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 N×N 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 × 8.
- Related tasks
- See also
11l
F magicSquareDoublyEven(n)
V bits = 1001'0110'0110'1001b
V size = n * n
V mult = n I/ 4
V result = [[0] * n] * n
V i = 0
L(r) 0 .< n
L(c) 0 .< n
V bitPos = c I/ mult + (r I/ mult) * 4
result[r][c] = I (bits [&] (1 << bitPos)) != 0 {i + 1} E size - i
i++
R result
V n = 8
L(row) magicSquareDoublyEven(n)
L(x) row
print(‘#2 ’.format(x), end' ‘’)
print()
print("\nMagic constant: "((n * n + 1) * n I/ 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
360 Assembly
* 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
ALGOL 60
begin
comment Magic squares of doubly even order - 10/02/2021;
integer array pattern[1:4,1:4];
integer n, r, c, s, m, i, b, t;
n:=8;
for r:=1 step 1 until 4 do
for c:=1 step 1 until 4 do
pattern[r,c]:=if
((c=1 or c=4) and (r=1 or r=4)) or
((c=2 or c=3) and (r=2 or r=3)) then 1 else 0;
s:=n*n; m:=n div 4;
outstring(1,"magic square -- n ="); outinteger(1,n); outstring(1,"\n");
i:=0;
for r:=1 step 1 until n do begin
for c:=1 step 1 until n do begin
b:=pattern[(r-1) div m+1,(c-1) div m+1];
if b=1 then t:=i+1 else t:=s-i;
if t less 10 then outstring(1," ");
outinteger(1,t);
i:=i+1
end;
outstring(1,"\n")
end;
outstring(1,"magic constant ="); outinteger(1,(s+1)*n div 2)
end- Output:
magic square -- n = 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
ALGOL 68
Using a procedure to generate the square for easier re-use.
BEGIN # Magic squares of doubly even order #
PROC doubly even magic square = ( INT n )[,]INT:
IF n MOD 4 /= 0 THEN
# not a doubly even number #
[ 1 : 0, 1 : 0 ]INT empty square;
empty square
ELSE
# ok to create the square #
[ 1 : 4, 1 : 4 ]INT pattern;
FOR r TO 4 DO
FOR c TO 4 DO
pattern[ r, c ] := IF ( ( c = 1 OR c = 4 ) AND ( r = 1 OR r = 4 ) )
OR ( ( c = 2 OR c = 3 ) AND ( r = 2 OR r = 3 ) )
THEN 1
ELSE 0
FI
OD
OD;
[ 1 : n, 1 : n ]INT result;
INT s = n * n, m = n OVER 4;
INT i := 0;
FOR r TO n DO
FOR c TO n DO
result[ r, c ] := IF pattern[ ( r - 1 ) OVER m + 1, ( c - 1 ) OVER m + 1 ] = 1
THEN i + 1
ELSE s - i
FI;
i +:= 1
OD
OD;
result
FI # doubly eben magic square # ;
# test doubly even magic square #
FOR order FROM 8 BY 4 TO 12 DO
# calculate the field width for the elements of the square #
INT w := 1, v := order * order;
WHILE ( v OVERAB 10 ) > 0 DO w +:= 1 OD;
# construct the square #
[,]INT square = doubly even magic square( order );
print( ( "magic square -- n = ", whole( order, 0 ), newline ) );
FOR r FROM 1 LWB square TO 1 UPB square DO
FOR c FROM 2 LWB square TO 2 UPB square DO
print( ( " ", whole( square[ r, c ], - w ) ) )
OD;
print( ( newline ) )
OD;
print( ( "magic constant = ", whole( ( ( ( order * order ) + 1 ) * order ) OVER 2, 0 ), newline ) )
OD
END- Output:
magic square -- n = 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 magic square -- n = 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 magic constant = 870
Amazing Hopper
/*
Magic squares of doubly even order. Rosettacode.org
By Mr. Dalien.
*/
#include <basico.h>
#proto cuadradomágico(_X_)
#synon _cuadradomágico generarcuadradomágico
principal {
decimales '0', fijar separador 'NULO'
malla de bits(bits,'1;0;0;1','0;1;1;0','0;1;1;0','1;0;0;1')
borrar pantalla
iterar para( i=1; n=4, #(i<=3), ++i; #(n=4*i) )
dim( #(n*n) ) matriz de ceros 'sqr'
generar cuadrado mágico (n);
ir a sub ( meter en tabla, sqr, #(n+1) ), para 'sqr'
imprimir '"Magic square order ",n,\
"\nMagic constant : ",#((n * n + 1) * n / 2),\
NL,sqr,NL,NL'
luego limpiar 'sqr'
siguiente
terminar
}
subrutinas
sub( meter en tabla, s, n ) {
i=n,rareti(i, 'i')
retener 'n', insertar columnas en 's'
insertar filas en 's'
#basic{
s[1:2:2*n-1,1:_end_] = "---"
s[1:_end_, 1:2:2*n-1] = "|"
s[1:2:_end_,1:2:_end_] = "+"
}
retornar 's'
}
cuadrado mágico ( n )
iterar para ( cr=0;i=0, #(cr<n), ++cr )
iterar para ( cc=0, #(cc<n), ++cc;++i )
#( sqr[ (cc+n*cr)+1 ] = (bits[(cr%4+1),(cc%4+1)]) ? (i+1) : (n^2-i); )
siguiente
siguiente
#( sqr = lpad(" ",3,string(sqr)))
redim ( sqr, n, n )
retornar
- Output:
Magic square order 4 Magic constant : 34 +---+---+---+---+ | 1| 15| 14| 4| +---+---+---+---+ | 12| 6| 7| 9| +---+---+---+---+ | 8| 10| 11| 5| +---+---+---+---+ | 13| 3| 2| 16| +---+---+---+---+ Magic square order 8 Magic constant : 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| +---+---+---+---+---+---+---+---+ Magic square order 12 Magic constant : 870 +---+---+---+---+---+---+---+---+---+---+---+---+ | 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| +---+---+---+---+---+---+---+---+---+---+---+---+
AppleScript
-- MAGIC SQUARE OF DOUBLY EVEN ORDER -----------------------------------------
-- magicSquare :: Int -> [[Int]]
on magicSquare(n)
if n mod 4 > 0 then
{}
else
set sqr to n * n
set maybePowerOfTwo to asPowerOfTwo(sqr)
if maybePowerOfTwo is not missing value then
-- For powers of 2, the (append not) 'magic' series directly
-- yields the truth table that we need
set truthSeries to magicSeries(maybePowerOfTwo)
else
-- where n is not a power of 2, we can replicate a
-- minimum truth table, horizontally and vertically
script scale
on |λ|(x)
replicate(n / 4, x)
end |λ|
end script
set truthSeries to ¬
flatten(scale's |λ|(map(scale, splitEvery(4, magicSeries(4)))))
end if
set limit to sqr + 1
script inOrderOrReversed
on |λ|(x, i)
cond(x, i, limit - i)
end |λ|
end script
-- Taken directly from an integer series [1..sqr] where True
-- and from the reverse of that series where False
splitEvery(n, map(inOrderOrReversed, truthSeries))
end if
end magicSquare
-- magicSeries :: Int -> [Bool]
on magicSeries(n)
script boolToggle
on |λ|(x)
not x
end |λ|
end script
if n ≤ 0 then
{true}
else
set xs to magicSeries(n - 1)
xs & map(boolToggle, xs)
end if
end magicSeries
-- TEST ----------------------------------------------------------------------
on run
formattedTable(magicSquare(8))
end run
-- formattedTable :: [[Int]] -> String
on formattedTable(lstTable)
set n to length of lstTable
set w to 2.5 * n
"magic(" & n & ")" & linefeed & wikiTable(lstTable, ¬
false, "text-align:center;width:" & ¬
w & "em;height:" & w & "em;table-layout:fixed;")
end formattedTable
-- wikiTable :: [Text] -> Bool -> Text -> Text
on wikiTable(lstRows, blnHdr, strStyle)
script fWikiRows
on |λ|(lstRow, iRow)
set strDelim to cond(blnHdr and (iRow = 0), "!", "|")
set strDbl to strDelim & strDelim
linefeed & "|-" & linefeed & strDelim & space & ¬
intercalate(space & strDbl & space, lstRow)
end |λ|
end script
linefeed & "{| class=\"wikitable\" " & ¬
cond(strStyle ≠ "", "style=\"" & strStyle & "\"", "") & ¬
intercalate("", ¬
map(fWikiRows, lstRows)) & linefeed & "|}" & linefeed
end wikiTable
-- GENERIC FUNCTIONS ---------------------------------------------------------
-- asPowerOfTwo :: Int -> maybe Int
on asPowerOfTwo(n)
if not isPowerOf(2, n) then
missing value
else
set strCMD to ("echo 'l(" & n as string) & ")/l(2)' | bc -l"
(do shell script strCMD) as integer
end if
end asPowerOfTwo
-- concatMap :: (a -> [b]) -> [a] -> [b]
on concatMap(f, xs)
script append
on |λ|(a, b)
a & b
end |λ|
end script
foldl(append, {}, map(f, xs))
end concatMap
-- cond :: Bool -> a -> a -> a
on cond(bool, f, g)
if bool then
f
else
g
end if
end cond
-- flatten :: Tree a -> [a]
on flatten(t)
if class of t is list then
concatMap(my flatten, t)
else
t
end if
end flatten
-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- intercalate :: Text -> [Text] -> Text
on intercalate(strText, lstText)
set {dlm, my text item delimiters} to {my text item delimiters, strText}
set strJoined to lstText as text
set my text item delimiters to dlm
return strJoined
end intercalate
-- isPowerOf :: Int -> Int -> Bool
on isPowerOf(k, n)
set v to k
script remLeft
on |λ|(x)
x mod v is not 0
end |λ|
end script
script integerDiv
on |λ|(x)
x div v
end |λ|
end script
|until|(remLeft, integerDiv, n) = 1
end isPowerOf
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- Egyptian multiplication - progressively doubling a list, appending
-- stages of doubling to an accumulator where needed for binary
-- assembly of a target length
-- replicate :: Int -> a -> [a]
on replicate(n, a)
set out to {}
if n < 1 then return out
set dbl to {a}
repeat while (n > 1)
if (n mod 2) > 0 then set out to out & dbl
set n to (n div 2)
set dbl to (dbl & dbl)
end repeat
return out & dbl
end replicate
-- splitAt :: Int -> [a] -> ([a],[a])
on splitAt(n, xs)
if n > 0 and n < length of xs then
if class of xs is text then
{items 1 thru n of xs as text, items (n + 1) thru -1 of xs as text}
else
{items 1 thru n of xs, items (n + 1) thru -1 of xs}
end if
else
if n < 1 then
{{}, xs}
else
{xs, {}}
end if
end if
end splitAt
-- splitEvery :: Int -> [a] -> [[a]]
on splitEvery(n, xs)
if length of xs ≤ n then
{xs}
else
set {gp, t} to splitAt(n, xs)
{gp} & splitEvery(n, t)
end if
end splitEvery
-- until :: (a -> Bool) -> (a -> a) -> a -> a
on |until|(p, f, x)
set mp to mReturn(p)
set v to x
tell mReturn(f)
repeat until mp's |λ|(v)
set v to |λ|(v)
end repeat
end tell
return v
end |until|
- Output:
magic(8)
| 1 | 63 | 62 | 4 | 60 | 6 | 7 | 57 |
| 56 | 10 | 11 | 53 | 13 | 51 | 50 | 16 |
| 48 | 18 | 19 | 45 | 21 | 43 | 42 | 24 |
| 25 | 39 | 38 | 28 | 36 | 30 | 31 | 33 |
| 32 | 34 | 35 | 29 | 37 | 27 | 26 | 40 |
| 41 | 23 | 22 | 44 | 20 | 46 | 47 | 17 |
| 49 | 15 | 14 | 52 | 12 | 54 | 55 | 9 |
| 8 | 58 | 59 | 5 | 61 | 3 | 2 | 64 |
AWK
Since standard awk does not support bitwise operators, we provide the function countup to do the magic.
# Usage: GAWK -f MAGIC_SQUARES_OF_DOUBLY_EVEN_ORDER.AWK
BEGIN {
n = 8
msquare[0, 0] = 0
if (magicsquaredoublyeven(msquare, n)) {
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
printf("%2d ", msquare[i, j])
}
printf("\n")
}
printf("\nMagic constant: %d\n", (n * n + 1) * n / 2)
exit 1
} else {
exit 0
}
}
function magicsquaredoublyeven(msquare, n, size, mult, r, c, i) {
if (n < 4 || n % 4 != 0) {
printf("Base must be a positive multiple of 4.\n")
return 0
}
size = n * n
mult = n / 4 # how many multiples of 4
i = 0
for (r = 0; r < n; r++) {
for (c = 0; c < n; c++) {
msquare[r, c] = countup(r, c, mult) ? i + 1 : size - i
i++
}
}
return 1
}
function countup(r, c, mult, pattern, bitpos) {
# Returns 1, if we are in a count-up zone (0 otherwise)
pattern = "1001011001101001"
bitpos = int(c / mult) + int(r / mult) * 4 + 1
return substr(pattern, bitpos, 1) + 0
}
- Output:
1 2 62 61 60 59 7 8 9 10 54 53 52 51 15 16 48 47 19 20 21 22 42 41 40 39 27 28 29 30 34 33 32 31 35 36 37 38 26 25 24 23 43 44 45 46 18 17 49 50 14 13 12 11 55 56 57 58 6 5 4 3 63 64 Magic constant: 260
Befunge
The size, N, is specified by the first value on the stack. The algorithm is loosely based on the Java implementation.
8>>>v>10p00g:*1-*\110g2*-*+1+.:00g%!9+,:#v_@
p00:<^:!!-!%3//4g00%g00\!!%3/*:g00*4:::-1<*:
- Output:
1 2 62 61 60 59 7 8 9 10 54 53 52 51 15 16 48 47 19 20 21 22 42 41 40 39 27 28 29 30 34 33 32 31 35 36 37 38 26 25 24 23 43 44 45 46 18 17 49 50 14 13 12 11 55 56 57 58 6 5 4 3 63 64
C
Takes number of rows from command line, prints out usage on incorrect invocation.
#include<stdlib.h>
#include<ctype.h>
#include<stdio.h>
int** doublyEvenMagicSquare(int n) {
if (n < 4 || n % 4 != 0)
return NULL;
int bits = 38505;
int size = n * n;
int mult = n / 4,i,r,c,bitPos;
int** result = (int**)malloc(n*sizeof(int*));
for(i=0;i<n;i++)
result[i] = (int*)malloc(n*sizeof(int));
for (r = 0, i = 0; r < n; r++) {
for (c = 0; c < n; c++, i++) {
bitPos = c / mult + (r / mult) * 4;
result[r][c] = (bits & (1 << bitPos)) != 0 ? i + 1 : size - i;
}
}
return result;
}
int numDigits(int n){
int count = 1;
while(n>=10){
n /= 10;
count++;
}
return count;
}
void printMagicSquare(int** square,int rows){
int i,j,baseWidth = numDigits(rows*rows) + 3;
printf("Doubly Magic Square of Order : %d and Magic Constant : %d\n\n",rows,(rows * rows + 1) * rows / 2);
for(i=0;i<rows;i++){
for(j=0;j<rows;j++){
printf("%*s%d",baseWidth - numDigits(square[i][j]),"",square[i][j]);
}
printf("\n");
}
}
int main(int argC,char* argV[])
{
int n;
if(argC!=2||isdigit(argV[1][0])==0)
printf("Usage : %s <integer specifying rows in magic square>",argV[0]);
else{
n = atoi(argV[1]);
printMagicSquare(doublyEvenMagicSquare(n),n);
}
return 0;
}
Invocation and Output :
C:\rosettaCode>doublyEvenMagicSquare 12
Doubly Magic Square of Order : 12 and Magic Constant : 870
1 2 3 141 140 139 138 137 136 10 11 12
13 14 15 129 128 127 126 125 124 22 23 24
25 26 27 117 116 115 114 113 112 34 35 36
108 107 106 40 41 42 43 44 45 99 98 97
96 95 94 52 53 54 55 56 57 87 86 85
84 83 82 64 65 66 67 68 69 75 74 73
72 71 70 76 77 78 79 80 81 63 62 61
60 59 58 88 89 90 91 92 93 51 50 49
48 47 46 100 101 102 103 104 105 39 38 37
109 110 111 33 32 31 30 29 28 118 119 120
121 122 123 21 20 19 18 17 16 130 131 132
133 134 135 9 8 7 6 5 4 142 143 144
C#
using System;
namespace MagicSquareDoublyEven
{
class Program
{
static void Main(string[] args)
{
int n = 8;
var result = MagicSquareDoublyEven(n);
for (int i = 0; i < result.GetLength(0); i++)
{
for (int j = 0; j < result.GetLength(1); j++)
Console.Write("{0,2} ", result[i, j]);
Console.WriteLine();
}
Console.WriteLine("\nMagic constant: {0} ", (n * n + 1) * n / 2);
Console.ReadLine();
}
private static int[,] MagicSquareDoublyEven(int n)
{
if (n < 4 || n % 4 != 0)
throw new ArgumentException("base must be a positive "
+ "multiple of 4");
// pattern of count-up vs count-down zones
int bits = 0b1001_0110_0110_1001;
int size = n * n;
int mult = n / 4; // how many multiples of 4
int[,] result = new int[n, n];
for (int r = 0, i = 0; r < n; r++)
{
for (int c = 0; c < n; c++, i++)
{
int bitPos = c / mult + (r / mult) * 4;
result[r, c] = (bits & (1 << bitPos)) != 0 ? i + 1 : size - i;
}
}
return result;
}
}
}
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
C++
#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
D
import std.stdio;
void main() {
int n=8;
foreach(row; magicSquareDoublyEven(n)) {
foreach(col; row) {
writef("%2s ", col);
}
writeln;
}
writeln("\nMagic constant: ", (n*n+1)*n/2);
}
int[][] magicSquareDoublyEven(int n) {
import std.exception;
enforce(n>=4 && n%4 == 0, "Base must be a positive multiple of 4");
int bits = 0b1001_0110_0110_1001;
int size = n * n;
int mult = n / 4; // how many multiples of 4
int[][] result;
result.length = n;
foreach(i; 0..n) {
result[i].length = n;
}
for (int r=0, i=0; r<n; r++) {
for (int c=0; c<n; c++, i++) {
int bitPos = c / mult + (r / mult) * 4;
result[r][c] = (bits & (1 << bitPos)) != 0 ? i + 1 : size - i;
}
}
return result;
}
- 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
EasyLang
func[][] mkarr s .
len q[][] s
for i = 1 to s
len q[i][] s
.
return q[][]
.
func[][] mk_oms s .
q[][] = mkarr s
r = 1
c = 1 + s div 2
for p = 1 to s * s
q[r][c] = p
tc = c + 1
if tc > s
tc = 1
.
tr = r - 1
if tr < 1
tr = s
.
if q[tr][tc] <> 0
tc = c
tr = r + 1
.
c = tc
r = tr
.
return q[][]
.
func[][] mk_dems s .
q[][] = mkarr s
tmp[][] = [ [ 1 0 0 1 ] [ 0 1 1 0 ] [ 0 1 1 0 ] [ 1 0 0 1 ] ]
tot = s * s
for r = 1 to s
for c = 1 to s
sx = c mod 4
if sx < 1
sx = 4
.
sy = r mod 4
if sy < 1
sy = 4
.
if tmp[sy][sx] = 1
q[r][c] = n + 1
else
q[r][c] = tot - n
.
n += 1
.
.
return q[][]
.
numfmt 0 2
print mk_dems 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 ] ]
EDSAC order code
[Magic squares of doubly even order, for Rosetta Code.
EDSAC program, Initial Orders 2.]
[====================================================================================
Certain cells in the square are marked, such that if a cell is marked then so is
its reflection in the centre point of the square. Cells are numbered 1, 2, 3 ...
from left to right and top to bottom, but if a cell is marked it is swapped with its
reflection. Two marking methods are used, illustrated below for an 8x8 square.
+ o o + + o o + + + o o o o + +
o + + o o + + o + + o o o o + +
o + + o o + + o o o + + + + o o
+ o o + + o o + o o + + + + o o o = unmarked
+ o o + + o o + o o + + + + o o + = marked
o + + o o + + o o o + + + + o o
o + + o o + + o + + o o o o + +
+ o o + + o o + + + o o o o + +
Diagonal method Rectangle method
====================================================================================]
[Arrange the storage]
T45K P56F [H parameter: subroutine to print string]
T46K P100F [N parameter: subroutine to print number]
T47K P204F [M parameter: main routine]
T54K P200F [C parameter: constants read by subroutine R2]
[Library subroutine R2: reads integers from tape and can then be overwritten.]
GK T20F VD L8F A40D UD TF I40F A40F S39F G@ S2F G23F A5@ T5@ E4@ E13Z
T#C [tell R2 where to store values]
13743895347F8589934592#TZ
[C parameter: masks read in by subroutine R2, not by the regular loader]
[0] PF PF [diagonal method, binary 01100110011001100110011001100110011]
[2] PF PF [rectangle method, binary 01000000000000000000000000000000000]
[M parameter Main routine + high-level subroutine]
E25K TM GK
[35-bit values, must be at even address]
[0] PF PF [initial value of x mask]
[2] PF [x-mask, low 17 bits]
[3] PF [x-mask, high 17 bits]
[4] PF [y-mask, low 17 bits]
[5] PF [y-mask, high 17 bits]
[17-bit values]
[6] PF [sign bit from y mask]
[7] PF [m, input by user]
[8] PF [n = 4*m = order of magic square]
[9] PF [n^2 + 1]
[10] PF [negative counter for x-values (columns)]
[11] PF [negative counter for y-values (rows)]
[12] PF [current entry 1, 2, 3, ...]
[13] PD [constant 1]
[14] K4096F [null]
[15] !F [space]
[16] @F [carriage return]
[17] &F [line feed]
[18] P10F [to check for user dialling '0']
[Strings to be printed]
[19] K2048FMFAFGFIFCF!FSFQFUFAFRFEF!FOFFF!FOFRFDFEFRF!F#FRF*FMF@F&FK4096F
[49] K2048FDFIFAFLF!FMF!F#FKFPF!F*FTFOF!FCFAFNFCFEFLF#FLF!FK4096F
[75] K2048FDFIFAFGFOFNFAFLF!FMFEFTFHFOFDF#FCF@F&FK4096F
[96] K2048FRFEFCFTFAFNFGFLFEF!FMFEFTFHFOFDF#FCF@F&FK4096F
[Enter with acc = 0]
[118] A118@ GH A19@ [print 'MAGIC SQUARE OF ORDER 4M']
[121] A121@ GH A49@ [print 'DIAL M (0 TO CANCEL)']
ZF [halt machine; restarts when user dials a number]
[Here acc holds number of pulses in address field]
S18@ E175@ [exit if dialled '0' (10 pulses)]
A18@ [restore acc after test; m is in address field]
L512F [shift 11 left for printing]
UF [temp to 0F]
OF O16@ O17@ [print m followed by CR, LF]
R1024F [shift 12 right, m is now right-justified]
U7@ [store m]
L1F T8@ [shift 2 left and store n = 4*m]
H8@ V8@ [acc := n^2]
L64F L64F [shift 16 left to adjust scaling after mult]
A13@ T9@ [store n^2 + 1 = sum of a cell and its reflection]
[143] A143@ GH A75@ [print 'DIAGONAL METHOD:']
A#C [acc := diagonal mask]
U#@ [store as x-mask at start of each row]
T4#@ [also as initial value of y-mask]
[149] A149@ G177@ [call subroutine to print magic square]
[151] A151@ GH A96@ [print 'RECTANGLE METHOD:']
A2#C U4D T6D [copy rectangke mask to 4D and 6D]
S7@ [initialize negative counter to -m]
[158] TF [loop: update negative counter in 0F]
A4D RD T4D [shift 4D 1 right]
A6D R2F T6D [shift 6D 3 right]
AF A13@ [inc negative counter]
G158@ [loop back till done m times]
A4D S6D L1F [acc := 4D - 6D, then 2 left]
[Mask in binary is now 0 (m times) 1 (2*m times) 0...0]
U#@ [store as x-mask at start of each row]
T4#@ [also as initial value of y-mask]
[173] A173@ G177@ [call subroutine to print magic square]
[175] O14@ [done; print null to flush teleprinter buffer]
ZF [halt machine]
[Subroutine to print magic square after x- and y-mask have been initialized.]
[It's assumed that strings printed by caller leave teleprinter in figures mode.]
[177] A3F T220@ [plant return link as usual]
A15@ T1F [space replaces leading 0 when printing]
T12@ [initialize cell entry to 0]
S8@ [initialize negative counter of rows to -n]
[Start of row]
[183] T11@ [update negative counter of rows]
A#@ T2#@ [reset x-mask for start of row]
H14@ C5@ T6@ [isolate sign bit of y-mask]
S8@ [initialize negative counter of columns to -n]
[Next cell in this row.
Cell is considered marked if sign bits in x- and y-masks are equal.
Or could say marked if sign bits are unequal; would also give a magic square.]
[190] T10@ [update negative counter of columns]
A12@ A13@ T12@ [inc cell entry]
A3@ A6@ [compare signs in x- and y-masks]
E200@ [jump if equal (or could replace E by G)]
TF [clear acc]
A12@ [acc := entry]
E203@ [join common code]
[200] TF [clear acc]
A9@ S12@ [acc := complement of entry]
[203] TF [to 0F for printing]
[204] A204@ GN [print number]
A2#@ LD T2#@ [shift x-mask 1 left]
A10@ A13@ [inc negative counter of cells]
G190@ [loop till row is complete]
[End of row]
O16@ O17@ [print CR, LF]
A4#@ LD T4#@ [shift y-mask 1 left]
A11@ A13@ [inc negative counter of rows]
G183@ [loop till magic square is complete]
[220] ZF [(planted) jump back to caller]
[H parameter: Subroutine to print a string.]
E25K TH
[Input: A order for first character must follow subroutine call (G order)
String is terminated with EDSAC null, which is sent to the teleprinter.]
GKA18@U17@S19@T4@AFT6@AFUFOFE12@A20@G16@TFA6@A2FG5@TFZFU3FU1FK2048F
[N parameter: Subroutine to print non-negative 17-bit integer.]
E25K TN
[Parameters: 0F = integer to be printed (not preserved)
1F = character for leading zero (preserved)
Workspace: 4F..7F, 38 locations]
GKA3FT34@A1FT7FS35@T6FT4#FAFT4FH36@V4FRDA4#FR1024FH37@E23@O7FA2F
T6FT5FV4#FYFL8FT4#FA5FL1024FUFA6FG16@OFTFT7FA6FG17@ZFP4FZ219DTF
[M parameter again]
E25K TM GK
E118Z [define entry point]
PF [acc = 0 on entry]- Output:
MAGIC SQUARE OF ORDER 4M
DIAL M (0 TO CANCEL) 2
DIAGONAL METHOD:
64 2 3 61 60 6 7 57
9 55 54 12 13 51 50 16
17 47 46 20 21 43 42 24
40 26 27 37 36 30 31 33
32 34 35 29 28 38 39 25
41 23 22 44 45 19 18 48
49 15 14 52 53 11 10 56
8 58 59 5 4 62 63 1
RECTANGLE METHOD:
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
Elena
ELENA 6.x :
import system'routines;
import extensions;
import extensions'routines;
MagicSquareDoublyEven(int n)
{
if(n < 4 || n.mod(4) != 0)
{ InvalidArgumentException.new("base must be a positive multiple of 4").raise() };
int bits := 09669h;
int size := n * n;
int mult := n / 4;
var result := IntMatrix.allocate(n,n);
int i := 0;
for (int r := 0; r < n; r += 1)
{
for(int c := 0; c < n; c += 1; i += 1)
{
int bitPos := c / mult + (r / mult) * 4;
result[r][c] := ((bits & (1 $shl bitPos)) != 0).iif(i+1,size - i)
}
};
^ result
}
public program()
{
int n := 8;
console.printLine(MagicSquareDoublyEven(n));
console.printLine().printLine("Magic constant: ",(n * n + 1) * n / 2)
}- Output:
1,2,62,61,60,59,7,8 9,10,54,53,52,51,15,16 48,47,19,20,21,22,42,41 40,39,27,28,29,30,34,33 32,31,35,36,37,38,26,25 24,23,43,44,45,46,18,17 49,50,14,13,12,11,55,56 57,58,6,5,4,3,63,64 Magic constant: 260
Elixir
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
Factor
USING: arrays combinators.short-circuit formatting fry
generalizations kernel math math.matrices prettyprint sequences
;
IN: rosetta-code.doubly-even-magic-squares
: top? ( loc n -- ? ) [ second ] dip 1/4 * < ;
: bottom? ( loc n -- ? ) [ second ] dip 3/4 * >= ;
: left? ( loc n -- ? ) [ first ] dip 1/4 * < ;
: right? ( loc n -- ? ) [ first ] dip 3/4 * >= ;
: corner? ( loc n -- ? )
{
[ { [ top? ] [ left? ] } ]
[ { [ top? ] [ right? ] } ]
[ { [ bottom? ] [ left? ] } ]
[ { [ bottom? ] [ right? ] } ]
} [ 2&& ] map-compose 2|| ;
: center? ( loc n -- ? )
{ [ top? ] [ bottom? ] [ left? ] [ right? ] } [ not ]
map-compose 2&& ;
: backward? ( loc n -- ? ) { [ corner? ] [ center? ] } 2|| ;
: forward ( loc n -- m ) [ first2 ] dip * 1 + + ;
: backward ( loc n -- m ) tuck forward [ sq ] dip - 1 + ;
: (doubly-even-magic-square) ( n -- matrix )
[ dup 2array matrix-coordinates flip ] [ 3 dupn ] bi
'[ dup _ backward? [ _ backward ] [ _ forward ] if ]
matrix-map ;
ERROR: invalid-order order ;
: check-order ( n -- )
dup { [ zero? not ] [ 4 mod zero? ] } 1&& [ drop ]
[ invalid-order ] if ;
: doubly-even-magic-square ( n -- matrix )
dup check-order (doubly-even-magic-square) ;
: main ( -- )
{ 4 8 12 } [
dup doubly-even-magic-square dup
[ "Order: %d\n" printf ]
[ simple-table. ]
[ first sum "Magic constant: %d\n\n" printf ] tri*
] each ;
MAIN: main
- Output:
Order: 4 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 Magic constant: 34 Order: 8 64 63 3 4 5 6 58 57 56 55 11 12 13 14 50 49 17 18 46 45 44 43 23 24 25 26 38 37 36 35 31 32 33 34 30 29 28 27 39 40 41 42 22 21 20 19 47 48 16 15 51 52 53 54 10 9 8 7 59 60 61 62 2 1 Magic constant: 260 Order: 12 144 143 142 4 5 6 7 8 9 135 134 133 132 131 130 16 17 18 19 20 21 123 122 121 120 119 118 28 29 30 31 32 33 111 110 109 37 38 39 105 104 103 102 101 100 46 47 48 49 50 51 93 92 91 90 89 88 58 59 60 61 62 63 81 80 79 78 77 76 70 71 72 73 74 75 69 68 67 66 65 64 82 83 84 85 86 87 57 56 55 54 53 52 94 95 96 97 98 99 45 44 43 42 41 40 106 107 108 36 35 34 112 113 114 115 116 117 27 26 25 24 23 22 124 125 126 127 128 129 15 14 13 12 11 10 136 137 138 139 140 141 3 2 1 Magic constant: 870
FreeBASIC
' 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
Go
package main
import (
"fmt"
"log"
"strings"
)
const dimensions int = 8
func setupMagicSquareData(d int) ([][]int, error) {
var output [][]int
if d < 4 || d%4 != 0 {
return [][]int{}, fmt.Errorf("Square dimension must be a positive number which is divisible by 4")
}
var bits uint = 0x9669 // 0b1001011001101001
size := d * d
mult := d / 4
for i, r := 0, 0; r < d; r++ {
output = append(output, []int{})
for c := 0; c < d; i, c = i+1, c+1 {
bitPos := c/mult + (r/mult)*4
if (bits & (1 << uint(bitPos))) != 0 {
output[r] = append(output[r], i+1)
} else {
output[r] = append(output[r], size-i)
}
}
}
return output, nil
}
func arrayItoa(input []int) []string {
var output []string
for _, i := range input {
output = append(output, fmt.Sprintf("%4d", i))
}
return output
}
func main() {
data, err := setupMagicSquareData(dimensions)
if err != nil {
log.Fatal(err)
}
magicConstant := (dimensions * (dimensions*dimensions + 1)) / 2
for _, row := range data {
fmt.Println(strings.Join(arrayItoa(row), " "))
}
fmt.Printf("\nMagic Constant: %d\n", magicConstant)
}
- Output:
1 2 62 61 60 59 7 8 9 10 54 53 52 51 15 16 48 47 19 20 21 22 42 41 40 39 27 28 29 30 34 33 32 31 35 36 37 38 26 25 24 23 43 44 45 46 18 17 49 50 14 13 12 11 55 56 57 58 6 5 4 3 63 64 Magic Constant: 260
Haskell
import Data.List (transpose, unfoldr, intercalate)
import Data.List.Split (chunksOf)
import Data.Bool (bool)
import Control.Monad (forM_)
magicSquare :: Int -> [[Int]]
magicSquare n
| rem n 4 > 0 = []
| otherwise =
chunksOf n $ zipWith (flip (bool =<< (-) limit)) series [1 .. sqr]
where
sqr = n * n
limit = sqr + 1
series
| isPowerOf 2 n = magicSeries $ floor (logBase 2 (fromIntegral sqr))
| otherwise =
concat . concat . concat . scale $ scale <$> chunksOf 4 (magicSeries 4)
where
scale = replicate $ quot n 4
magicSeries :: Int -> [Bool]
magicSeries = (iterate ((++) <*> fmap not) [True] !!)
isPowerOf :: Int -> Int -> Bool
isPowerOf k n = until ((0 /=) . flip rem k) (`quot` k) n == 1
-- TEST AND DISPLAY FUNCTIONS --------------------------------------------------
checked :: [[Int]] -> (Int, Bool)
checked square =
let diagonals =
fmap (flip (zipWith (!!)) [0 ..]) . ((:) <*> (return . reverse))
h:t =
sum <$>
square ++ -- rows
transpose square ++ -- cols
diagonals square -- diagonals
in (h, all (h ==) t)
table :: String -> [[String]] -> [String]
table delim rows =
let justifyRight c n s = drop (length s) (replicate n c ++ s)
in intercalate delim <$>
transpose
((fmap =<< justifyRight ' ' . maximum . fmap length) <$> transpose rows)
main :: IO ()
main =
forM_ [4, 8, 16] $
\n -> do
let test = magicSquare n
putStrLn $ unlines (table " " (fmap show <$> test))
print $ checked test
putStrLn []
- Output:
1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16 (34,True) 1 63 62 4 60 6 7 57 56 10 11 53 13 51 50 16 48 18 19 45 21 43 42 24 25 39 38 28 36 30 31 33 32 34 35 29 37 27 26 40 41 23 22 44 20 46 47 17 49 15 14 52 12 54 55 9 8 58 59 5 61 3 2 64 (260,True) 1 255 254 4 252 6 7 249 248 10 11 245 13 243 242 16 240 18 19 237 21 235 234 24 25 231 230 28 228 30 31 225 224 34 35 221 37 219 218 40 41 215 214 44 212 46 47 209 49 207 206 52 204 54 55 201 200 58 59 197 61 195 194 64 192 66 67 189 69 187 186 72 73 183 182 76 180 78 79 177 81 175 174 84 172 86 87 169 168 90 91 165 93 163 162 96 97 159 158 100 156 102 103 153 152 106 107 149 109 147 146 112 144 114 115 141 117 139 138 120 121 135 134 124 132 126 127 129 128 130 131 125 133 123 122 136 137 119 118 140 116 142 143 113 145 111 110 148 108 150 151 105 104 154 155 101 157 99 98 160 161 95 94 164 92 166 167 89 88 170 171 85 173 83 82 176 80 178 179 77 181 75 74 184 185 71 70 188 68 190 191 65 193 63 62 196 60 198 199 57 56 202 203 53 205 51 50 208 48 210 211 45 213 43 42 216 217 39 38 220 36 222 223 33 32 226 227 29 229 27 26 232 233 23 22 236 20 238 239 17 241 15 14 244 12 246 247 9 8 250 251 5 253 3 2 256 (2056,True)
J
masksq=: >:@#@, | _1&^ * 1 + i.@$
pat4=: ,:(+.|.)=i.4
mask=: ,/@(,./"3@$&pat4)@] ,~ % 4:
demsq=: masksq@mask
- Output:
demsq 8 64 2 3 61 60 6 7 57 9 55 54 12 13 51 50 16 17 47 46 20 21 43 42 24 40 26 27 37 36 30 31 33 32 34 35 29 28 38 39 25 41 23 22 44 45 19 18 48 49 15 14 52 53 11 10 56 8 58 59 5 4 62 63 1 ~.(+/,&,+/"1)demsq 128 1048640
Java
public class MagicSquareDoublyEven {
public static void main(String[] args) {
int n = 8;
for (int[] row : magicSquareDoublyEven(n)) {
for (int x : row)
System.out.printf("%2s ", x);
System.out.println();
}
System.out.printf("\nMagic constant: %d ", (n * n + 1) * n / 2);
}
static int[][] magicSquareDoublyEven(final int n) {
if (n < 4 || n % 4 != 0)
throw new IllegalArgumentException("base must be a positive "
+ "multiple of 4");
// pattern of count-up vs count-down zones
int bits = 0b1001_0110_0110_1001;
int size = n * n;
int mult = n / 4; // how many multiples of 4
int[][] result = new int[n][n];
for (int r = 0, i = 0; r < n; r++) {
for (int c = 0; c < n; c++, i++) {
int bitPos = c / mult + (r / mult) * 4;
result[r][c] = (bits & (1 << bitPos)) != 0 ? i + 1 : size - i;
}
}
return result;
}
}
1 2 62 61 60 59 7 8 9 10 54 53 52 51 15 16 48 47 19 20 21 22 42 41 40 39 27 28 29 30 34 33 32 31 35 36 37 38 26 25 24 23 43 44 45 46 18 17 49 50 14 13 12 11 55 56 57 58 6 5 4 3 63 64 Magic constant: 260
JavaScript
ES6
(() => {
'use strict';
// doublyEvenMagicSquare :: Int -> [[Int]]
const doublyEvenMagicSquare = n =>
0 === n % 4 ? (() => {
const
sqr = n * n,
power = Math.log2(sqr),
scale = replicate(n / 4);
return chunksOf(n)(
map((x, i) => x ? 1 + i : sqr - i)(
isInt(power) ? truthSeries(power) : (
compose(
flatten,
scale,
map(scale),
chunksOf(4)
)(truthSeries(4))
)
)
);
})() : undefined;
// truthSeries :: Int -> [Bool]
const truthSeries = n =>
0 >= n ? (
[true]
) : (() => {
const xs = truthSeries(n - 1);
return xs.concat(xs.map(x => !x));
})();
// TEST -----------------------------------------------
const main = () =>
// Magic squares of orders 4, 8 and 12, with
// checks of row, column and diagonal sums.
intercalate('\n\n')(
map(n => {
const
lines = doublyEvenMagicSquare(n),
sums = map(sum)(
lines.concat(
transpose(lines)
.concat(diagonals(lines))
)
),
total = sums[0];
return unlines([
"Order: " + str(n),
"Summing to: " + str(total),
"Row, column and diagonal sums checked: " +
str(all(eq(total))(sums)) + '\n',
unlines(map(compose(
intercalate(' '),
map(compose(justifyRight(3)(' '), str))
))(lines))
]);
})([4, 8, 12])
);
// GENERIC FUNCTIONS ----------------------------------
// all :: (a -> Bool) -> [a] -> Bool
const all = p =>
// True if p(x) holds for every x in xs.
xs => xs.every(p);
// chunksOf :: Int -> [a] -> [[a]]
const chunksOf = n => xs =>
enumFromThenTo(0)(n)(
xs.length - 1
).reduce(
(a, i) => a.concat([xs.slice(i, (n + i))]),
[]
);
// compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
const compose = (...fs) =>
x => fs.reduceRight((a, f) => f(a), x);
// diagonals :: [[a]] -> [[a], [a]]
const diagonals = rows =>
// Two diagonal sequences,
// from top left and bottom left
// respectively, of a given matrix.
map(flip(zipWith(index))(
enumFromTo(0)(pred(
0 < rows.length ? (
rows[0].length
) : 0
))
))([rows, reverse(rows)]);
// enumFromThenTo :: Int -> Int -> Int -> [Int]
const enumFromThenTo = x1 => x2 => y => {
const d = x2 - x1;
return Array.from({
length: Math.floor(y - x2) / d + 2
}, (_, i) => x1 + (d * i));
};
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = m => n =>
Array.from({
length: 1 + n - m
}, (_, i) => m + i);
// eq (==) :: Eq a => a -> a -> Bool
const eq = a => b => a === b;
// flatten :: NestedList a -> [a]
const flatten = nest => nest.flat(Infinity);
// flip :: (a -> b -> c) -> b -> a -> c
const flip = f =>
x => y => f(y)(x);
// index (!!) :: [a] -> Int -> a
const index = xs => i => xs[i];
// intercalate :: String -> [String] -> String
const intercalate = s =>
xs => xs.join(s);
// isInt :: Int -> Bool
const isInt = x => x === Math.floor(x);
// justifyRight :: Int -> Char -> String -> String
const justifyRight = n => cFiller => s =>
n > s.length ? (
s.padStart(n, cFiller)
) : s;
// map :: (a -> b) -> [a] -> [b]
const map = f => xs =>
(Array.isArray(xs) ? (
xs
) : xs.split('')).map(f);
// pred :: Enum a => a -> a
const pred = x => x - 1;
// replicate :: Int -> a -> [a]
const replicate = n => x =>
Array.from({
length: n
}, () => x);
// reverse :: [a] -> [a]
const reverse = xs =>
'string' !== typeof xs ? (
xs.slice(0).reverse()
) : xs.split('').reverse().join('');
// show :: a -> String
const show = x => JSON.stringify(x);
// str :: a -> String
const str = x => x.toString();
// sum :: [Num] -> Num
const sum = xs => xs.reduce((a, x) => a + x, 0);
// transpose :: [[a]] -> [[a]]
const transpose = xs =>
xs[0].map((_, iCol) => xs.map((row) => row[iCol]));
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = f => xs => ys =>
xs.slice(
0, Math.min(xs.length, ys.length)
).map((x, i) => f(x)(ys[i]));
// MAIN ------------------------------------------------
return main();
})();
- 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
jq
Works with gojq, the Go implementation of jq
def lpad($len):
def l: tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
if type == "array" then map(l) else l end;
def magicSquareDoublyEven:
if . < 4 or .%4 != 0 then "Base must be a positive multiple of 4" | error else . end
| . as $n
# pattern of count-up vs count-down zones
| [1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1] as $bits
| ($n * $n) as $size
| ($n / 4 | floor) as $mult # how many multiples of 4
| { i:0, result: null }
| reduce range(0; $n) as $r (.;
reduce range(0; $n) as $c (.;
( (($c/$mult)|floor) + (($r/$mult)|floor) * 4) as $bitPos
| .result[$r][$c] =
(if ($bits[$bitPos] != 0) then .i + 1 else $size - .i end)
| .i += 1 ) )
| .result ;
# Input: the order
def task:
. as $n
| (.*.|tostring|length+1) as $width
| (magicSquareDoublyEven[] | lpad($width) | join(" ")),
"\nMagic constant for order \($n): \(($n*$n + 1) * $n / 2)\n\n" ;
8, 12 | task- 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 for order 8: 260 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 Magic constant for order 12: 870
Julia
using Printf
function magicsquaredoubleeven(order::Int)
if order % 4 != 0; error("the order must be divisible by 4") end
sqr = Matrix{Int}(order, order)
mul = div(order, 4)
ext = vcat(1:mul, order-mul+1:order)
isext(i::Int, j::Int) = (i in ext) == (j in ext)
boolsqr = collect(isext(i, j) for i in 1:order, j in 1:order)
for i in linearindices(sqr)
if boolsqr[i]; sqr[i] = i end
if !boolsqr[end+1-i]; sqr[end+1-i] = i end
end
return sqr
end
for n in (4, 8, 12)
magicconst = div(n ^ 3 + n, 2)
sq = magicsquaredoubleeven(n)
println("Order: $n; magic constant: $magicconst.\nSquare:")
for r in 1:n, c in 1:n
@printf("%4i", sq[r, c])
if c == n; println() end
end
println()
end
- Output:
Order: 4; magic constant: 34. Square: 1 12 8 13 15 6 10 3 14 7 11 2 4 9 5 16 Order: 8; magic constant: 260. Square: 1 9 48 40 32 24 49 57 2 10 47 39 31 23 50 58 62 54 19 27 35 43 14 6 61 53 20 28 36 44 13 5 60 52 21 29 37 45 12 4 59 51 22 30 38 46 11 3 7 15 42 34 26 18 55 63 8 16 41 33 25 17 56 64 Order: 12; magic constant: 870. Square: 1 13 25 108 96 84 72 60 48 109 121 133 2 14 26 107 95 83 71 59 47 110 122 134 3 15 27 106 94 82 70 58 46 111 123 135 141 129 117 40 52 64 76 88 100 33 21 9 140 128 116 41 53 65 77 89 101 32 20 8 139 127 115 42 54 66 78 90 102 31 19 7 138 126 114 43 55 67 79 91 103 30 18 6 137 125 113 44 56 68 80 92 104 29 17 5 136 124 112 45 57 69 81 93 105 28 16 4 10 22 34 99 87 75 63 51 39 118 130 142 11 23 35 98 86 74 62 50 38 119 131 143 12 24 36 97 85 73 61 49 37 120 132 144
Kotlin
// 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 ${(n * n + 1) * n / 2}")
}
- Output:
1 2 62 61 60 59 7 8 9 10 54 53 52 51 15 16 48 47 19 20 21 22 42 41 40 39 27 28 29 30 34 33 32 31 35 36 37 38 26 25 24 23 43 44 45 46 18 17 49 50 14 13 12 11 55 56 57 58 6 5 4 3 63 64 Magic constant 260
Lua
For all three kinds of Magic Squares(Odd, singly and doubly even)
See Magic_squares/Lua.
Nim
import bitops, sequtils, strutils
type Square = seq[seq[int]]
func magicSquareDoublyEven(n: int): Square =
## Build a magic square of doubly even order.
assert n >= 4 and (n and 3) == 0, "base must be a positive multiple of 4."
result = newSeqWith(n, newSeq[int](n))
const bits = 0b1001_0110_0110_1001 # Pattern of count-up vs count-down zones.
let size = n * n
let mult = n div 4 # How many multiples of 4.
var i = 0
for r in 0..<n:
for c in 0..<n:
let bitPos = c div mult + r div mult * 4
result[r][c] = if bits.testBit(bitPos): i + 1 else: size - i
inc i
func `$`(square: Square): string =
## Return the string representation of a magic square.
let length = len($(square.len * square.len))
for row in square:
result.add row.mapIt(($it).align(length)).join(" ") & '\n'
when isMainModule:
let n = 8
echo magicSquareDoublyEven(n)
echo "Magic constant = ", n * (n * n + 1) div 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
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
See Magic squares/Perl for a general magic square generator.
Phix
with javascript_semantics constant t = {{1,1,0,0}, {1,1,0,0}, {0,0,1,1}, {0,0,1,1}} function magic_square(integer n) if n<4 or mod(n,4)!=0 then return false end if sequence square = repeat(repeat(0,n),n) integer i = 0 for r=1 to n do for c=1 to n do square[r,c] = iff(t[mod(r,4)+1,mod(c,4)+1]?i+1:n*n-i) i += 1 end for end for return square end function procedure check(sequence sq) integer n = length(sq) integer magic = n*(n*n+1)/2 integer bd = 0, fd = 0 for i=1 to length(sq) do if sum(sq[i])!=magic then ?9/0 end if if sum(columnize(sq,i))!=magic then ?9/0 end if bd += sq[i,i] fd += sq[n-i+1,n-i+1] end for if bd!=magic or fd!=magic then ?9/0 end if end procedure --for i=4 to 16 by 4 do for i=8 to 8 by 4 do sequence square = magic_square(i) printf(1,"maqic square of order %d, sum: %d\n", {i,sum(square[i])}) string fmt = sprintf("%%%dd",length(sprintf("%d",i*i))) pp(square,{pp_Nest,1,pp_IntFmt,fmt,pp_StrFmt,3,pp_IntCh,false,pp_Pause,0}) check(square) end for
- Output:
maqic square of order 8, sum: 260
{{ 1,63,62, 4, 5,59,58, 8},
{56,10,11,53,52,14,15,49},
{48,18,19,45,44,22,23,41},
{25,39,38,28,29,35,34,32},
{33,31,30,36,37,27,26,40},
{24,42,43,21,20,46,47,17},
{16,50,51,13,12,54,55, 9},
{57, 7, 6,60,61, 3, 2,64}}
PureBasic
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 64Input [4,8,12..n] (0=Exit)
>
Python
Procedural
def MagicSquareDoublyEven(order):
sq = [range(1+n*order,order + (n*order)+1) for n in range(order) ]
n1 = order/4
for r in range(n1):
r1 = sq[r][n1:-n1]
r2 = sq[order -r - 1][n1:-n1]
r1.reverse()
r2.reverse()
sq[r][n1:-n1] = r2
sq[order -r - 1][n1:-n1] = r1
for r in range(n1, order-n1):
r1 = sq[r][:n1]
r2 = sq[order -r - 1][order-n1:]
r1.reverse()
r2.reverse()
sq[r][:n1] = r2
sq[order -r - 1][order-n1:] = r1
return sq
def printsq(s):
n = len(s)
bl = len(str(n**2))+1
for i in range(n):
print ''.join( [ ("%"+str(bl)+"s")%(str(x)) for x in s[i]] )
print "\nMagic constant = %d"%sum(s[0])
printsq(MagicSquareDoublyEven(8))
- 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 64Magic constant = 260
>>>
Composition of pure functions
Generating test results and a magic square in the form of a wiki table:
'''Magic squares of doubly even order'''
from itertools import chain, repeat
from functools import reduce
from math import log
# doublyEvenMagicSquare :: Int -> [[Int]]
def doublyEvenMagicSquare(n):
'''Magic square of order n'''
# magic :: Int -> [Bool]
def magic(n):
'''Truth-table series'''
if 0 < n:
xs = magic(n - 1)
return xs + [not x for x in xs]
else:
return [True]
sqr = n * n
power = log(sqr, 2)
scale = replicate(n / 4)
return chunksOf(n)([
succ(i) if bln else sqr - i for i, bln in
enumerate(magic(power) if isInteger(power) else (
flatten(scale(
map(scale, chunksOf(4)(magic(4)))
))
))
])
# TEST ----------------------------------------------------
# main :: IO()
def main():
'''Tests'''
order = 8
magicSquare = doublyEvenMagicSquare(order)
print(
'Row sums: ',
[sum(xs) for xs in magicSquare],
'\nCol sums:',
[sum(xs) for xs in transpose(magicSquare)],
'\n1st diagonal sum:',
sum(magicSquare[i][i] for i in range(0, order)),
'\n2nd diagonal sum:',
sum(magicSquare[i][(order - 1) - i] for i in range(0, order)),
'\n'
)
print(wikiTable({
'class': 'wikitable',
'style': cssFromDict({
'text-align': 'center',
'color': '#605B4B',
'border': '2px solid silver'
}),
'colwidth': '3em'
})(magicSquare))
# GENERIC -------------------------------------------------
# chunksOf :: Int -> [a] -> [[a]]
def chunksOf(n):
'''A series of lists of length n,
subdividing the contents of xs.
Where the length of xs is not evenly divible
the final list will be shorter than n.'''
return lambda xs: reduce(
lambda a, i: a + [xs[i:n + i]],
range(0, len(xs), n), []
) if 0 < n else []
# concatMap :: (a -> [b]) -> [a] -> [b]
def concatMap(f):
'''Concatenated list over which a function has been mapped.
The list monad can be derived by using a function f which
wraps its output a in list
(using an empty list to represent computational failure).'''
return lambda xs: list(
chain.from_iterable(
map(f, xs)
)
)
# cssFromDict :: Dict -> String
def cssFromDict(dct):
'''CSS string serialized from key values in a Dictionary.'''
return reduce(
lambda a, k: a + k + ':' + dct[k] + '; ', dct.keys(), ''
)
# flatten :: NestedList a -> [a]
def flatten(x):
'''A list of atoms resulting from fully flattening
an arbitrarily nested list.'''
return concatMap(flatten)(x) if isinstance(x, list) else [x]
# isInteger :: Num -> Bool
def isInteger(n):
'''Divisible by one without remainder ?'''
return 0 == (n - int(n))
# replicate :: Int -> a -> [a]
def replicate(n):
'''A list of length n in which every element
has the value x.'''
return lambda x: list(repeat(x, n))
# succ :: Enum a => a -> a
def succ(x):
'''The successor of a value. For numeric types, (1 +).'''
return 1 + x if isinstance(x, int) else (
chr(1 + ord(x))
)
# transpose :: Matrix a -> Matrix a
def transpose(m):
'''The rows and columns of the argument transposed.
(The matrix containers and rows can be lists or tuples).'''
if m:
inner = type(m[0])
z = zip(*m)
return (type(m))(
map(inner, z) if tuple != inner else z
)
else:
return m
# wikiTable :: Dict -> [[a]] -> String
def wikiTable(opts):
'''List of lists rendered as a wiki table string.'''
def colWidth():
return 'width:' + opts['colwidth'] + '; ' if (
'colwidth' in opts
) else ''
def cellStyle():
return opts['cell'] if 'cell' in opts else ''
return lambda rows: '{| ' + reduce(
lambda a, k: (
a + k + '="' + opts[k] + '" ' if k in opts else a
),
['class', 'style'],
''
) + '\n' + '\n|-\n'.join(
'\n'.join(
('|' if (0 != i and ('cell' not in opts)) else (
'|style="' + colWidth() + cellStyle() + '"|'
)) + (
str(x) or ' '
) for x in row
) for i, row in enumerate(rows)
) + '\n|}\n\n'
if __name__ == '__main__':
main()
- Output:
Row sums: [260, 260, 260, 260, 260, 260, 260, 260] Col sums: [260, 260, 260, 260, 260, 260, 260, 260] 1st diagonal sum: 260 2nd diagonal sum: 260
| 1 | 63 | 62 | 4 | 60 | 6 | 7 | 57 |
| 56 | 10 | 11 | 53 | 13 | 51 | 50 | 16 |
| 48 | 18 | 19 | 45 | 21 | 43 | 42 | 24 |
| 25 | 39 | 38 | 28 | 36 | 30 | 31 | 33 |
| 32 | 34 | 35 | 29 | 37 | 27 | 26 | 40 |
| 41 | 23 | 22 | 44 | 20 | 46 | 47 | 17 |
| 49 | 15 | 14 | 52 | 12 | 54 | 55 | 9 |
| 8 | 58 | 59 | 5 | 61 | 3 | 2 | 64 |
R
Translation of the magic square code example from Jama, which is released to the public domain. This includes all three cases.
magic <- function(n) {
if (n %% 2 == 1) {
p <- (n + 1) %/% 2 - 2
ii <- seq(n)
outer(ii, ii, function(i, j) n * ((i + j + p) %% n) + (i + 2 * (j - 1)) %% n + 1)
} else if (n %% 4 == 0) {
p <- n * (n + 1) + 1
ii <- seq(n)
outer(ii, ii, function(i, j) ifelse((i %/% 2 - j %/% 2) %% 2 == 0, p - n * i - j, n * (i - 1) + j))
} else {
p <- n %/% 2
q <- p * p
k <- (n - 2) %/% 4 + 1
a <- Recall(p)
a <- rbind(cbind(a, a + 2 * q), cbind(a + 3 * q, a + q))
ii <- seq(p)
jj <- c(seq(k - 1), seq(length.out=k - 2, to=n))
m <- a[ii, jj]; a[ii, jj] <- a[ii + p, jj]; a[ii + p, jj] <- m
jj <- c(1, k)
m <- a[k, jj]; a[k, jj] <- a[k + p, jj]; a[k + p, jj] <- m
a
}
}
Example
> magic(8)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
[1,] 64 2 3 61 60 6 7 57
[2,] 9 55 54 12 13 51 50 16
[3,] 17 47 46 20 21 43 42 24
[4,] 40 26 27 37 36 30 31 33
[5,] 32 34 35 29 28 38 39 25
[6,] 41 23 22 44 45 19 18 48
[7,] 49 15 14 52 53 11 10 56
[8,] 8 58 59 5 4 62 63 1
Raku
(formerly Perl 6)
See Magic squares/Raku 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
REXX
"Marked" numbers (via the diag subroutine) indicate that those (sequentially generated) numbers don't get
swapped (and thusly, stay in place in the magic square).
/*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) /*negate*/
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
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
Rust
use std::env;
fn main() {
let n: usize = match env::args()
.nth(1)
.and_then(|arg| arg.parse().ok())
.ok_or("Please specify the size of the magic square, as a positive multiple of 4.")
{
Ok(arg) if arg >= 4 && arg % 4 == 0 => arg,
Err(e) => panic!(e),
_ => panic!("Argument must be a positive multiple of 4."),
};
let mc = (n * n + 1) * n / 2;
println!("Magic constant: {}\n", mc);
let bits = 0b1001_0110_0110_1001u32;
let size = n * n;
let width = size.to_string().len() + 1;
let mult = n / 4;
let mut i = 0;
for r in 0..n {
for c in 0..n {
let bit_pos = c / mult + (r / mult) * 4;
print!(
"{e:>w$}",
e = if bits & (1 << bit_pos) != 0 {
i + 1
} else {
size - i
},
w = width
);
i += 1;
}
println!();
}
}
- Output:
Magic constant: 260 1 2 62 61 60 59 7 8 9 10 54 53 52 51 15 16 48 47 19 20 21 22 42 41 40 39 27 28 29 30 34 33 32 31 35 36 37 38 26 25 24 23 43 44 45 46 18 17 49 50 14 13 12 11 55 56 57 58 6 5 4 3 63 64
Scala
- Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).
object MagicSquareDoublyEven extends App {
private val n = 8
private def magicSquareDoublyEven(n: Int): Array[Array[Int]] = {
require(n >= 4 || n % 4 == 0, "Base must be a positive multiple of 4.")
// pattern of count-up vs count-down zones
val (bits, mult, result, size) = (38505, n / 4, Array.ofDim[Int](n, n), n * n)
var i = 0
for (r <- result.indices; c <- result(0).indices) {
def bitPos = c / mult + (r / mult) * 4
result(r)(c) = if ((bits & (1 << bitPos)) != 0) i + 1 else size - i
i += 1
}
result
}
magicSquareDoublyEven(n).foreach(row => println(row.map(x => f"$x%2s ").mkString))
println(f"---%nMagic constant: ${(n * n + 1) * n / 2}%d")
}
Sidef
func double_even_magic_square(n) {
assert(n%4 == 0, "Need multiple of four")
var (bsize, max) = (n/4, n*n)
var pre_pat = [true, false, false, true,
false, true, true, false]
pre_pat += pre_pat.flip
var pattern = (pre_pat.map{|b| bsize.of(b)... } * bsize)
pattern.map_kv{|k,v| v ? k+1 : max-k }.slices(n)
}
func format_matrix(a) {
var fmt = "%#{a.len**2 -> len}s"
a.map { .map { fmt % _ }.join(' ') }.join("\n")
}
say format_matrix(double_even_magic_square(8))
- 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
Stata
mata
function magic(n) {
if (mod(n,2)==1) {
p = (n+1)/2-2
a = J(n,n,.)
for (i=1; i<=n; i++) {
for (j=1; j<=n; j++) {
a[i,j] = n*mod(i+j+p,n)+mod(i+2*j-2,n)+1
}
}
} else if (mod(n,4)==0) {
p = n^2+n+1
a = J(n,n,.)
for (i=1; i<=n; i++) {
for (j=1; j<=n; j++) {
a[i,j] = mod(floor(i/2)-floor(j/2),2)==0 ? p-n*i-j : n*(i-1)+j
}
}
} else {
p = n/2
q = p*p
k = (n-2)/4+1
a = magic(p)
a = a,a:+2*q\a:+3*q,a:+q
i = 1..p
j = 1..k-1
if (k>2) j = j,n-k+3..n
m = a[i,j]; a[i,j] = a[i:+p,j]; a[i:+p,j] = m
j = 1,k
m = a[k,j]; a[k,j] = a[k:+p,j]; a[k:+p,j] = m
}
return(a)
}
end
. mata magic(8)
1 2 3 4 5 6 7 8
+-----------------------------------------+
1 | 64 2 3 61 60 6 7 57 |
2 | 9 55 54 12 13 51 50 16 |
3 | 17 47 46 20 21 43 42 24 |
4 | 40 26 27 37 36 30 31 33 |
5 | 32 34 35 29 28 38 39 25 |
6 | 41 23 22 44 45 19 18 48 |
7 | 49 15 14 52 53 11 10 56 |
8 | 8 58 59 5 4 62 63 1 |
+-----------------------------------------+
VBScript
' 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
Visual Basic .NET
Module MagicSquares
Function MagicSquareDoublyEven(n As Integer) As Integer(,)
If n < 4 OrElse n Mod 4 <> 0 Then
Throw New ArgumentException("base must be a positive multiple of 4")
End If
'pattern of count-up vs count-down zones
Dim bits = Convert.ToInt32("1001011001101001", 2)
Dim size = n * n
Dim mult As Integer = n / 4 ' how many multiples of 4
Dim result(n - 1, n - 1) As Integer
Dim i = 0
For r = 0 To n - 1
For c = 0 To n - 1
Dim bitPos As Integer = Math.Floor(c / mult) + Math.Floor(r / mult) * 4
Dim test = (bits And (1 << bitPos)) <> 0
If test Then
result(r, c) = i + 1
Else
result(r, c) = size - i
End If
i = i + 1
Next
Console.WriteLine()
Next
Return result
End Function
Sub Main()
Dim n = 8
Dim result = MagicSquareDoublyEven(n)
For i = 0 To result.GetLength(0) - 1
For j = 0 To result.GetLength(1) - 1
Console.Write("{0,2} ", result(i, j))
Next
Console.WriteLine()
Next
Console.WriteLine()
Console.WriteLine("Magic constant: {0} ", (n * n + 1) * n / 2)
Console.ReadLine()
End Sub
End Module
- 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
Wren
import "./fmt" for Conv, Fmt
var magicSquareDoublyEven = Fn.new { |n|
if (n < 4 || n%4 != 0) Fiber.abort("Base must be a positive multiple of 4")
// pattern of count-up vs count-down zones
var bits = Conv.atoi("1001011001101001", 2)
var size = n * n
var mult = (n/4).floor // how many multiples of 4
var result = List.filled(n, null)
for (i in 0...n) result[i] = List.filled(n, 0)
var i = 0
for (r in 0...n) {
for (c in 0...n) {
var bitPos = (c/mult).floor + (r/mult).floor * 4
result[r][c] = ((bits & (1<<bitPos)) != 0) ? i + 1 : size - i
i = i + 1
}
}
return result
}
var n = 8
for (ia in magicSquareDoublyEven.call(n)) {
for (i in ia) Fmt.write("$2d ", i)
System.print()
}
System.print("\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
XPL0
\Magic squares of doubly even order - 11/30/2024;
integer Pattern(1+4,1+4);
integer N, R, C, S, M, I, B, T;
begin
N:=8;
for R:=1 to 4 do
for C:=1 to 4 do
Pattern(R,C):=if
((C=1 or C=4) and (R=1 or R=4)) or
((C=2 or C=3) and (R=2 or R=3)) then 1 else 0;
S:=N*N; M:=N/4;
Text(1,"magic square -- n = "); IntOut(1,N); Text(1,"^m^j");
I:=0;
for R:=1 to N do begin
for C:=1 to N do begin
B:=Pattern((R-1)/M+1, (C-1)/M+1);
if B=1 then T:=I+1 else T:=S-I;
if T < 10 then Text(1," ");
Text(1," ");
IntOut(1,T);
I:=I+1
end;
Text(1,"^m^j")
end;
Text(1,"magic constant = "); IntOut(1,(S+1)*N/2)
end- Output:
magic square -- n = 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
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
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