M.fromList . concat) $
zipWith
(\y xs -> (zipWith (\x c -> ((x, y), c)) [0 ..] xs))
[0 ..]
(luxPattern n)
 
-- LUX dimension -> square of L|U|X cells with two mixed rows
luxPattern :: Int -> [String]
luxPattern n =
let d = (2 * n) + 1
[ls, us] = replicate n <$> "LU"
[lRow, xRow] = replicate d <$> "LX"
in replicate n lRow <> [ls <> ('U' : ls)] <> [us <> ('L' : us)] <>
replicate (n - 1) xRow
 
-- Highest zero-based index of grid -> Siamese indices keyed by coordinates
siamMap :: Int -> M.Map (Int, Int) Int
siamMap n =
let uBound = (2 * n)
sPath uBound sMap (x, y) n =
let newMap = M.insert (x, y) n sMap
in if y == uBound && x == quot uBound 2
then newMap
else sPath uBound newMap (nextSiam uBound sMap (x, y)) (n + 1)
in sPath uBound (M.fromList []) (n, 0) 1
 
-- Highest index of square -> Siam xys so far -> xy -> next xy coordinate
nextSiam :: Int -> M.Map (Int, Int) Int -> (Int, Int) -> (Int, Int)
nextSiam uBound sMap (x, y) =
let alt (a, b)
| a > uBound && b < 0 = (uBound, 1) -- Top right corner ?
| a > uBound = (0, b) -- beyond right edge ?
| b < 0 = (a, uBound) -- above top edge ?
| isJust (M.lookup (a, b) sMap) = (a - 1, b + 2) -- already filled ?
| otherwise = (a, b) -- Up one, right one.
in alt (x + 1, y - 1)
 
-- LUX cell coordinate -> four coordinates at higher resolution
hiRes :: (Int, Int) -> [(Int, Int)]
hiRes (x, y) =
let [col, row] = (* 2) <$> [x, y]
[col1, row1] = succ <$> [col, row]
in [(col, row), (col1, row), (col, row1), (col1, row1)]
 
-- TESTS ----------------------------------------------------------------------
checked :: [[Int]] -> (Int, Bool)
checked square = (h, all (h ==) t)
where
diagonals = fmap (flip (zipWith (!!)) [0 ..]) . ((:) <*> (return . reverse))
h:t = sum <$> square <> transpose square <> diagonals square
 
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_ [1, 2, 3] $
\n -> do
let test = magic n
putStrLn $ unlines (table " " (fmap show <$> test))
print $ checked test
putStrLn ""
Output:
32 29  4  1 24 21
30 31  2  3 22 23
12  9 17 20 28 25
10 11 18 19 26 27
13 16 36 33  5  8
14 15 34 35  6  7

(111,True)

68 65 96 93  4   1 32 29 60 57
66 67 94 95  2   3 30 31 58 59
92 89 20 17 28  25 56 53 64 61
90 91 18 19 26  27 54 55 62 63
16 13 24 21 49  52 80 77 88 85
14 15 22 23 50  51 78 79 86 87
37 40 45 48 76  73 81 84  9 12
38 39 46 47 74  75 82 83 10 11
41 44 69 72 97 100  5  8 33 36
43 42 71 70 99  98  7  6 35 34

(505,True)

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

(1379,True)

Java[edit]

public class MagicSquareSinglyEven {
 
public static void main(String[] args) {
int n = 6;
for (int[] row : magicSquareSinglyEven(n)) {
for (int x : row)
System.out.printf("%2s ", x);
System.out.println();
}
System.out.printf("\nMagic constant: %d ", (n * n + 1) * n / 2);
}
 
public static int[][] magicSquareOdd(final int n) {
if (n < 3 || n % 2 == 0)
throw new IllegalArgumentException("base must be odd and > 2");
 
int value = 0;
int gridSize = n * n;
int c = n / 2, r = 0;
 
int[][] result = new int[n][n];
 
while (++value <= gridSize) {
result[r][c] = value;
if (r == 0) {
if (c == n - 1) {
r++;
} else {
r = n - 1;
c++;
}
} else if (c == n - 1) {
r--;
c = 0;
} else if (result[r - 1][c + 1] == 0) {
r--;
c++;
} else {
r++;
}
}
return result;
}
 
static int[][] magicSquareSinglyEven(final int n) {
if (n < 6 || (n - 2) % 4 != 0)
throw new IllegalArgumentException("base must be a positive "
+ "multiple of 4 plus 2");
 
int size = n * n;
int halfN = n / 2;
int subSquareSize = size / 4;
 
int[][] subSquare = magicSquareOdd(halfN);
int[] quadrantFactors = {0, 2, 3, 1};
int[][] result = new int[n][n];
 
for (int r = 0; r < n; r++) {
for (int c = 0; c < n; c++) {
int quadrant = (r / halfN) * 2 + (c / halfN);
result[r][c] = subSquare[r % halfN][c % halfN];
result[r][c] += quadrantFactors[quadrant] * subSquareSize;
}
}
 
int nColsLeft = halfN / 2;
int nColsRight = nColsLeft - 1;
 
for (int r = 0; r < halfN; r++)
for (int c = 0; c < n; c++) {
if (c < nColsLeft || c >= n - nColsRight
|| (c == nColsLeft && r == nColsLeft)) {
 
if (c == 0 && r == nColsLeft)
continue;
 
int tmp = result[r][c];
result[r][c] = result[r + halfN][c];
result[r + halfN][c] = tmp;
}
}
 
return result;
}
}
35  1  6 26 19 24 
 3 32  7 21 23 25 
31  9  2 22 27 20 
 8 28 33 17 10 15 
30  5 34 12 14 16 
 4 36 29 13 18 11 

Magic constant: 111

Kotlin[edit]

Translation of: Java
// version 1.0.6
 
fun magicSquareOdd(n: Int): Array<IntArray> {
if (n < 3 || n % 2 == 0)
throw IllegalArgumentException("Base must be odd and > 2")
 
var value = 0
val gridSize = n * n
var c = n / 2
var r = 0
val result = Array(n) { IntArray(n) }
while (++value <= gridSize) {
result[r][c] = value
if (r == 0) {
if (c == n - 1) r++
else {
r = n - 1
c++
}
}
else if (c == n - 1) {
r--
c = 0
}
else if (result[r - 1][c + 1] == 0) {
r--
c++
}
else r++
}
return result
}
 
fun magicSquareSinglyEven(n: Int): Array<IntArray> {
if (n < 6 || (n - 2) % 4 != 0)
throw IllegalArgumentException("Base must be a positive multiple of 4 plus 2")
 
val size = n * n
val halfN = n / 2
val subSquareSize = size / 4
val subSquare = magicSquareOdd(halfN)
val quadrantFactors = intArrayOf(0, 2, 3, 1)
val result = Array(n) { IntArray(n) }
for (r in 0 until n)
for (c in 0 until n) {
val quadrant = r / halfN * 2 + c / halfN
result[r][c] = subSquare[r % halfN][c % halfN]
result[r][c] += quadrantFactors[quadrant] * subSquareSize
}
val nColsLeft = halfN / 2
val nColsRight = nColsLeft - 1
for (r in 0 until halfN)
for (c in 0 until n)
if (c < nColsLeft || c >= n - nColsRight || (c == nColsLeft && r == nColsLeft)) {
if (c == 0 && r == nColsLeft) continue
val tmp = result[r][c]
result[r][c] = result[r + halfN][c]
result[r + halfN][c] = tmp
}
return result
}
 
fun main(args: Array<String>) {
val n = 6
for (ia in magicSquareSinglyEven(n)) {
for (i in ia) print("%2d ".format(i))
println()
}
println("\nMagic constant ${(n * n + 1) * n / 2}")
}
Output:
35   1   6  26  19  24
 3  32   7  21  23  25
31   9   2  22  27  20
 8  28  33  17  10  15
30   5  34  12  14  16
 4  36  29  13  18  11

Magic constant 111

Lua[edit]

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

Perl 6[edit]

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

Output:

With a parameter of 6:

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

The magic number is 111

With a parameter of 10:

 92  99   1   8  15  67  74  51  58  40
 98  80   7  14  16  73  55  57  64  41
  4  81  88  20  22  54  56  63  70  47
 85  87  19  21   3  60  62  69  71  28
 86  93  25   2   9  61  68  75  52  34
 17  24  76  83  90  42  49  26  33  65
 23   5  82  89  91  48  30  32  39  66
 79   6  13  95  97  29  31  38  45  72
 10  12  94  96  78  35  37  44  46  53
 11  18 100  77  84  36  43  50  27  59

The magic number is 505

Ruby[edit]

def odd_magic_square(n)
n.times.map{|i| n.times.map{|j| n*((i+j+1+n/2)%n) + ((i+2*j-5)%n) + 1} }
end
 
def single_even_magic_square(n)
raise ArgumentError, "must be even, but not divisible by 4." unless (n-2) % 4 == 0
raise ArgumentError, "2x2 magic square not possible." if n == 2
 
order = (n-2)/4
odd_square = odd_magic_square(n/2)
to_add = (0..3).map{|f| f*n*n/4}
quarts = to_add.map{|f| odd_square.dup.map{|row|row.map{|el| el+f}} }
 
sq = []
quarts[0].zip(quarts[2]){|d1,d2| sq << [d1,d2].flatten}
quarts[3].zip(quarts[1]){|d1,d2| sq << [d1,d2].flatten}
 
sq = sq.transpose
order.times{|i| sq[i].rotate!(n/2)}
swap(sq[0][order], sq[0][-order-1])
swap(sq[order][order], sq[order][-order-1])
(order-1).times{|i| sq[-(i+1)].rotate!(n/2)}
randomize(sq)
end
 
def swap(a,b)
a,b = b,a
end
 
def randomize(square)
square.shuffle.transpose.shuffle
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(single_even_magic_square(6))
Output:
 23  7  5 21 30 25
 18 29 36 13  4 11
 14 34 32 12  3 16
 19  6  1 26 35 24
 27  2  9 22 31 20
 10 33 28 17  8 15

LUX method[edit]

wp:Conway's LUX method for magic squares

class Magic_square
attr_reader :square
LUX = { L: [[4, 1], [2, 3]], U: [[1, 4], [2, 3]], X: [[1, 4], [3, 2]] }
 
def initialize(n)
raise ArgumentError, "must be even, but not divisible by 4." unless (n-2) % 4 == 0
raise ArgumentError, "2x2 magic square not possible." if n == 2
@n = n
oms = odd_magic_square(n/2)
mat = make_lux_matrix(n/2)
@square = synthesize(oms, mat)
puts to_s
end
 
def odd_magic_square(n) # zero beginning, it is 4 multiples.
n.times.map{|i| n.times.map{|j| (n*((i+j+1+n/2)%n) + ((i+2*j-5)%n)) * 4} }
end
 
def make_lux_matrix(n)
center = n / 2
lux = [*[:L]*(center+1), :U, *[:X]*(n-center-2)]
matrix = lux.map{|x| Array.new(n, x)}
matrix[center][center] = :U
matrix[center+1][center] = :L
matrix
end
 
def synthesize(oms, mat)
range = 0...@n/2
range.inject([]) do |matrix,i|
row = [[], []]
range.each do |j|
x = oms[i][j]
LUX[mat[i][j]].each_with_index{|lux,k| row[k] << lux.map{|y| x+y}}
end
matrix << row[0].flatten << row[1].flatten
end
end
 
def to_s
format = "%#{(@n*@n).to_s.size}d " * @n + "\n"
@square.map{|row| format % row}.join
end
end
 
sq = Magic_square.new(6).square
Output:
32 29  4  1 24 21 
30 31  2  3 22 23 
12  9 17 20 28 25 
10 11 18 19 26 27 
13 16 36 33  5  8 
14 15 34 35  6  7 

zkl[edit]

Translation of: Java
class MagicSquareSinglyEven{
fcn init(n){ var result=magicSquareSinglyEven(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 magicSquareOdd(n){
if (n<3 or n%2==0) throw(Exception.ValueError("base must be odd and > 2"));
value,gridSize,c,r:=0, n*n, n/2, 0;
result:=n.pump(List(),n.pump(List(),0).copy); // array[n,n] of zero
 
while((value+=1)<=gridSize){
result[r][c]=value;
if(r==0){
if(c==n-1) r+=1;
else{ r=n-1; c+=1; }
}
else if(c==n-1){ r-=1; c=0; }
else if(result[r-1][c+1]==0){ r-=1; c+=1; }
else r+=1;
}
result;
}
fcn magicSquareSinglyEven(n){
if (n<6 or (n-2)%4!=0)
throw(Exception.ValueError("base must be a positive multiple of 4 +2"));
size,halfN,subSquareSize:=n*n, n/2, size/4;
 
subSquare:=magicSquareOdd(halfN);
quadrantFactors:=T(0, 2, 3, 1);
result:=n.pump(List(),n.pump(List(),0).copy); // array[n,n] of zero
 
foreach r,c in (n,n){
quadrant:=(r/halfN)*2 + (c/halfN);
result[r][c]=subSquare[r%halfN][c%halfN];
result[r][c]+=quadrantFactors[quadrant]*subSquareSize;
}
nColsLeft,nColsRight:=halfN/2, nColsLeft-1;
foreach r,c in (halfN,n){
if ( c<nColsLeft or c>=(n-nColsRight) or
(c==nColsLeft and r==nColsLeft) ){
if(c==0 and r==nColsLeft) continue;
tmp:=result[r][c];
result[r][c]=result[r+halfN][c];
result[r+halfN][c]=tmp;
}
}
result
}
}
MagicSquareSinglyEven(6).println();
Output:
35  1  6 26 19 24 
 3 32  7 21 23 25 
31  9  2 22 27 20 
 8 28 33 17 10 15 
30  5 34 12 14 16 
 4 36 29 13 18 11 

Magic constant: 111