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Magic squares of odd order

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

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

The numbers are usually (but not always) the first   N2   positive integers.

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

8 1 6
3 5 7
4 9 2


Task

For any odd   N,   generate a magic square with the integers   1 ──► N,   and show the results here.

Optionally, show the magic number.

You should demonstrate the generator by showing at least a magic square for   N = 5.


Related tasks


See also


360 Assembly[edit]

Translation of: C
*        Magic squares of odd order - 20/10/2015
MAGICS CSECT
USING MAGICS,R15 set base register
LA R6,1 i=1
LOOPI C R6,N do i=1 to n
BH ELOOPI
LR R8,R6 i
SLA R8,1 i*2
LA R9,PG [email protected]
LA R7,1 j=1
LOOPJ C R7,N do j=1 to n
BH ELOOPJ
LR R5,R8 i*2
SR R5,R7 -j
A R5,N +n
BCTR R5,0 -1
XR R4,R4 clear high reg
D R4,N /n
LR R5,R4 //n
M R4,N *n
LR R2,R5 (i*2-j+n-1)//n*n
LR R5,R8 i*2
AR R5,R7 -j
S R5,=F'2' -2
XR R4,R4 clear high reg
D R4,N /n
AR R2,R4 +(i*2+j-2)//n
LA R2,1(R2) +1
XDECO R2,PG+80 (i*2-j+n-1)//n*n+(i*2+j-2)//n+1
MVC 0(5,R9),PG+87 put in buffer
LA R9,5(R9) pgi=pgi+5
LA R7,1(R7) j=j+1
B LOOPJ
ELOOPJ XPRNT PG,80
LA R6,1(R6) i=i+1
B LOOPI
ELOOPI XR R15,R15 set return code
BR R14 return to caller
N DC F'9' <== input
PG DC CL92' ' buffer
YREGS
END MAGICS
Output:
    2   75   67   59   51   43   35   27   10
   22   14    6   79   71   63   46   38   30
   42   34   26   18    1   74   66   58   50
   62   54   37   29   21   13    5   78   70
   73   65   57   49   41   33   25   17    9
   12    4   77   69   61   53   45   28   20
   32   24   16    8   81   64   56   48   40
   52   44   36   19   11    3   76   68   60
   72   55   47   39   31   23   15    7   80

Ada[edit]

with Ada.Text_IO, Ada.Command_Line;
 
procedure Magic_Square is
N: constant Positive := Positive'Value(Ada.Command_Line.Argument(1));
 
subtype Constants is Natural range 1 .. N*N;
package CIO is new Ada.Text_IO.Integer_IO(Constants);
Undef: constant Natural := 0;
 
subtype Index is Natural range 0 .. N-1;
function Inc(I: Index) return Index is (if I = N-1 then 0 else I+1);
function Dec(I: Index) return Index is (if I = 0 then N-1 else I-1);
 
A: array(Index, Index) of Natural := (others => (others => Undef));
-- initially undefined; at the end holding the magic square
 
X: Index := 0; Y: Index := N/2; -- start position for the algorithm
begin
for I in Constants loop -- write 1, 2, ..., N*N into the magic array
A(X, Y) := I; -- write I into the magic array
if A(Dec(X), Inc(Y)) = Undef then
X := Dec(X); Y := Inc(Y); -- go right-up
else
X := Inc(X); -- go down
end if;
end loop;
 
for Row in Index loop -- output the magic array
for Collumn in Index loop
CIO.Put(A(Row, Collumn),
Width => (if N*N < 10 then 2 elsif N*N < 100 then 3 else 4));
end loop;
Ada.Text_IO.New_Line;
end loop;
end Magic_Square;
Output:
>./magic_square 3
 8 1 6
 3 5 7
 4 9 2
>./magic_square 11
  68  81  94 107 120   1  14  27  40  53  66
  80  93 106 119  11  13  26  39  52  65  67
  92 105 118  10  12  25  38  51  64  77  79
 104 117   9  22  24  37  50  63  76  78  91
 116   8  21  23  36  49  62  75  88  90 103
   7  20  33  35  48  61  74  87  89 102 115
  19  32  34  47  60  73  86  99 101 114   6
  31  44  46  59  72  85  98 100 113   5  18
  43  45  58  71  84  97 110 112   4  17  30
  55  57  70  83  96 109 111   3  16  29  42
  56  69  82  95 108 121   2  15  28  41  54

ALGOL W[edit]

begin
 % construct a magic square of odd order - as a procedure can't return an %
 % array, the caller must supply one that is big enough  %
logical procedure magicSquare( integer array square ( *, * )
 ; integer value order
) ;
if not odd( order ) or order < 1 then begin
 % can't make a magic square of the specified order  %
false
end
else begin
 % order is OK - construct the square using de la Loubère's  %
 % algorithm as in the wikipedia page  %
 
 % ensure a row/col position is on the square  %
integer procedure inSquare( integer value pos ) ;
if pos < 1 then order else if pos > order then 1 else pos;
 % move "up" a row in the square  %
integer procedure up ( integer value row ) ; inSquare( row - 1 );
 % move "accross right" in the square  %
integer procedure right( integer value col ) ; inSquare( col + 1 );
 
integer row, col;
 % initialise square  %
for i := 1 until order do for j := 1 until order do square( i, j ) := 0;
 
 % initial position is the middle of the top row  %
col := ( order + 1 ) div 2;
row := 1;
 % construct square  %
for i := 1 until ( order * order ) do begin
square( row, col ) := i;
if square( up( row ), right( col ) ) not = 0 then begin
 % the up/right position is already taken, move down  %
row := row + 1;
end
else begin
 % can move up/right  %
row := up( row );
col := right( col );
end
end for_i;
 % sucessful result  %
true
end magicSquare ;
 
 % prints the magic square  %
procedure printSquare( integer array square ( *, * )
 ; integer value order
) ;
begin
integer sum, w;
 
 % set integer width to accomodate the largest number in the square  %
w := ( order * order ) div 10;
i_w := s_w := 1;
while w > 0 do begin i_w := i_w + 1; w := w div 10 end;
 
for i := 1 until order do sum := sum + square( 1, i );
write( "maqic square of order ", order, ": sum: ", sum );
for i := 1 until order do begin
write( square( i, 1 ) );
for j := 2 until order do writeon( square( i, j ) )
end for_i
 
end printSquare ;
 
 % test the magic square generation  %
 
integer array sq ( 1 :: 11, 1 :: 11 );
 
for i := 1, 3, 5, 7 do begin
if magicSquare( sq, i ) then printSquare( sq, i )
else write( "can't generate square" );
end for_i
 
end.
Output:
maqic square of order 1 : sum: 1 
1 
maqic square of order 3 : sum: 15 
8 1 6 
3 5 7 
4 9 2 
maqic square of order  5 : sum: 65 
17 24  1  8 15 
23  5  7 14 16 
 4  6 13 20 22 
10 12 19 21  3 
11 18 25  2  9 
maqic square of order  7 : sum: 175 
30 39 48  1 10 19 28 
38 47  7  9 18 27 29 
46  6  8 17 26 35 37 
 5 14 16 25 34 36 45 
13 15 24 33 42 44  4 
21 23 32 41 43  3 12 
22 31 40 49  2 11 20 

ALGOL 68[edit]

# construct a magic square of odd order                                      #
PROC magic square = ( INT order ) [,]INT:
IF NOT ODD order OR order < 1
THEN
# can't make a magic square of the specified order #
LOC [ 1 : 0, 1 : 0 ]INT
ELSE
# order is OK - construct the square using de la Loubère's #
# algorithm as in the wikipedia page #
 
[ 1 : order, 1 : order ]INT square;
FOR i TO order DO FOR j TO order DO square[ i, j ] := 0 OD OD;
 
# as square [ 1, 1 ] if the top-left, moving "up" reduces the row #
# operator to advance "up" the square #
OP PREV = ( INT pos )INT: IF pos = 1 THEN order ELSE pos - 1 FI;
# operator to advance "across right" or "down" the square #
OP NEXT = ( INT pos )INT: ( pos MOD order ) + 1;
 
# fill in the square, starting from the middle of the top row #
INT col := ( order + 1 ) OVER 2;
INT row := 1;
FOR i TO order * order DO
square[ row, col ] := i;
IF square[ PREV row, NEXT col ] /= 0
THEN
# the up/right position is already taken, move down #
row := NEXT row
ELSE
# can move up and right #
row := PREV row;
col := NEXT col
FI
OD;
 
square
FI # magic square # ;
 
# prints the magic square #
PROC print square = ( [,]INT square )VOID:
BEGIN
INT order = 1 UPB square;
# calculate print width: negative so a leading "+" is not printed #
INT width := -1;
INT mag := order * order;
WHILE mag >= 10 DO mag OVERAB 10; width MINUSAB 1 OD;
# calculate the "magic sum" #
INT sum := 0;
FOR i TO order DO sum +:= square[ 1, i ] OD;
# print the square #
print( ( "maqic square of order ", whole( order, 0 ), ": sum: ", whole( sum, 0 ), newline ) );
FOR i TO order DO
FOR j TO order DO write( ( " ", whole( square[ i, j ], width ) ) ) OD;
write( ( newline ) )
OD
END # print square # ;
 
# test the magic square generation #
FOR order BY 2 TO 7 DO print square( magic square( order ) ) OD
Output:
maqic square of order 1: sum: 1
 1
maqic square of order 3: sum: 15
 8 1 6
 3 5 7
 4 9 2
maqic square of order 5: sum: 65
 17 24  1  8 15
 23  5  7 14 16
  4  6 13 20 22
 10 12 19 21  3
 11 18 25  2  9
maqic square of order 7: sum: 175
 30 39 48  1 10 19 28
 38 47  7  9 18 27 29
 46  6  8 17 26 35 37
  5 14 16 25 34 36 45
 13 15 24 33 42 44  4
 21 23 32 41 43  3 12
 22 31 40 49  2 11 20


AppleScript[edit]

Translation of: JavaScript


Built by composing functions ( rotate . transpose . rotate ), and by lifting AppleScript handlers into first class script objects, to allow for first class functions and closures.

-- oddMagicSquare :: Int -> [[Int]]
on oddMagicSquare(n)
cond((n mod 2) > 0, ¬
rotate(transpose(rotate(table(n)))), ¬
missing value)
end oddMagicSquare
 
 
-- TEST
on run
-- Orders 3, 5, 11
 
-- wikiTableMagic :: Int -> String
script wikiTableMagic
on lambda(n)
formattedTable(oddMagicSquare(n))
end lambda
end script
 
intercalate(linefeed & linefeed, map(wikiTableMagic, {3, 5, 11}))
end run
 
-- table :: Int -> [[Int]]
on table(n)
set lstTop to range(1, n)
 
script cols
on lambda(row)
script rows
on lambda(x)
(row * n) + x
end lambda
end script
 
map(rows, lstTop)
end lambda
end script
 
map(cols, range(0, n - 1))
end table
 
-- rotation :: [[a]] -> [[a]]
on rotate(lst)
script rotationRow
-- rotatedList :: [a] -> Int -> [a]
on rotatedList(lst, n)
if n = 0 then return lst
 
set lng to length of lst
set m to (n + lng) mod lng
items -m thru -1 of lst & items 1 thru (lng - m) of lst
end rotatedList
 
on lambda(row, i)
rotatedList(row, (((length of row) + 1) div 2) - (i))
end lambda
end script
 
map(rotationRow, lst)
end rotate
 
 
-- GENERIC FUNCTIONS
 
-- 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
 
-- 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
 
-- 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
 
-- range :: Int -> Int -> [Int]
on range(m, n)
if n < m then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end range
 
-- splitOn :: Text -> Text -> [Text]
on splitOn(strDelim, strMain)
set {dlm, my text item delimiters} to {my text item delimiters, strDelim}
set xs to text items of strMain
set my text item delimiters to dlm
return xs
end splitOn
 
-- transpose :: [[a]] -> [[a]]
on transpose(xss)
script column
on lambda(_, iCol)
script row
on lambda(xs)
item iCol of xs
end lambda
end script
 
map(row, xss)
end lambda
end script
 
map(column, item 1 of xss)
end transpose
 
-- WIKI DISPLAY
 
-- formattedTable :: [[Int]] -> String
on formattedTable(lstTable)
set n to length of lstTable
set w to 2.5 * n
"magic(" & n & ")" & linefeed & 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
 
-- cond :: Bool -> a -> a -> a
on cond(bool, f, g)
if bool then
f
else
g
end if
end cond
Output:

magic(3)

8 3 4
1 5 9
6 7 2

magic(5)

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

magic(11)

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

AutoHotkey[edit]

 
msgbox % OddMagicSquare(5)
msgbox % OddMagicSquare(7)
return
 
OddMagicSquare(oddN){
sq := oddN**2
obj := {}
loop % oddN
obj[A_Index] := {} ; dis is row
mid := Round((oddN+1)/2)
sum := Round(sq*(sq+1)/2/oddN)
obj[1][mid] := 1
cR := 1 , cC := mid
loop % sq-1
{
done := 0 , a := A_index+1
while !done {
nR := cR-1 , nC := cC+1
if !nR
nR := oddN
if (nC>oddN)
nC := 1
if obj[nR][nC] ;filled
cR += 1
else cR := nR , cC := nC
if !obj[cR][cC]
obj[cR][cC] := a , done := 1
}
}
 
str := "Magic Constant for " oddN "x" oddN " is " sum "`n"
for k,v in obj
{
for k2,v2 in v
str .= " " v2
str .= "`n"
}
return str
}
 
Output:
Magic Constant for 5x5 is 65
 17 24 1 8 15
 23 5 7 14 16
 4 6 13 20 22
 10 12 19 21 3
 11 18 25 2 9


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

AWK[edit]

 
# syntax: GAWK -f MAGIC_SQUARES_OF_ODD_ORDER.AWK
BEGIN {
build(5)
build(3,1) # verify sum
build(7)
exit(0)
}
function build(n,check, arr,i,width,x,y) {
if (n !~ /^[0-9]*[13579]$/ || n < 3) {
printf("error: %s is invalid\n",n)
return
}
printf("\nmagic constant for %dx%d is %d\n",n,n,(n*n+1)*n/2)
x = 0
y = int(n/2)
for (i=1; i<=(n*n); i++) {
arr[x,y] = i
if (arr[(x+n-1)%n,(y+n+1)%n]) {
x = (x+n+1) % n
}
else {
x = (x+n-1) % n
y = (y+n+1) % n
}
}
width = length(n*n)
for (x=0; x<n; x++) {
for (y=0; y<n; y++) {
printf("%*s ",width,arr[x,y])
}
printf("\n")
}
if (check) { verify(arr,n) }
}
function verify(arr,n, total,x,y) { # verify sum of each row, column and diagonal
print("\nverify")
# horizontal
for (x=0; x<n; x++) {
total = 0
for (y=0; y<n; y++) {
printf("%d ",arr[x,y])
total += arr[x,y]
}
printf("\t: %d row %d\n",total,x+1)
}
# vertical
for (y=0; y<n; y++) {
total = 0
for (x=0; x<n; x++) {
printf("%d ",arr[x,y])
total += arr[x,y]
}
printf("\t: %d column %d\n",total,y+1)
}
# left diagonal
total = 0
for (x=y=0; x<n; x++ y++) {
printf("%d ",arr[x,y])
total += arr[x,y]
}
printf("\t: %d diagonal top left to bottom right\n",total)
# right diagonal
x = n - 1
total = 0
for (y=0; y<n; y++ x--) {
printf("%d ",arr[x,y])
total += arr[x,y]
}
printf("\t: %d diagonal bottom left to top right\n",total)
}
 
Output:
magic constant for 5x5 is 65
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

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

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

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

Batch File[edit]

@echo off
rem Magic squares of odd order
setlocal EnableDelayedExpansion
set n=9
echo The square order is: %n%
for /l %%i in (1,1,%n%) do (
set w=
for /l %%j in (1,1,%n%) do (
set /a v1=%%i*2-%%j+n-1
set /a v1=v1%%n*n
set /a v2=%%i*2+%%j+n-2
set /a v2=v2%%n
set /a v=v1+v2+1
set v= !v!
set w=!w!!v:~-5!)
echo !w!)
set /a w=n*(n*n+1)/2
echo The magic number is: %w%
pause
Output:
The square order is: 9
    2   75   67   59   51   43   35   27   10
   22   14    6   79   71   63   46   38   30
   42   34   26   18    1   74   66   58   50
   62   54   37   29   21   13    5   78   70
   73   65   57   49   41   33   25   17    9
   12    4   77   69   61   53   45   28   20
   32   24   16    8   81   64   56   48   40
   52   44   36   19   11    3   76   68   60
   72   55   47   39   31   23   15    7   80
The magic number is: 369
Press any key to continue ...


bc[edit]

Works with: GNU bc
define magic_constant(n) {
return(((n * n + 1) / 2) * n)
}
 
define print_magic_square(n) {
auto i, x, col, row, len, old_scale
 
old_scale = scale
scale = 0
len = length(n * n)
 
print "Magic constant for n=", n, ": ", magic_constant(n), "\n"
for (row = 1; row <= n; row++) {
for (col = 1; col <= n; col++) {
x = n * ((row + col - 1 + (n / 2)) % n) + \
((row + 2 * col - 2) % n) + 1
for (i = 0; i < len - length(x); i++) {
print " "
}
print x
if (col != n) print " "
}
print "\n"
}
 
scale = old_scale
}
 
temp = print_magic_square(5)
Output:
Magic constant for n=5: 65
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

Befunge[edit]

Translation of: C

The size, n, is specified by the first value on the stack.

500p0>:::00g%00g\-1-\00g/2*+1+00g%00g*\:00g%v
@<$<_^#!-*:g00:,+9!%g00:+1.+1+%g00+1+*2/g00\<
Output:
2 	23 	19 	15 	6 
14 	10 	1 	22 	18 
21 	17 	13 	9 	5 
8 	4 	25 	16 	12 
20 	11 	7 	3 	24 

C[edit]

Generates an associative magic square. If the size is larger than 3, the square is also panmagic.

#include <stdio.h>
#include <stdlib.h>
 
int f(int n, int x, int y)
{
return (x + y*2 + 1)%n;
}
 
int main(int argc, char **argv)
{
int i, j, n;
 
//Edit: Add argument checking
if(argc!=2) return 1;
 
//Edit: Input must be odd and not less than 3.
n = atoi(argv[1]);
if (n < 3 || (n%2) == 0) return 2;
 
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++)
printf("% 4d", f(n, n - j - 1, i)*n + f(n, j, i) + 1);
putchar('\n');
}
printf("\n Magic Constant: %d.\n", (n*n+1)/2*n);
 
return 0;
}
Output:
$ ./magic 5
   2  23  19  15   6
  14  10   1  22  18
  21  17  13   9   5
   8   4  25  16  12
  20  11   7   3  24

 Magic Constant: 65.

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++; sz = d;
sqr = new int[sz * sz];
memset( sqr, 0, sz * sz * sizeof( int ) );
fillSqr();
}
 
void display()
{
cout << "Odd Magic Square: " << sz << " x " << sz << "\n";
cout << "It's Magic Sum is: " << magicNumber() << "\n\n";
ostringstream cvr; cvr << sz * sz;
int l = cvr.str().size();
 
for( int y = 0; y < sz; y++ )
{
int yy = y * sz;
for( int x = 0; x < sz; x++ )
cout << setw( l + 2 ) << sqr[yy + x];
 
cout << "\n";
}
cout << "\n\n";
}
 
private:
void fillSqr()
{
int sx = sz / 2, sy = 0, c = 0;
while( c < sz * sz )
{
if( !sqr[sx + sy * sz] )
{
sqr[sx + sy * sz]= c + 1;
inc( sx ); dec( sy );
c++;
}
else
{
dec( sx ); inc( sy ); inc( sy );
}
}
}
 
int magicNumber()
{ return sz * ( ( sz * 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( 5 );
s.display();
return 0;
}
 
Output:
Odd Magic Square: 5 x 5
It's Magic Sum is: 65

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

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

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

Common Lisp[edit]

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

D[edit]

Translation of: Python
void main(in string[] args)
{
import std.stdio, std.conv, std.range, std.algorithm, std.exception;
 
immutable n = args.length == 2 ? args[1].to!uint : 5;
enforce(n > 0 && n % 2 == 1, "Only odd n > 1");
immutable len = text(n ^^ 2).length.text;
// writeln(len);
 
foreach (immutable r; 1 .. n + 1)
{
foreach (immutable c; 1 .. n + 1)
{
auto a = (n * ((r + c - 1 + (n / 2)) % n)) + ((r + (2 * c) - 2) % n) + 1;
// n(( I + J - 1 + ( n / 2 ) ) mod n ) + (( I + 2J - 2 ) mod n ) + 1
// writeln("n = ",n, " r = ",r," c = ",c, " a = ",a );
writef("%" ~ len ~ "d%s",a, " ");
}
writeln("");
}
;
 
writeln("\nMagic constant: ", ((n * n + 1) * n) / 2);
}}
Output:
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

Magic constant: 65

Alternative Version[edit]

Translation of: C
import std.stdio, std.conv, std.string, std.range, std.algorithm;
 
uint[][] magicSquare(immutable uint n) pure nothrow @safe
in {
assert(n > 0 && n % 2 == 1);
} out(mat) {
// mat is square of the right size.
assert(mat.length == n);
assert(mat.all!(row => row.length == n));
 
immutable magic = mat[0].sum;
 
// The sum of all rows is the same magic number.
assert(mat.all!(row => row.sum == magic));
 
// The sum of all columns is the same magic number.
//assert(mat.transposed.all!(col => col.sum == magic));
assert(mat.dup.transposed.all!(col => col.sum == magic));
 
// The sum of the main diagonals is the same magic number.
assert(mat.enumerate.map!(ir => ir[1][ir[0]]).sum == magic);
//assert(mat.enumerate.map!({i, r} => r[i]).sum == magic);
assert(mat.enumerate.map!(ir => ir[1][ir[0]]).sum == magic);
} body {
enum M = (in uint x) pure nothrow @safe @nogc => (x + n - 1) % n;
auto m = new uint[][](n, n);
 
uint i = 0;
uint j = n / 2;
foreach (immutable uint k; 1 .. n ^^ 2 + 1) {
m[i][j] = k;
if (m[M(i)][M(j)]) {
i = (i + 1) % n;
} else {
i = M(i);
j = M(j);
}
}
 
return m;
}
 
void showSquare(in uint[][] m)
in {
assert(m.all!(row => row.length == m[0].length));
} body {
immutable maxLen = text(m.length ^^ 2).length.text;
writefln("%(%(%" ~ maxLen ~ "d %)\n%)", m);
writeln("\nMagic constant: ", m[0].sum);
}
 
int main(in string[] args) {
if (args.length == 1) {
5.magicSquare.showSquare;
return 0;
} else if (args.length == 2) {
immutable n = args[1].to!uint;
if (n > 0 && n % 2 == 1) {
n.magicSquare.showSquare;
return 0;
}
}
 
stderr.writefln("Requires n odd and larger than 0.");
return 1;
}
Output:
15  8  1 24 17
16 14  7  5 23
22 20 13  6  4
 3 21 19 12 10
 9  2 25 18 11

Magic constant: 65

EchoLisp[edit]

The make-ms procedure allows to construct different magic squares for a same n, by modifying the grid filling moves. (see MathWorld reference)

 
(lib 'matrix)
 
;; compute next i,j = f(move,i,j)
(define-syntax-rule (path imove jmove)
(begin (set! i (imove i n)) (set! j (jmove j n))))
 
;; We define the ordinary and break moves
;; (1 , -1), (0, 1) King's move
(define (inext i n) (modulo (1+ i) n))
(define (jnext j n) (modulo (1- j) n))
(define (ibreak i n) i)
(define (jbreak j n) (modulo (1+ j) n))
 
(define (make-ms n)
(define n2+1 (1+ (* n n)))
(define ms (make-array n n))
(define i (quotient n 2))
(define j 0)
(array-set! ms i j 1)
 
(for ((ns (in-range 2 n2+1)))
(if (zero? (array-ref ms (inext i n ) (jnext j n )))
(path inext jnext) ;; ordinary move if empty target
(path ibreak jbreak)) ;; else break move
 
(if (zero? (array-ref ms i j))
(array-set! ms i j ns)
(error ns "illegal path"))
)
(writeln 'order n 'magic-number (/ ( * n n2+1) 2))
(array-print ms))
 
 
Output:
(make-ms 7)
order     7     magic-number     175    
  30   38   46   5    13   21   22 
  39   47   6    14   15   23   31 
  48   7    8    16   24   32   40 
  1    9    17   25   33   41   49 
  10   18   26   34   42   43   2  
  19   27   35   36   44   3    11 
  28   29   37   45   4    12   20 

;; Changing the moves allow to generate other magic squares
;; (2 ,1) (1,-2) Knight's move !
(define (inext i n) (modulo (+ 2 i) n))
(define (jnext j n) (modulo (1+ j) n))
(define (ibreak i n) (modulo (1+ i) n))
(define (jbreak j n) (modulo (- j 2) n))
(make-ms 7)

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

;; (2 ,1) (1,-1)
(define (inext i n) (modulo (+ 2 i) n))
(define (jnext j n) (modulo (1+ j) n))
(define (ibreak i n) (modulo (1+ i) n))
(define (jbreak j n) (modulo (1- j) n))
(make-ms 7)

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


Elixir[edit]

Translation of: Ruby
defmodule RC do
def odd_magic_square(n) when rem(n,2)==1 do
for i <- 0..n-1 do
for j <- 0..n-1, do: n * rem(i+j+1+div(n,2),n) + rem(i+2*j+2*n-5,n) + 1
end
end
 
def print_square(sq) do
width = List.flatten(sq) |> Enum.max |> to_char_list |> length
fmt = String.duplicate(" ~#{width}w", length(sq)) <> "~n"
Enum.each(sq, fn row -> :io.format fmt, row end)
end
end
 
Enum.each([3,5,11], fn n ->
IO.puts "\nSize #{n}, magic sum #{div(n*n+1,2)*n}"
RC.odd_magic_square(n) |> RC.print_square
end)
Output:
Size 3, magic sum 15
 8 1 6
 3 5 7
 4 9 2

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

Size 11, magic sum 671
  73  86  99 101 114   6  19  32  34  47  60
  85  98 100 113   5  18  31  44  46  59  72
  97 110 112   4  17  30  43  45  58  71  84
 109 111   3  16  29  42  55  57  70  83  96
 121   2  15  28  41  54  56  69  82  95 108
   1  14  27  40  53  66  68  81  94 107 120
  13  26  39  52  65  67  80  93 106 119  11
  25  38  51  64  77  79  92 105 118  10  12
  37  50  63  76  78  91 104 117   9  22  24
  49  62  75  88  90 103 116   8  21  23  36
  61  74  87  89 102 115   7  20  33  35  48

ERRE[edit]

 
PROGRAM MAGIC_SQUARE
 
!$INTEGER
 
PROCEDURE Magicsq(size,filename$)
 
LOCAL DIM sq[25,25] ! array to hold square
 
IF (size AND 1)=0 OR size<3 THEN
PRINT PRINT(CHR$(7)) ! beep
PRINT("error: size is not odd or size is smaller then 3")
PAUSE(3)
EXIT PROCEDURE
END IF
 
! filename$ <> "" then save magic square in a file
! filename$ can contain directory name
! if filename$ exist it will be overwriten, no error checking
 
! start in the middle of the first row
nr=1 x=size-(size DIV 2) y=1
max=size*size
 
! create format string for using
frmt$=STRING$(LEN(STR$(max)),"#")
 
! main loop for creating magic square
REPEAT
IF sq[x,y]=0 THEN
sq[x,y]=nr
IF nr MOD size=0 THEN
y=y+1
ELSE
x=x+1
y=y-1
END IF
nr=nr+1
END IF
IF x>size THEN
x=1
WHILE sq[x,y]<>0 DO
x=x+1
END WHILE
END IF
IF y<1 THEN
y=size
WHILE sq[x,y]<>0 DO
y=y-1
END WHILE
END IF
UNTIL nr>max
 
! printing square's bigger than 19 result in a wrapping of the line
PRINT("Odd magic square size:";size;"*";size)
PRINT("The magic sum =";((max+1) DIV 2)*size)
PRINT
 
FOR y=1 TO size DO
FOR x=1 TO size DO
WRITE(frmt$;sq[x,y];)
END FOR
PRINT
END FOR
 
 ! output magic square to a file with the name provided
IF filename$<>"" THEN
OPEN("O",1,filename$)
PRINT(#1,"Odd magic square size:";size;" *";size)
PRINT(#1,"The magic sum =";((max+1) DIV 2)*size)
PRINT(#1,)
 
FOR y=1 TO size DO
FOR x=1 TO size DO
WRITE(#1,frmt$;sq[x,y];)
END FOR
PRINT(#1,)
END FOR
END IF
CLOSE(1)
 
END PROCEDURE
 
BEGIN
PRINT(CHR$(12);)  ! CLS
Magicsq(5,"")
Magicsq(11,"")
!----------------------------------------------------
! the next line will also print the square to a file
! called 'magic_square_19txt'
!----------------------------------------------------
Magicsq(19,"msq_19.txt")
 
END PROGRAM
 
Output:

Same as FreeBasic version

Odd magic square size: 5 * 5        Odd magic square size: 11 * 11
The magic sum = 65                  The magic sum = 671           
             
 17 24  1  8 15                       68  81  94 107 120   1  14  27  40  53  66
 23  5  7 14 16                       80  93 106 119  11  13  26  39  52  65  67
  4  6 13 20 22                       92 105 118  10  12  25  38  51  64  77  79
 10 12 19 21  3                      104 117   9  22  24  37  50  63  76  78  91
 11 18 25  2  9                      116   8  21  23  36  49  62  75  88  90 103
                                       7  20  33  35  48  61  74  87  89 102 115
                                      19  32  34  47  60  73  86  99 101 114   6
                                      31  44  46  59  72  85  98 100 113   5  18
                                      43  45  58  71  84  97 110 112   4  17  30
                                      55  57  70  83  96 109 111   3  16  29  42
Only the first 2 square shown.        56  69  82  95 108 121   2  15  28  41  54


Fortran[edit]

Works with: Fortran version 95 and later
program Magic_Square
implicit none
 
integer, parameter :: order = 15
integer :: i, j
 
write(*, "(a, i0)") "Magic Square Order: ", order
write(*, "(a)") "----------------------"
do i = 1, order
do j = 1, order
write(*, "(i4)", advance = "no") f1(order, i, j)
end do
write(*,*)
end do
write(*, "(a, i0)") "Magic number = ", f2(order)
 
contains
 
integer function f1(n, x, y)
integer, intent(in) :: n, x, y
 
f1 = n * mod(x + y - 1 + n/2, n) + mod(x + 2*y - 2, n) + 1
end function
 
integer function f2(n)
integer, intent(in) :: n
 
f2 = n * (1 + n * n) / 2
end function
end program

Output:

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

FreeBASIC[edit]

' version 23-06-2015
' compile with: fbc -s console
 
Sub magicsq(size As Integer, filename As String ="")
 
If (size And 1) = 0 Or size < 3 Then
Print : Beep ' alert
Print "error: size is not odd or size is smaller then 3"
Sleep 3000,1 'wait 3 seconds, ignore key press
Exit Sub
End If
 
' filename <> "" then save magic square in a file
' filename can contain directory name
' if filename exist it will be overwriten, no error checking
 
Dim As Integer sq(size,size) ' array to hold square
' start in the middle of the first row
Dim As Integer nr = 1, x = size - (size \ 2), y = 1
Dim As Integer max = size * size
' create format string for using
Dim As String frmt = String(Len(Str(max)) +1, "#")
 
' main loop for creating magic square
Do
If sq(x, y) = 0 Then
sq(x, y) = nr
If nr Mod size = 0 Then
y += 1
Else
x += 1
y -= 1
End If
nr += 1
End If
If x > size Then
x = 1
Do While sq(x,y) <> 0
x += 1
Loop
End If
If y < 1 Then
y = size
Do While sq(x,y) <> 0
y -= 1
Loop
EndIf
Loop Until nr > max
 
' printing square's bigger than 19 result in a wrapping of the line
Print "Odd magic square size:"; size; " *"; size
Print "The magic sum ="; ((max +1) \ 2) * size
Print
 
For y = 1 To size
For x = 1 To size
Print Using frmt; sq(x,y);
Next
Print
Next
print
 
' output magic square to a file with the name provided
If filename <> "" Then
nr = FreeFile
Open filename For Output As #nr
Print #nr, "Odd magic square size:"; size; " *"; size
Print #nr, "The magic sum ="; ((max +1) \ 2) * size
Print #nr,
 
For y = 1 To size
For x = 1 To size
Print #nr, Using frmt; sq(x,y);
Next
Print #nr,
Next
End If
Close
 
End Sub
 
' ------=< MAIN >=------
 
magicsq(5)
magicsq(11)
' the next line will also print the square to a file called: magic_square_19.txt
magicsq(19, "magic_square_19.txt")
 
 
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Output:
Odd magic square size: 5 * 5        Odd magic square size: 11 * 11
The magic sum = 65                  The magic sum = 671           
             
 17 24  1  8 15                       68  81  94 107 120   1  14  27  40  53  66
 23  5  7 14 16                       80  93 106 119  11  13  26  39  52  65  67
  4  6 13 20 22                       92 105 118  10  12  25  38  51  64  77  79
 10 12 19 21  3                      104 117   9  22  24  37  50  63  76  78  91
 11 18 25  2  9                      116   8  21  23  36  49  62  75  88  90 103
                                       7  20  33  35  48  61  74  87  89 102 115
                                      19  32  34  47  60  73  86  99 101 114   6
                                      31  44  46  59  72  85  98 100 113   5  18
                                      43  45  58  71  84  97 110 112   4  17  30
                                      55  57  70  83  96 109 111   3  16  29  42
Only the first 2 square shown.        56  69  82  95 108 121   2  15  28  41  54

Go[edit]

Translation of: C
package main
 
import (
"fmt"
"log"
)
 
func ms(n int) (int, []int) {
M := func(x int) int { return (x + n - 1) % n }
if n <= 0 || n&1 == 0 {
n = 5
log.Println("forcing size", n)
}
m := make([]int, n*n)
i, j := 0, n/2
for k := 1; k <= n*n; k++ {
m[i*n+j] = k
if m[M(i)*n+M(j)] != 0 {
i = (i + 1) % n
} else {
i, j = M(i), M(j)
}
}
return n, m
}
 
func main() {
n, m := ms(5)
i := 2
for j := 1; j <= n*n; j *= 10 {
i++
}
f := fmt.Sprintf("%%%dd", i)
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
fmt.Printf(f, m[i*n+j])
}
fmt.Println()
}
}
Output:
  15   8   1  24  17
  16  14   7   5  23
  22  20  13   6   4
   3  21  19  12  10
   9   2  25  18  11

Haskell[edit]

Translation of: cpp
 
-- as a translation from imperative code, this is probably not a "good" implementation
import Data.List
 
type Var = (Int, Int, Int, Int) -- sx sy sz c
 
magicSum :: Int -> Int
magicSum x = ((x * x + 1) `div` 2) * x
 
wrapInc :: Int -> Int -> Int
wrapInc max x
| x + 1 == max = 0
| otherwise = x + 1
 
wrapDec :: Int -> Int -> Int
wrapDec max x
| x == 0 = max - 1
| otherwise = x - 1
 
isZero :: [[Int]] -> Int -> Int -> Bool
isZero m x y = m !! x !! y == 0
 
setAt :: (Int,Int) -> Int -> [[Int]] -> [[Int]]
setAt (x, y) val table
| (upper, current : lower) <- splitAt x table,
(left, this : right) <- splitAt y current
= upper ++ (left ++ val : right) : lower
| otherwise = error "Outside"
 
create :: Int -> [[Int]]
create x = replicate x $ replicate x 0
 
cells :: [[Int]] -> Int
cells m = x*x where x = length m
 
fill :: Var -> [[Int]] -> [[Int]]
fill (sx, sy, sz, c) m
| c < cells m =
if isZero m sx sy
then fill ((wrapInc sz sx), (wrapDec sz sy), sz, c + 1) (setAt (sx, sy) (c + 1) m)
else fill ((wrapDec sz sx), (wrapInc sz(wrapInc sz sy)), sz, c) m
| otherwise = m
 
magicNumber :: Int -> [[Int]]
magicNumber d = transpose $ fill (d `div` 2, 0, d, 0) (create d)
 
display :: [[Int]] -> String
display (x:xs)
| null xs = vdisplay x
| otherwise = vdisplay x ++ ('\n' : display xs)
 
vdisplay :: [Int] -> String
vdisplay (x:xs)
| null xs = show x
| otherwise = show x ++ " " ++ vdisplay xs
 
 
magicSquare x = do
putStr "Magic Square of "
putStr $ show x
putStr " = "
putStrLn $ show $ magicSum x
putStrLn $ display $ magicNumber x
 


Or, defining the magic square as cycledRows . transpose . cycledRows . rangeSquare

import Data.List (transpose, unfoldr, intercalate)
 
magicSquare :: Int -> [[Int]]
magicSquare n
| mod n 2 /= 0 = cycledRows . transpose . cycledRows . rangeSquare $ n
| otherwise = []
 
-- N * N square with cells numbered sequentially
-- left right, top down
rangeSquare :: Int -> [[Int]]
rangeSquare n = splitEvery n [1 .. (n ^ 2)]
 
-- Numbers in each row cycled to the right
-- The first row by (N div 2) cells, and each subsequent row
-- by one less, down to minus (N div 2) in the last row
cycledRows :: [[Int]] -> [[Int]]
cycledRows rows = uncurry listCycle <$> zip [d,d - 1 .. -d] rows
where
d = quot (length rows) 2
listCycle n xs = zipWith const (drop (length xs - n) (cycle xs)) xs
 
splitEvery :: Int -> [a] -> [[a]]
splitEvery n = takeWhile (not . null) . unfoldr (Just . splitAt n)
 
-- Magic squares of dimension 3, 5 and 7
main :: IO ()
main =
putStr $
intercalate "\n\n" ((unlines . (show <$>) . magicSquare) <$> [3, 5, 7])
Output:
[8,3,4]
[1,5,9]
[6,7,2]

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

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

Icon and Unicon[edit]

This is a Unicon-specific solution because of the use of the [: ... :] construct.

procedure main(A)
n := integer(!A) | 3
write("Magic number: ",n*(n*n+1)/2)
sq := buildSquare(n)
showSquare(sq)
end
 
procedure buildSquare(n)
sq := [: |list(n)\n :]
r := 0
c := n/2
every i := !(n*n) do {
/sq[r+1,c+1] := i
nr := (n+r-1)%n
nc := (c+1)%n
if /sq[nr+1,nc+1] then (r := nr,c := nc) else r := (r+1)%n
}
return sq
end
 
procedure showSquare(sq)
n := *sq
s := *(n*n)+2
every r := !sq do every writes(right(!r,s)|"\n")
end
Output:
->ms 5
Magic number: 65
  17  24   1   8  15
  23   5   7  14  16
   4   6  13  20  22
  10  12  19  21   3
  11  18  25   2   9
->

J[edit]

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

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

In other words, generate a square of counting integers, like this:

   >:@i.@,~ 3
1 2 3
4 5 6
7 8 9

Then generate a list of integers centering on 0 up to half of that value, like this:

   i:@<.@-: 3
_1 0 1

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

Example use:

   ms 5
9 15 16 22 3
20 21 2 8 14
1 7 13 19 25
12 18 24 5 6
23 4 10 11 17
~.+/ms 5
65
~.+/ms 101
515201

Java[edit]

public class MagicSquare {
 
public static void main(String[] args) {
int n = 5;
for (int[] row : magicSquareOdd(n)) {
for (int x : row)
System.out.format("%2s ", x);
System.out.println();
}
System.out.printf("\nMagic constant: %d ", (n * n + 1) * n / 2);
}
 
public static int[][] magicSquareOdd(final int base) {
if (base % 2 == 0 || base < 3)
throw new IllegalArgumentException("base must be odd and > 2");
 
int[][] grid = new int[base][base];
int r = 0, number = 0;
int size = base * base;
 
int c = base / 2;
while (number++ < size) {
grid[r][c] = number;
if (r == 0) {
if (c == base - 1) {
r++;
} else {
r = base - 1;
c++;
}
} else {
if (c == base - 1) {
r--;
c = 0;
} else {
if (grid[r - 1][c + 1] == 0) {
r--;
c++;
} else {
r++;
}
}
}
}
return grid;
}
}
Output:
17 24  1  8 15 
23  5  7 14 16 
 4  6 13 20 22 
10 12 19 21  3 
11 18 25  2  9

Magic constant: 65 


JavaScript[edit]

ES5[edit]

Translation of: Mathematica

( and referring to http://www.jsoftware.com/papers/eem/magicsq.htm )

(function () {
 
// n -> [[n]]
function magic(n) {
return n % 2 ? rotation(
transposed(
rotation(
table(n)
)
)
) : null;
}
 
// [[a]] -> [[a]]
function rotation(lst) {
return lst.map(function (row, i) {
return rotated(
row, ((row.length + 1) / 2) - (i + 1)
);
})
}
 
// [[a]] -> [[a]]
function transposed(lst) {
return lst[0].map(function (col, i) {
return lst.map(function (row) {
return row[i];
})
});
}
 
// [a] -> n -> [a]
function rotated(lst, n) {
var lng = lst.length,
m = (typeof n === 'undefined') ? 1 : (
n < 0 ? lng + n : (n > lng ? n % lng : n)
);
 
return m ? (
lst.slice(-m).concat(lst.slice(0, lng - m))
) : lst;
}
 
// n -> [[n]]
function table(n) {
var rngTop = rng(1, n);
 
return rng(0, n - 1).map(function (row) {
return rngTop.map(function (x) {
return row * n + x;
});
});
}
 
// [m..n]
function rng(m, n) {
return Array.apply(null, Array(n - m + 1)).map(
function (x, i) {
return m + i;
});
}
 
/******************** TEST WITH 3, 5, 11 ***************************/
 
// Results as right-aligned wiki tables
function wikiTable(lstRows, blnHeaderRow, strStyle) {
var css = strStyle ? 'style="' + strStyle + '"' : '';
 
return '{| class="wikitable" ' + css + lstRows.map(
function (lstRow, iRow) {
var strDelim = ((blnHeaderRow && !iRow) ? '!' : '|'),
strDbl = strDelim + strDelim;
 
return '\n|-\n' + strDelim + ' ' + lstRow.join(' ' + strDbl + ' ');
}).join('') + '\n|}';
}
 
return [3, 5, 11].map(
function (n) {
var w = 2.5 * n;
return 'magic(' + n + ')\n\n' + wikiTable(
magic(n), false, 'text-align:center;width:' + w + 'em;height:' + w + 'em;table-layout:fixed;'
)
}
).join('\n\n')
})();

Output:

magic(3)

8 3 4
1 5 9
6 7 2

magic(5)

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

magic(11)

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

ES6[edit]

Translation of: Haskell
(2nd Haskell version: cycledRows . transpose . cycledRows . rangeSquare)
(() => {
'use strict';
 
// magicSquare :: Int -> [[Int]]
const magicSquare = n =>
n % 2 !== 0 ? (
compose([cycledRows, transpose, cycledRows, rangeSquare])(n)
) : [];
 
// rangeSquare :: Int -> [[Int]]
const rangeSquare = n =>
splitEvery(n, range(1, n * n));
 
// Numbers in each row cycled to the right
// The first row by (N div 2) cells, and each subsequent row
// by one less, down to minus (N div 2) in the last row
// cycledRows :: [[Int]] -> [[Int]]
const cycledRows = rows => {
const d = Math.floor(rows.length / 2);
return zip(range(d, -d), rows)
.map(x => listCycle(...x));
};
 
// listCycle :: Int -> [a] -> [a]
const listCycle = (n, xs) => {
const d = -(n % xs.length);
return (d !== 0 ? xs.slice(d)
.concat(xs.slice(0, d)) : xs);
};
 
// GENERIC FUNCTIONS
 
// transpose :: [[a]] -> [[a]]
const transpose = xs =>
xs[0].map((_, iCol) => xs.map(row => row[iCol]));
 
// compose :: [(a -> a)] -> (a -> a)
const compose = fs => x => fs.reduceRight((a, f) => f(a), x);
 
// zip :: [a] -> [b] -> [(a,b)]
const zip = (xs, ys) =>
xs.slice(0, Math.min(xs.length, ys.length))
.map((x, i) => [x, ys[i]]);
 
// range :: Int -> Int -> Maybe Int -> [Int]
const range = (m, n, step) => {
const d = (step || 1) * (n >= m ? 1 : -1);
return Array.from({
length: Math.floor((n - m) / d) + 1
}, (_, i) => m + (i * d));
};
 
// 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));
};
 
// TEST AND DISPLAY (orders 3, 5, 7)
 
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
 
// intercalate :: String -> [a] -> String
const intercalate = (s, xs) => xs.join(s);
 
// show :: a -> String
const show = JSON.stringify;
 
return intercalate('\n\n', [3, 5, 7]
.map(magicSquare)
.map(xs => unlines(xs.map(show))));
})();
Output:
[8,3,4]
[1,5,9]
[6,7,2]

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

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

jq[edit]

Adapted from #AWK

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

Examples

def task:
def pp: if length == 0 then empty
else "\(.[0])", (.[1:] | pp )
end;
"The magic sum for a square of size \(.) is \( (.*. + 1)*./2 ):",
(odd_magic_square | pp)
;
 
(3, 5, 9) | task
Output:
$ jq -n -r -M -c -f odd_magic_square.jq
The magic sum for a square of size 3 is 15:
[8,1,6]
[3,5,7]
[4,9,2]
The magic sum for a square of size 5 is 65:
[17,24,1,8,15]
[23,5,7,14,16]
[4,6,13,20,22]
[10,12,19,21,3]
[11,18,25,2,9]
The magic sum for a square of size 9 is 369:
[47,58,69,80,1,12,23,34,45]
[57,68,79,9,11,22,33,44,46]
[67,78,8,10,21,32,43,54,56]
[77,7,18,20,31,42,53,55,66]
[6,17,19,30,41,52,63,65,76]
[16,27,29,40,51,62,64,75,5]
[26,28,39,50,61,72,74,4,15]
[36,38,49,60,71,73,3,14,25]
[37,48,59,70,81,2,13,24,35]

Kotlin[edit]

Translation of: C
// version 1.0.6
 
fun f(n: Int, x: Int, y: Int) = (x + y * 2 + 1) % n
 
fun main(args: Array<String>) {
var n: Int
while (true) {
print("Enter the order of the magic square : ")
n = readLine()!!.toInt()
if (n < 1 || n % 2 == 0) println("Must be odd and >= 1, try again")
else break
}
println()
for (i in 0 until n) {
for (j in 0 until n) print("%4d".format(f(n, n - j - 1, i) * n + f(n, j, i) + 1))
println()
}
println("\nThe magic constant is ${(n * n + 1) / 2 * n}")
}

Sample input/output:

Output:
Enter the order of the magic square : 9

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

The magic constant is 369

Liberty BASIC[edit]

 
Dim m(1,1)
 
Call magicSquare 5
Call magicSquare 17
 
End
 
Sub magicSquare n
ReDim m(n,n)
inc = 1
count = 1
row = 1
col=(n+1)/2
While count <= n*n
m(row,col) = count
count = count + 1
If inc < n Then
inc = inc + 1
row = row - 1
col = col + 1
If row <> 0 Then
If col > n Then col = 1
Else
row = n
End If
Else
inc = 1
row = row + 1
End If
Wend
Call printSquare n
End Sub
 
Sub printSquare n
'Arbitrary limit to fit width of A4 paper
If n < 23 Then
Print n;" x ";n;" Magic Square --- ";
Print "Magic constant is ";Int((n*n+1)/2*n)
For row = 1 To n
For col = 1 To n
Print Using("####",m(row,col));
Next col
Print
Print
Next row
Else
Notice "Magic Square will not fit on one sheet of paper."
End If
End Sub
 
Output:
5 x 5 Magic Square --- Magic constant is 65
  17  24   1   8  15

  23   5   7  14  16

   4   6  13  20  22

  10  12  19  21   3

  11  18  25   2   9

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Mathematica[edit]

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

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

 
rp[v_, pos_] := RotateRight[v, (Length[v] + 1)/2 - pos];
rho[m_] := MapIndexed[rp, m];
magic[n_] :=
rho[Transpose[rho[Table[i*n + j, {i, 0, n - 1}, {j, 1, n}]]]];
 
square = magic[11] // Grid
Print["Magic number is ", Total[square[[1, 1]]]]
 
Output:
(alignment lost in translation to text):
{68, 80, 92, 104, 116, 7, 19, 31, 43, 55, 56},
{81, 93, 105, 117, 8, 20, 32, 44, 45, 57, 69},
{94, 106, 118, 9, 21, 33, 34, 46, 58, 70, 82},
{107, 119, 10, 22, 23, 35, 47, 59, 71, 83, 95},
{120, 11, 12, 24, 36, 48, 60, 72, 84, 96, 108},
{1, 13, 25, 37, 49, 61, 73, 85, 97, 109, 121},
{14, 26, 38, 50, 62, 74, 86, 98, 110, 111, 2},
{27, 39, 51, 63, 75, 87, 99, 100, 112, 3, 15},
{40, 52, 64, 76, 88, 89, 101, 113, 4, 16, 28},
{53, 65, 77, 78, 90, 102, 114, 5, 17, 29, 41},
{66, 67, 79, 91, 103, 115, 6, 18, 30, 42, 54}

Magic number is 671

Output from code that checks the results Rows

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

Columns

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

Diagonals

671

671

Maxima[edit]

wrap1(i):= if i>%n% then 1 else if i<1 then %n% else i;
wrap(P):=maplist('wrap1, P);
 
uprigth(P):= wrap(P + [-1, 1]);
down(P):= wrap(P + [1, 0]);
 
magic(n):=block([%n%: n,
M: zeromatrix (n, n),
P: [1, (n + 1)/2],
m: 1, Pc],
do (
M[P[1],P[2]]: m,
m: m + 1,
if m>n^2 then return(M),
Pc: uprigth(P),
if M[Pc[1],Pc[2]]=0 then P: Pc
else while(M[P[1],P[2]]#0) do P: down(P)));

Usage:

(%i6) magic(3);
[ 8 1 6 ]
[ ]
(%o6) [ 3 5 7 ]
[ ]
[ 4 9 2 ]
(%i7) magic(5);
[ 17 24 1 8 15 ]
[ ]
[ 23 5 7 14 16 ]
[ ]
(%o7) [ 4 6 13 20 22 ]
[ ]
[ 10 12 19 21 3 ]
[ ]
[ 11 18 25 2 9 ]
(%i8) magic(7);
[ 30 39 48 1 10 19 28 ]
[ ]
[ 38 47 7 9 18 27 29 ]
[ ]
[ 46 6 8 17 26 35 37 ]
[ ]
(%o8) [ 5 14 16 25 34 36 45 ]
[ ]
[ 13 15 24 33 42 44 4 ]
[ ]
[ 21 23 32 41 43 3 12 ]
[ ]
[ 22 31 40 49 2 11 20 ]
/* magic number for n=7 */
(%i9) lsum(q, q, first(magic(7)));
(%o9) 175

Nim[edit]

Translation of: Python
import strutils
 
proc `^`*(base: int, exp: int): int =
var (base, exp) = (base, exp)
result = 1
 
while exp != 0:
if (exp and 1) != 0:
result *= base
exp = exp shr 1
base *= base
 
proc magic(n) =
for row in 1 .. n:
for col in 1 .. n:
let cell = (n * ((row + col - 1 + n div 2) mod n) +
((row + 2 * col - 2) mod n) + 1)
stdout.write align($cell, len($(n^2)))," "
echo ""
echo "\nAll sum to magic number ", ((n * n + 1) * n div 2)
 
for n in [5, 3, 7]:
echo "\nOrder ",n,"\n======="
magic(n)
Output:
Order 5
=======
17 24  1  8 15 
23  5  7 14 16 
 4  6 13 20 22 
10 12 19 21  3 
11 18 25  2  9 

All sum to magic number 65

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

All sum to magic number 15

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

All sum to magic number 175

Oforth[edit]

: magicSquare(n)
| i j wd |
n sq log asInteger 1+ ->wd
n loop: i [
n loop: j [
i j + 1- n 2 / + n mod n *
i j + j + 2 - n mod 1 + +
System.Out swap <<w(wd) " " << drop
]
printcr
]
System.Out "Magic constant is : " << n sq 1 + 2 / n * << cr ;
Output:
5 magicSquare
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9
Magic constant is : 65

Pascal[edit]

Works with: Free Pascal version 1.0
Translation of: C
PROGRAM magic;
(* Magic squares of odd order *)
CONST
n=9;
VAR
i,j :INTEGER;
BEGIN (*magic*)
WRITELN('The square order is: ',n);
FOR i:=1 TO n DO
BEGIN
FOR j:=1 TO n DO
WRITE((i*2-j+n-1) MOD n*n + (i*2+j-2) MOD n+1:5);
WRITELN
END;
WRITELN('The magic number is: ',n*(n*n+1) DIV 2)
END (*magic*).
Output:
The square order is: 9
    2   75   67   59   51   43   35   27   10
   22   14    6   79   71   63   46   38   30
   42   34   26   18    1   74   66   58   50
   62   54   37   29   21   13    5   78   70
   73   65   57   49   41   33   25   17    9
   12    4   77   69   61   53   45   28   20
   32   24   16    8   81   64   56   48   40
   52   44   36   19   11    3   76   68   60
   72   55   47   39   31   23   15    7   80
The magic number is: 369

improved[edit]

shuffles columns and rows and changed col<-> row to get different looks. n! x n! * 2 different arrangements. See last column of version before moved to the top row.

PROGRAM magic;
{$IFDEF FPC }{$MODE DELPHI}{$ELSE}{$APPTYPE CONSOLE}{$ENDIF}
uses
sysutils;
(* Magic squares of odd order *)
type
tsquare = array of array of LongInt;
trowcol = array of NativeInt;
 
function GenShuffleRowCol(n: nativeInt):trowcol;
var
i,j,tmp: NativeInt;
begin
setlength(result,0);
IF n > 0 then
Begin
setlength(result,n);
For i := 0 to n-1 do
result[i] := i;
//shuffle
For i := n-1 downto 1 do
Begin
j := random(i+1);//j == [0..i]
tmp := result[i];result[i]:= result[j];result[j]:= tmp;
end;
end;
end;
 
function MagicSqrOdd(n:nativeInt;SwapColRoW:boolean):tsquare;
VAR
rowIdx,colIdx,row,col,num :NativeInt;
cols,rows :trowcol;
BEGIN
rows:= GenShuffleRowCol(n);
cols:= GenShuffleRowCol(n);
setlength(result,n,n);
FOR rowIdx:= 0 TO n-1 DO
BEGIN
row := rows[rowIdx];
FOR colIdx:=0 TO n-1 DO
Begin
col := cols[colIdx];
//corrected formula cause row :0..n*1-> corrected to 1..n
num := (row*2-col+n+2) MOD n*n + (row*2+col+1) MOD n+1;
IF SwapColRoW then
result[colIdx,rowIdx] := num
else
result[rowIdx,colIdx] := num;
end;
END;
END;
 
function MagicSqrCheck(const Mq:tsquare):boolean;
var
row,col,rowsum,mn,n,itm: NativeInt;
colSum:trowcol;
begin
n := length(Mq[0]);
mn := n*(n*n+1) DIV 2;
setlength(colsum,n);//automatic initialised to zero
For row := n-1 downto 0 do
Begin
//check one row
rowsum := 0;
For col := n-1 downto 0 do
Begin
itm := Mq[row,col];
write(itm:4);
inc(rowsum,itm);
//sum up the columns too, for I'm just here
inc(colSum[col],itm);
end;
writeln;
result := (rowsum=mn);
IF Not(result) then begin writeln(row:4,col:4,rowsum:10);EXIT;end;
end;
//check columns
For col := n-1 downto 0 do
Begin
result := (colSum[col]=mn);
IF Not(result) then begin writeln(col:4,colSum[col]:10);EXIT;end;
end;
writeln;
end;
 
 
var
n,mn : nativeInt;
Mq : tsquare;
Begin
randomize;
n := 9;
mn := n*(n*n+1) DIV 2;
WRITELN('The square order is: ',n);
WRITELN('The magic number is: ',mn);
Mq := MagicSqrOdd(n,random(2)=0);
writeln(MagicSqrCheck(Mq));
end.
Output:
The square order is: 9
The magic number is: 369
  70  30  20   9  40  50  10  80  60
  13  63  53  33  64  74  43  23   3
  54  14   4  65  24  34  75  55  44
   5  46  45  25  56  66  35  15  76
  37   6  77  57  16  26  67  47  36
  29  79  69  49   8  18  59  39  19
  78  38  28  17  48  58  27   7  68
  62  22  12  73  32  42   2  72  52
  21  71  61  41  81   1  51  31  11

TRUE

PARI/GP[edit]

Translation of: Perl

The index-fiddling differs from Perl since GP vectors start at 1.

magicSquare(n)={
my(M=matrix(n,n),j=n\2+1,i=1);
for(l=1,n^2,
M[i,j]=l;
if(M[(i-2)%n+1,j%n+1],
i=i%n+1
,
i=(i-2)%n+1;
j=j%n+1
)
);
M;
}
magicSquare(7)
Output:
[30 39 48  1 10 19 28]

[38 47  7  9 18 27 29]

[46  6  8 17 26 35 37]

[ 5 14 16 25 34 36 45]

[13 15 24 33 42 44  4]

[21 23 32 41 43  3 12]

[22 31 40 49  2 11 20]

Perl[edit]

 
sub magic_square {
my $n = shift;
my $i = 0;
my $j = $n / 2;
my @magic_square;
 
for my $l ( 1 .. $n**2 ) {
$magic_square[$i][$j] = $l;
 
if ( $magic_square[ ( $i - 1 ) % $n ][ ( $j + 1 ) % $n ] ) {
$i = ( $i + 1 ) % $n;
}
else {
$i = ( $i - 1 ) % $n;
$j = ( $j + 1 ) % $n;
}
}
return @magic_square;
}
 
my $n = 7;
 
for ( magic_square($n) ) {
printf '%8d' x $n . qq{\n}, @{$_};
}
 

Perl 6[edit]

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

Output:

With a parameter of 5:

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

The magic number is 65

With a parameter of 19:

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

The magic number is 3439

PL/I[edit]

magic: procedure options (main);  /* 18 April 2014 */
declare n fixed binary;
 
put skip list ('What is the order of the magic square?');
get list (n);
if n < 3 | iand(n, 1) = 0 then
do; put skip list ('The value is out of range'); stop; end;
put skip list ('The order is ' || trim(n));
 
begin;
declare m(n, n) fixed, (i, j, k) fixed binary;
 
on subrg snap put data (i, j, k);
m = 0;
i = 1; j = (n+1)/2;
 
do k = 1 to n*n;
if m(i,j) = 0 then
m(i,j) = k;
else
do;
i = i + 2; j = j + 1;
if i > n then i = mod(i,n);
if j > n then j = 1;
m(i,j) = k;
end;
i = i - 1; j = j - 1;
if i < 1 then i = n;
if j < 1 then j = n;
end;
 
do i = 1 to n;
put skip edit (m(i, *)) (f(4));
end;
 
put skip list ('The magic number is' || sum(m(1,*)));
end;
end magic;
Output:
What is the order of the magic square? 

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

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

PureBasic[edit]

Translation of: Pascal
#N=9
Define.i i,j
 
If OpenConsole("Magic squares")
PrintN("The square order is: "+Str(#N))
For i=1 To #N
For j=1 To #N
Print(RSet(Str((i*2-j+#N-1) % #N*#N + (i*2+j-2) % #N+1),5))
Next
PrintN("")
Next
PrintN("The magic number is: "+Str(#N*(#N*#N+1)/2))
EndIf
Input()
Output:
The square order is: 9
    2   75   67   59   51   43   35   27   10
   22   14    6   79   71   63   46   38   30
   42   34   26   18    1   74   66   58   50
   62   54   37   29   21   13    5   78   70
   73   65   57   49   41   33   25   17    9
   12    4   77   69   61   53   45   28   20
   32   24   16    8   81   64   56   48   40
   52   44   36   19   11    3   76   68   60
   72   55   47   39   31   23   15    7   80
The magic number is: 369

Python[edit]

>>> def magic(n):
for row in range(1, n + 1):
print(' '.join('%*i' % (len(str(n**2)), cell) for cell in
(n * ((row + col - 1 + n // 2) % n) +
((row + 2 * col - 2) % n) + 1
for col in range(1, n + 1))))
print('\nAll sum to magic number %i' % ((n * n + 1) * n // 2))
 
 
>>> for n in (5, 3, 7):
print('\nOrder %i\n=======' % n)
magic(n)
 
 
 
Order 5
=======
17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9
 
All sum to magic number 65
 
Order 3
=======
8 1 6
3 5 7
4 9 2
 
All sum to magic number 15
 
Order 7
=======
30 39 48 1 10 19 28
38 47 7 9 18 27 29
46 6 8 17 26 35 37
5 14 16 25 34 36 45
13 15 24 33 42 44 4
21 23 32 41 43 3 12
22 31 40 49 2 11 20
 
All sum to magic number 175
>>>

Racket[edit]

#lang racket
;; Using "helpful formulae" in:
;; http://en.wikipedia.org/wiki/Magic_square#Method_for_constructing_a_magic_square_of_odd_order
(define (squares n) n)
 
(define (last-no n) (sqr n))
 
(define (middle-no n) (/ (add1 (sqr n)) 2))
 
(define (M n) (* n (middle-no n)))
 
(define ((Ith-row-Jth-col n) I J)
(+ (* (modulo (+ I J -1 (exact-floor (/ n 2))) n) n)
(modulo (+ I (* 2 J) -2) n)
1))
 
(define (magic-square n)
(define IrJc (Ith-row-Jth-col n))
(for/list ((I (in-range 1 (add1 n)))) (for/list ((J (in-range 1 (add1 n)))) (IrJc I J))))
 
(define (fmt-list-of-lists l-o-l width)
(string-join
(for/list ((row l-o-l))
(string-join (map (λ (x) (~a #:align 'right #:width width x)) row) " "))
"\n"))
 
(define (show-magic-square n)
(format "MAGIC SQUARE ORDER:~a~%~a~%MAGIC NUMBER:~a~%"
n (fmt-list-of-lists (magic-square n) (+ (order-of-magnitude (last-no n)) 1)) (M n)))
 
(displayln (show-magic-square 3))
(displayln (show-magic-square 5))
(displayln (show-magic-square 9))
Output:
MAGIC SQUARE ORDER:3
8  1  6
3  5  7
4  9  2
Magic Number:15

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

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

REXX[edit]

This REXX version will also generate a square of an even order, but it'll not be a magic square.

/*REXX program generates and displays magic squares (odd N will be a true magic square).*/
parse arg N . /*obtain the optional argument from CL.*/
if N=='' | N=="," then N=5 /*Not specified? Then use the default.*/
NN=N*N; w=length(NN) /*W: width of largest number (output).*/
r=1; c=(n+1) % 2 /*define the initial row and column.*/
@.=. /*assign a default value for entire @.*/
do j=1 for NN /* [↓] filling uses the Siamese method*/
if r<1 & c>N then do; r=r+2; c=c-1; end /*the row is under, column is over.*/
if r<1 then r=N /* " " " " make row=last. */
if r>N then r=1 /* " " " over, " " first.*/
if c>N then c=1 /* " column " over, " col=first.*/
if @.r.c\==. then do; r=min(N,r+2); c=max(1,c-1); end /*at the previous cell? */
@.r.c=j; r=r-1; c=c+1 /*assign # ───► cell; next row & column*/
end /*j*/
/* [↓] display square with aligned #'s*/
do r=1 for N; _= /*display one matrix row at a time. */
do c=1 for N; _=_ right(@.r.c, w) /*construct a row of the magic square. */
end /*c*/
say substr(_, 2) /*display a row of the magic square. */
end /*c*/
say /* [↓] If an odd square, show magic #.*/
if N//2 then say 'The magic number (or magic constant is): ' N * (NN+1) % 2
/*stick a fork in it, we're all done. */

output   when using the default input of:   5

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

The magic number  (or magic constant is):  65

output   when using the input of:   3

8 1 6
3 5 7
4 9 2

The magic number  (or magic constant is):  15

output   when using the input of:   19

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

The magic number  (or magic constant is):  3439

Ring[edit]

 
n=9
see "the square order is : " + n + nl
for i=1 to n
for j = 1 to n
x = (i*2-j+n-1) % n*n + (i*2+j-2) % n + 1
see "" + x + " "
next
see nl
next
see "'the magic number is : " + n*(n*n+1) / 2 + nl
 

Output:

the square order is : 9

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

the magic number is : 369

Ruby[edit]

def odd_magic_square(n)
raise ArgumentError "Need odd positive number" if n.even? || n <= 0
n.times.map{|i| n.times.map{|j| n*((i+j+1+n/2)%n) + ((i+2*j-5)%n) + 1} }
end
 
[3, 5, 9].each do |n|
puts "\nSize #{n}, magic sum #{(n*n+1)/2*n}"
fmt = "%#{(n*n).to_s.size + 1}d" * n
odd_magic_square(n).each{|row| puts fmt % row}
end
 
Output:
Size 3, magic sum 15
 8 1 6
 3 5 7
 4 9 2

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

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

Rust[edit]

fn main() {
let n = 9;
let mut square = vec![vec![0; n]; n];
for (i, row) in square.iter_mut().enumerate() {
for (j, e) in row.iter_mut().enumerate() {
*e = n * (((i + 1) + (j + 1) - 1 + (n >> 1)) % n) + (((i + 1) + (2 * (j + 1)) - 2) % n) + 1;
print!("{:3} ", e);
}
println!("");
}
let sum = n * (((n * n) + 1) / 2);
println!("The sum of the square is {}.", sum);
}
Output:
 47  58  69  80   1  12  23  34  45 
 57  68  79   9  11  22  33  44  46 
 67  78   8  10  21  32  43  54  56 
 77   7  18  20  31  42  53  55  66 
  6  17  19  30  41  52  63  65  76 
 16  27  29  40  51  62  64  75   5 
 26  28  39  50  61  72  74   4  15 
 36  38  49  60  71  73   3  14  25 
 37  48  59  70  81   2  13  24  35 
The sum of the square is 369.

Scala[edit]

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

Seed7[edit]

$ include "seed7_05.s7i";
 
const func integer: succ (in integer: num, in integer: max) is
return succ(num mod max);
 
const func integer: pred (in integer: num, in integer: max) is
return succ((num - 2) mod max);
 
const proc: main is func
local
var integer: size is 3;
var array array integer: magic is 0 times 0 times 0;
var integer: row is 1;
var integer: column is 1;
var integer: number is 0;
begin
if length(argv(PROGRAM)) >= 1 then
size := integer parse (argv(PROGRAM)[1]);
end if;
magic := size times size times 0;
column := succ(size div 2);
for number range 1 to size ** 2 do
magic[row][column] := number;
if magic[pred(row, size)][succ(column, size)] = 0 then
row := pred(row, size);
column := succ(column, size);
else
row := succ(row, size);
end if;
end for;
for key row range magic do
for key column range magic[row] do
write(magic[row][column] lpad 4);
end for;
writeln;
end for;
end func;
Output:
> s7 magicSquaresOfOddOrder 7
SEED7 INTERPRETER Version 5.0.5203  Copyright (c) 1990-2014 Thomas Mertes
  30  39  48   1  10  19  28
  38  47   7   9  18  27  29
  46   6   8  17  26  35  37
   5  14  16  25  34  36  45
  13  15  24  33  42  44   4
  21  23  32  41  43   3  12
  22  31  40  49   2  11  20

Sidef[edit]

func magic_square(n {.is_pos && .is_odd}) {
var i = 0
var j = int(n/2)
 
var magic_square = []
for l in (1 .. n**2) {
magic_square[i][j] = l
 
if (magic_square[i.dec % n][j.inc % n]) {
i = (i.inc % n)
}
else {
i = (i.dec % n)
j = (j.inc % n)
}
}
 
return magic_square
}
 
func print_square(sq) {
var f = "%#{(sq.len**2).len}d";
for row in sq {
say row.map{ f % _ }.join(' ')
}
}
 
var(n=5) = ARGV»to_i»()...
var sq = magic_square(n)
print_square(sq)
 
say "\nThe magic number is: #{sq[0].sum}"
Output:
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9

The magic number is: 65

Tcl[edit]

proc magicSquare {order} {
if {!($order & 1) || $order < 0} {
error "order must be odd and positive"
}
set s [lrepeat $order [lrepeat $order 0]]
set x [expr {$order / 2}]
set y 0
for {set i 1} {$i <= $order**2} {incr i} {
lset s $y $x $i
set x [expr {($x + 1) % $order}]
set y [expr {($y - 1) % $order}]
if {[lindex $s $y $x]} {
set x [expr {($x - 1) % $order}]
set y [expr {($y + 2) % $order}]
}
}
return $s
}

Demonstrating:

Works with: Tcl version 8.6
package require Tcl 8.6
 
set square [magicSquare 5]
puts [join [lmap row $square {join [lmap n $row {format "%2s" $n}]}] "\n"]
puts "magic number = [tcl::mathop::+ {*}[lindex $square 0]]"
Output:
17 24  1  8 15
23  5  7 14 16
 4  6 13 20 22
10 12 19 21  3
11 18 25  2  9
magic number = 65

TI-83 BASIC[edit]

Translation of: C
Works with: TI-83 BASIC version TI-84Plus 2.55MP
9→N
DelVar [A]:{N,N}→dim([A])
For(I,1,N)
For(J,1,N)
Remainder(I*2-J+N-1,N)*N+Remainder(I*2+J-2,N)+1→[A](I,J)
End
End
[A]
Output:
[[2  75 67 59 51 43 35 27 10]
 [22 14 6  79 71 63 46 38 30]
 [42 34 26 18 1  74 66 58 50]
 [62 54 37 29 21 13 5  78 70]
 [73 65 57 49 41 33 25 17 9 ]
 [12 4  77 69 61 53 45 28 20]
 [32 24 16 8  81 64 56 48 40]
 [52 44 36 19 11 3  76 68 60]
 [72 55 47 39 31 23 15 7  80]]

uBasic/4tH[edit]

Translation of: FreeBASIC
' ------=< MAIN >=------
 
Proc _magicsq(5)
Proc _magicsq(11)
End
 
_magicsq Param (1) Local (4)
 
' reset the array
For b@ = 0 to 255
@(b@) = 0
Next
 
If ((a@ % 2) = 0) + (a@ < 3) + (a@ > 15) Then
Print "error: size is not odd or size is smaller then 3 or bigger than 15"
Return
EndIf
 
' start in the middle of the first row
b@ = 1
c@ = a@ - (a@ / 2)
d@ = 1
e@ = a@ * a@
 
' main loop for creating magic square
Do
If @(c@*a@+d@) = 0 Then
@(c@*a@+d@) = b@
If (b@ % a@) = 0 Then
d@ = d@ + 1
Else
c@ = c@ + 1
d@ = d@ - 1
EndIf
b@ = b@ + 1
EndIf
If c@ > a@ Then
c@ = 1
Do While @(c@*a@+d@) # 0
c@ = c@ + 1
Loop
EndIf
If d@ < 1 Then
d@ = a@
Do While @(c@*a@+d@) # 0
d@ = d@ - 1
Loop
EndIf
Until b@ > e@
Loop
 
Print "Odd magic square size: "; a@; " * "; a@
Print "The magic sum = "; ((e@+1) / 2) * a@
Print
 
For d@ = 1 To a@
For c@ = 1 To a@
Print Using "____"; @(c@*a@+d@);
Next
Print
Next
Print
Return
 
Output:
Odd magic square size: 5 * 5
The magic sum = 65

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

Odd magic square size: 11 * 11
The magic sum = 671

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


0 OK, 0:64

VBA[edit]

Translation of: C

Works with Excel VBA.

Sub magicsquare()
'Magic squares of odd order
Const n = 9
Dim i As Integer, j As Integer, v As Integer
Debug.Print "The square order is: " & n
For i = 1 To n
For j = 1 To n
Cells(i, j) = ((i * 2 - j + n - 1) Mod n) * n + ((i * 2 + j - 2) Mod n) + 1
Next j
Next i
Debug.Print "The magic number of"; n; "x"; n; "square is:"; n * (n * n + 1) \ 2
End Sub 'magicsquare


VBScript[edit]

Translation of: Liberty BASIC
 
Sub magic_square(n)
Dim ms()
ReDim ms(n-1,n-1)
inc = 0
count = 1
row = 0
col = Int(n/2)
Do While count <= n*n
ms(row,col) = count
count = count + 1
If inc < n-1 Then
inc = inc + 1
row = row - 1
col = col + 1
If row >= 0 Then
If col > n-1 Then
col = 0
End If
Else
row = n-1
End If
Else
inc = 0
row = row + 1
End If
Loop
For i = 0 To n-1
For j = 0 To n-1
If j = n-1 Then
WScript.StdOut.Write ms(i,j)
Else
WScript.StdOut.Write ms(i,j) & vbTab
End If
Next
WScript.StdOut.WriteLine
Next
End Sub
 
magic_square(5)
 
Output:
17	24	1	8	15
23	5	7	14	16
4	6	13	20	22
10	12	19	21	3
11	18	25	2	9

Visual Basic[edit]

Translation of: C
Works with: Visual Basic version VB6 Standard
Sub magicsquare()
'Magic squares of odd order
Const n = 9
Dim i As Integer, j As Integer, v As Integer
Debug.Print "The square order is: " & n
For i = 1 To n
For j = 1 To n
v = ((i * 2 - j + n - 1) Mod n) * n + ((i * 2 + j - 2) Mod n) + 1
Debug.Print Right(Space(5) & v, 5);
Next j
Debug.Print
Next i
Debug.Print "The magic number is: " & n * (n * n + 1) \ 2
End Sub 'magicsquare
Output:
The square order is: 9
    2   75   67   59   51   43   35   27   10
   22   14    6   79   71   63   46   38   30
   42   34   26   18    1   74   66   58   50
   62   54   37   29   21   13    5   78   70
   73   65   57   49   41   33   25   17    9
   12    4   77   69   61   53   45   28   20
   32   24   16    8   81   64   56   48   40
   52   44   36   19   11    3   76   68   60
   72   55   47   39   31   23   15    7   80
The magic number is: 369


Visual Basic .NET[edit]

Works with: Visual Basic .NET version 2011
Sub magicsquare()
'Magic squares of odd order
Const n = 9
Dim i, j, v As Integer
Console.WriteLine("The square order is: " & n)
For i = 1 To n
For j = 1 To n
v = ((i * 2 - j + n - 1) Mod n) * n + ((i * 2 + j - 2) Mod n) + 1
Console.Write(" " & Right(Space(5) & v, 5))
Next j
Console.WriteLine("")
Next i
Console.WriteLine("The magic number is: " & n * (n * n + 1) \ 2)
End Sub 'magicsquare
Output:
The square order is: 9
    2   75   67   59   51   43   35   27   10
   22   14    6   79   71   63   46   38   30
   42   34   26   18    1   74   66   58   50
   62   54   37   29   21   13    5   78   70
   73   65   57   49   41   33   25   17    9
   12    4   77   69   61   53   45   28   20
   32   24   16    8   81   64   56   48   40
   52   44   36   19   11    3   76   68   60
   72   55   47   39   31   23   15    7   80
The magic number is: 369

zkl[edit]

Translation of: Ruby
fcn rmod(n,m){ n=n%m; if (n<0) n+=m; n } // Ruby: -5%3-->1
fcn odd_magic_square(n){ //-->list of n*n numbers, row order
if (n.isEven or n <= 0) throw(Exception.ValueError("Need odd positive number"));
[[(i,j); n; n; '{ n*((i+j+1+n/2):rmod(_,n)) + ((i+2*j-5):rmod(_,n)) + 1 }]]
}
 
T(3, 5, 9).pump(Void,fcn(n){
"\nSize %d, magic sum %d".fmt(n,(n*n+1)/2*n).println();
fmt := "%%%dd".fmt((n*n).toString().len() + 1) * n;
odd_magic_square(n).pump(Console.println,T(Void.Read,n-1),fmt.fmt);
});
Output:
Size 3, magic sum 15
 8 1 6
 3 5 7
 4 9 2

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

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