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Magic squares of doubly even order

From Rosetta Code
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[edit]

Translation of: Java
*        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
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[edit]

Translation of: JavaScript
-- 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 lambda(x)
replicate(n / 4, x)
end lambda
end script
 
set truthSeries to ¬
flatten(scale's lambda(map(scale, splitEvery(4, magicSeries(4)))))
end if
 
set limit to sqr + 1
script inOrderOrReversed
on lambda(x, i)
cond(x, i, limit - i)
end lambda
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 lambda(x)
not x
end lambda
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 lambda(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 lambda
end script
 
linefeed & "{| class=\"wikitable\" " & ¬
cond(strStyle ≠ "", "style=\"" & strStyle & "\"", "") & ¬
intercalate("", ¬
map(fWikiRows, lstRows)) & linefeed & "|}" & linefeed
end wikiTable
 
 
-- GENERIC FUNCTIONS ---------------------------------------------------------------------------
 
-- 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
 
-- isPowerOf :: Int -> Int -> Bool
on isPowerOf(k, n)
set v to k
script remLeft
on lambda(x)
x mod v is not 0
end lambda
end script
 
script integerDiv
on lambda(x)
x div v
end lambda
end script
 
|until|(remLeft, integerDiv, n) = 1
end isPowerOf
 
-- 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
 
-- 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 lambda(item i of xs, i, xs)
end repeat
return lst
end tell
end map
 
-- 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
 
-- concatMap :: (a -> [b]) -> [a] -> [b]
on concatMap(f, xs)
script append
on lambda(a, b)
a & b
end lambda
end script
 
foldl(append, {}, map(f, xs))
end concatMap
 
-- 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 lambda(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
 
-- 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 lambda(v)
set v to lambda(v)
end repeat
end tell
return v
end |until|
 
-- 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 lambda : 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
 
-- 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
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

C++[edit]

#include <iostream>
#include <sstream>
#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;
}
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

Elixir[edit]

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)
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

FreeBASIC[edit]

' 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
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

Haskell[edit]

import Data.List (transpose, unfoldr)
 
magicSquare :: Int -> [[Int]]
magicSquare n
| rem n 4 > 0 = []
| otherwise =
splitEvery n $
-- Taken directly from the integer series where True
-- and from the reverse of that series where False
zipWith
(\x i ->
if x
then i
else limit - i)
series
[1 .. sqr]
where
sqr = n * n
limit = sqr + 1
series
-- For integer powers of 2, the (append not) 'magic' series directly
-- yields the truth table that we need
| isPowerOf 2 n = magicSeries $ floor (logBase 2 (fromIntegral sqr))
-- where n is not an integer power of 2, we can replicate a
-- minimum truth table, horizontally and vertically
| otherwise =
concat . concat . concat . scale $
scale <$> splitEvery 4 (magicSeries 4)
where
scale = replicate $ quot n 4
 
------------------------------------------------------------------------
magicSeries :: Int -> [Bool]
magicSeries = (iterate (\xs -> xs ++ (not <$> xs)) [True] !!)
 
splitEvery :: Int -> [a] -> [[a]]
splitEvery n = takeWhile (not . null) . unfoldr (Just . splitAt n)
 
isPowerOf :: Int -> Int -> Bool
isPowerOf k n = until (\x -> rem x k /= 0) (`quot` k) n == 1
 
main :: IO ()
main = mapM_ print $ magicSquare 8
 
-- Summed and checked---------------------------------------------------
checked :: Int -> (Int, Bool)
checked n = (h, all (h ==) t)
where
square = magicSquare n
h:t =
sum <$>
square ++ -- rows
transpose square ++ -- cols
diagonals square -- diagonals
 
diagonals :: [[Int]] -> [[Int]]
diagonals xs = flip (zipWith (!!)) [0 ..] <$> [xs, reverse xs]
 
main2 :: IO ()
main2 = print $ checked 8
Output:
main
[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]

main2
(260,True)

Java[edit]

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;
}
}
 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[edit]

ES6[edit]

Translation of: Haskell
(() => {
'use strict';
 
// doubleEvenMagicSquare :: Int -> [[Int]]
const doubleEvenMagicSquare = n => {
if (n % 4 > 0) return undefined;
 
// truthSeries :: Int -> [Int]
const truthSeries = n => {
if (n <= 0) return [true];
const xs = truthSeries(n - 1);
return xs.concat(xs.map(x => !x));
};
 
const sqr = n * n,
scale = curry(replicate)(n / 4),
power = Math.log2(sqr),
sequence = isInt(power) ? truthSeries(power) : (
flatten(
scale(
splitEvery(4, truthSeries(4))
.map(scale)
)
)
);
 
return splitEvery(n, sequence
.map((x, i) => x ? i + 1 : sqr - i));
};
 
 
// GENERIC FUNCTIONS ----------------------------------------------------
 
// flatten :: Tree a -> [a]
const flatten = t => (t instanceof Array ? concatMap(flatten, t) : [t]);
 
// concatMap :: (a -> [b]) -> [a] -> [b]
const concatMap = (f, xs) => [].concat.apply([], xs.map(f));
 
// splitEvery :: Int -> [a] -> [][a]]
const splitEvery = (n, xs) => {
if (xs.length <= n) return [xs];
const [h, t] = [xs.slice(0, n), xs.slice(n)];
return [h].concat(splitEvery(n, t));
}
 
// curry :: ((a, b) -> c) -> a -> b -> c
const curry = f => a => b => f(a, b);
 
// replicate :: Int -> a -> [a]
const replicate = (n, a) => {
let v = [a],
o = [];
if (n < 1) return o;
while (n > 1) {
if (n & 1) o = o.concat(v);
n >>= 1;
v = v.concat(v);
}
return o.concat(v);
};
 
// isInt :: Int -> Bool
const isInt = x => x === Math.floor(x);
 
 
// TEST AND DISPLAY FUNCTIONS -------------------------------------------
 
// transpose :: [[a]] -> [[a]]
const transpose = xs =>
xs[0].map((_, iCol) => xs.map((row) => row[iCol]));
 
// diagonals :: [[a]] -> ([a], [a])
const diagonals = xs => {
const nRows = xs.length,
nCols = (nRows > 0 ? xs[0].length : 0);
const cell = (x, y) => xs[y][x];
 
if (nRows === nCols) {
const ns = range(0, nCols - 1);
return [zipWith(cell, ns, ns), zipWith(cell, ns, reverse(ns))];
} else return [
[],
[]
];
};
 
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = (f, xs, ys) => {
const ny = ys.length;
return (xs.length <= ny ? xs : xs.slice(0, ny))
.map((x, i) => f(x, ys[i]));
}
 
// reverse :: [a] -> [a]
const reverse = (xs) => xs.slice(0)
.reverse()
 
// range :: Int -> Int -> [Int]
const range = (m, n) =>
Array.from({
length: Math.floor(n - m) + 1
}, (_, i) => m + i);
 
// all :: (a -> Bool) -> [a] -> Bool
const all = (f, xs) => xs.every(f);
 
// show :: a -> String
const show = x => JSON.stringify(x);
 
// justifyRight :: Int -> Char -> Text -> Text
const justifyRight = (n, cFiller, strText) =>
n > strText.length ? (
(cFiller.repeat(n) + strText)
.slice(-n)
) : strText;
 
// TEST -----------------------------------------------------------------
 
//return doubleEvenMagicSquare(8)
 
return [4, 8, 12]
.map(n => {
const lines = doubleEvenMagicSquare(n);
const sums = lines.concat(
transpose(lines)
.concat(diagonals(lines))
)
.map(xs => xs.reduce((a, b) => a + b, 0));
const sum = sums[0];
return [
"Order: " + n.toString(),
"Summing to: " + sum.toString(),
"Row, column and diagonal sums checked: " +
all(x => x === sum, sums)
.toString() + '\n',
lines.map(
xs => xs.map(
x => justifyRight(3, ' ', x.toString())
)
.join(' '))
.join('\n')
].join('\n')
})
.join('\n\n');
})();
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

Kotlin[edit]

Translation of: Java
// 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}")
}
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

PARI/GP[edit]

A magic one-liner:

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))
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 6[edit]

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

PureBasic[edit]

Translation of: FreeBasic
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
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)

>

REXX[edit]

"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).

/*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

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[edit]

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))
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[edit]

Translation of: Java
' 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
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

zkl[edit]

Translation of: Java
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();
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