Magic squares of singly even order
You are encouraged to solve this task according to the task description, using any language you may know.
A magic square is an NxN square matrix whose numbers consist of consecutive numbers arranged so that the sum of each row and column, and both diagonals are equal to the same sum (which is called the magic number or magic constant).
A magic square of singly even order has a size that is a multiple of 4, plus 2 (e.g. 6, 10, 14). This means that the subsquares have an odd size, which plays a role in the construction.
- Task
Create a magic square of 6 x 6.
- Related tasks
- See also
C++
<lang cpp>
- include <iostream>
- include <sstream>
- include <iomanip>
using namespace std;
class magicSqr { public:
magicSqr() { sqr = 0; }
~magicSqr() { if( sqr ) delete [] sqr; }
void create( int d ) {
if( sqr ) delete [] sqr;
if( d & 1 ) d++;
while( d % 4 == 0 ) { d += 2; }
sz = d;
sqr = new int[sz * sz];
memset( sqr, 0, sz * sz * sizeof( int ) );
fillSqr();
}
void display() {
cout << "Singly Even Magic Square: " << sz << " x " << sz << "\n";
cout << "It's Magic Sum is: " << magicNumber() << "\n\n";
ostringstream cvr; cvr << sz * sz;
int l = cvr.str().size();
for( int y = 0; y < sz; y++ ) {
int yy = y * sz;
for( int x = 0; x < sz; x++ ) {
cout << setw( l + 2 ) << sqr[yy + x];
}
cout << "\n";
}
cout << "\n\n";
}
private:
void siamese( int from, int to ) {
int oneSide = to - from, curCol = oneSide / 2, curRow = 0, count = oneSide * oneSide, s = 1;
while( count > 0 ) {
bool done = false;
while ( false == done ) {
if( curCol >= oneSide ) curCol = 0;
if( curRow < 0 ) curRow = oneSide - 1;
done = true;
if( sqr[curCol + sz * curRow] != 0 ) {
curCol -= 1; curRow += 2;
if( curCol < 0 ) curCol = oneSide - 1;
if( curRow >= oneSide ) curRow -= oneSide;
done = false;
}
}
sqr[curCol + sz * curRow] = s;
s++; count--; curCol++; curRow--;
}
}
void fillSqr() {
int n = sz / 2, ns = n * sz, size = sz * sz, add1 = size / 2, add3 = size / 4, add2 = 3 * add3;
siamese( 0, n );
for( int r = 0; r < n; r++ ) {
int row = r * sz;
for( int c = n; c < sz; c++ ) {
int m = sqr[c - n + row];
sqr[c + row] = m + add1;
sqr[c + row + ns] = m + add3;
sqr[c - n + row + ns] = m + add2;
}
}
int lc = ( sz - 2 ) / 4, co = sz - ( lc - 1 );
for( int r = 0; r < n; r++ ) {
int row = r * sz;
for( int c = co; c < sz; c++ ) {
sqr[c + row] -= add3;
sqr[c + row + ns] += add3;
}
}
for( int r = 0; r < n; r++ ) {
int row = r * sz;
for( int c = 0; c < lc; c++ ) {
int cc = c;
if( r == lc ) cc++;
sqr[cc + row] += add2;
sqr[cc + row + ns] -= add2;
}
}
}
int magicNumber() { return sz * ( ( sz * sz ) + 1 ) / 2; }
void inc( int& a ) { if( ++a == sz ) a = 0; }
void dec( int& a ) { if( --a < 0 ) a = sz - 1; }
bool checkPos( int x, int y ) { return( isInside( x ) && isInside( y ) && !sqr[sz * y + x] ); }
bool isInside( int s ) { return ( s < sz && s > -1 ); }
int* sqr;
int sz;
}; int main( int argc, char* argv[] ) {
magicSqr s; s.create( 6 ); s.display(); return 0;
} </lang>
- Output:
Singly Even Magic Square: 6 x 6 It's Magic Sum is: 111 35 1 6 26 19 24 3 32 7 21 23 25 31 9 2 22 27 20 8 28 33 17 10 15 30 5 34 12 14 16 4 36 29 13 18 11
D
<lang d> import std.exception; import std.stdio;
void main() {
int n = 6;
foreach (row; magicSquareSinglyEven(n)) {
foreach (x; row) {
writef("%2s ", x);
}
writeln();
}
writeln("\nMagic constant: ", (n * n + 1) * n / 2);
}
int[][] magicSquareOdd(const int n) {
enforce(n >= 3 && n % 2 != 0, "Base must be odd and >2");
int value = 0; int gridSize = n * n; int c = n / 2; int r = 0;
int[][] result = new int[][](n, n);
while(++value <= gridSize) {
result[r][c] = value;
if (r == 0) {
if (c == n - 1) {
r++;
} else {
r = n - 1;
c++;
}
} else if (c == n - 1) {
r--;
c = 0;
} else if (result[r - 1][c + 1] == 0) {
r--;
c++;
} else {
r++;
}
}
return result;
}
int[][] magicSquareSinglyEven(const int n) {
enforce(n >= 6 && (n - 2) % 4 == 0, "Base must be a positive multiple of four plus 2");
int size = n * n; int halfN = n / 2; int subSquareSize = size / 4;
int[][] subSquare = magicSquareOdd(halfN); int[] quadrantFactors = [0, 2, 3, 1]; int[][] result = new int[][](n, n);
for (int r = 0; r < n; r++) {
for (int c = 0; c < n; c++) {
int quadrant = (r / halfN) * 2 + (c / halfN);
result[r][c] = subSquare[r % halfN][c % halfN];
result[r][c] += quadrantFactors[quadrant] * subSquareSize;
}
}
int nColsLeft = halfN / 2; int nColsRight = nColsLeft - 1;
for (int r = 0; r < halfN; r++) {
for (int c = 0; c < n; c++) {
if (c < nColsLeft || c >= n - nColsRight
|| (c == nColsLeft && r == nColsLeft)) {
if (c == 0 && r == nColsLeft) {
continue;
}
int tmp = result[r][c];
result[r][c] = result[r + halfN][c];
result[r + halfN][c] = tmp;
}
}
}
return result;
} </lang>
Elixir
wp:Conway's LUX method for magic squares: <lang elixir>defmodule Magic_square do
@lux %{ L: [4, 1, 2, 3], U: [1, 4, 2, 3], X: [1, 4, 3, 2] }
def singly_even(n) when rem(n-2,4)!=0, do: raise ArgumentError, "must be even, but not divisible by 4."
def singly_even(2), do: raise ArgumentError, "2x2 magic square not possible."
def singly_even(n) do
n2 = div(n, 2)
oms = odd_magic_square(n2)
mat = make_lux_matrix(n2)
square = synthesis(n2, oms, mat)
IO.puts to_string(n, square)
square
end
defp odd_magic_square(m) do # zero beginning, it is 4 multiples.
for i <- 0..m-1, j <- 0..m-1, into: %{},
do: {{i,j}, (m*(rem(i+j+1+div(m,2),m)) + rem(i+2*j-5+2*m, m)) * 4}
end
defp make_lux_matrix(m) do
center = div(m, 2)
lux = List.duplicate(:L, center+1) ++ [:U] ++ List.duplicate(:X, m-center-2)
(for {x,i} <- Enum.with_index(lux), j <- 0..m-1, into: %{}, do: {{i,j}, x})
|> Map.put({center, center}, :U)
|> Map.put({center+1, center}, :L)
end
defp synthesis(m, oms, mat) do
range = 0..m-1
Enum.reduce(range, [], fn i,acc ->
{row0, row1} = Enum.reduce(range, {[],[]}, fn j,{r0,r1} ->
x = oms[{i,j}]
[lux0, lux1, lux2, lux3] = @lux[mat[{i,j}]]
{[x+lux0, x+lux1 | r0], [x+lux2, x+lux3 | r1]}
end)
[row0, row1 | acc]
end)
end
defp to_string(n, square) do
format = String.duplicate("~#{length(to_char_list(n*n))}w ", n) <> "\n"
Enum.map_join(square, fn row ->
:io_lib.format(format, row)
end)
end
end
Magic_square.singly_even(6)</lang>
- Output:
5 8 36 33 13 16 6 7 34 35 14 15 28 25 17 20 12 9 26 27 18 19 10 11 24 21 4 1 32 29 22 23 2 3 30 31
FreeBASIC
<lang freebasic>' version 18-03-2016 ' compile with: fbc -s console ' singly even magic square 6, 10, 14, 18...
Sub Err_msg(msg As String)
Print msg Beep : Sleep 5000, 1 : Exit Sub
End Sub
Sub se_magicsq(n As UInteger, filename As String = "")
' filename <> "" then save square in a file ' filename can contain directory name ' if filename exist it will be overwriten, no error checking
If n < 6 Then
Err_msg( "Error: n is to small")
Exit Sub
End If
If ((n -2) Mod 4) <> 0 Then
Err_msg "Error: not possible to make singly" + _
" even magic square size " + Str(n)
Exit Sub
End If
Dim As UInteger sq(1 To n, 1 To n) Dim As UInteger magic_sum = n * (n ^ 2 +1) \ 2 Dim As UInteger sq_d_2 = n \ 2, q2 = sq_d_2 ^ 2 Dim As UInteger l = (n -2) \ 4 Dim As UInteger x = sq_d_2 \ 2 + 1, y = 1, nr = 1 Dim As String frmt = String(Len(Str(n * n)) +1, "#")
' fill pattern A C
' D B
' main loop for creating magic square in section A
' the value for B,C and D is derived from A
' uses the FreeBASIC odd order magic square routine
Do
If sq(x, y) = 0 Then
sq(x , y ) = nr ' A
sq(x + sq_d_2, y + sq_d_2) = nr + q2 ' B
sq(x + sq_d_2, y ) = nr + q2 * 2 ' C
sq(x , y + sq_d_2) = nr + q2 * 3 ' D
If nr Mod sq_d_2 = 0 Then
y += 1
Else
x += 1 : y -= 1
End If
nr += 1
End If
If x > sq_d_2 Then
x = 1
Do While sq(x,y) <> 0
x += 1
Loop
End If
If y < 1 Then
y = sq_d_2
Do While sq(x,y) <> 0
y -= 1
Loop
End If
Loop Until nr > q2
' swap left side
For y = 1 To sq_d_2
For x = 1 To l
Swap sq(x, y), sq(x,y + sq_d_2)
Next
Next
' make indent
y = (sq_d_2 \ 2) +1
Swap sq(1, y), sq(1, y + sq_d_2) ' was swapped, restore to orignal value
Swap sq(l +1, y), sq(l +1, y + sq_d_2)
' swap right side
For y = 1 To sq_d_2
For x = n - l +2 To n
Swap sq(x, y), sq(x,y + sq_d_2)
Next
Next
' check columms and rows
For y = 1 To n
nr = 0 : l = 0
For x = 1 To n
nr += sq(x,y)
l += sq(y,x)
Next
If nr <> magic_sum Or l <> magic_sum Then
Err_msg "Error: value <> magic_sum"
Exit Sub
End If
Next
' check diagonals
nr = 0 : l = 0
For x = 1 To n
nr += sq(x, x)
l += sq(n - x +1, n - x +1)
Next
If nr <> magic_sum Or l <> magic_sum Then
Err_msg "Error: value <> magic_sum"
Exit Sub
End If
' printing square's on screen bigger when
' n > 19 results in a wrapping of the line
Print "Single even magic square size: "; n; "*"; n
Print "The magic sum = "; magic_sum
Print
For y = 1 To n
For x = 1 To n
Print Using frmt; sq(x, y);
Next
Print
Next
' output magic square to a file with the name provided
If filename <> "" Then
nr = FreeFile
Open filename For Output As #nr
Print #nr, "Single even magic square size: "; n; "*"; n
Print #nr, "The magic sum = "; magic_sum
Print #nr,
For y = 1 To n
For x = 1 To n
Print #nr, Using frmt; sq(x,y);
Next
Print #nr,
Next
Close #nr
End If
End Sub
' ------=< MAIN >=------
se_magicsq(6, "magicse6.txt") : Print
' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
Single even magic square size: 6*6 The magic sum = 111 35 1 6 26 19 24 3 32 7 21 23 25 31 9 2 22 27 20 8 28 33 17 10 15 30 5 34 12 14 16 4 36 29 13 18 11
Java
<lang java>public class MagicSquareSinglyEven {
public static void main(String[] args) {
int n = 6;
for (int[] row : magicSquareSinglyEven(n)) {
for (int x : row)
System.out.printf("%2s ", x);
System.out.println();
}
System.out.printf("\nMagic constant: %d ", (n * n + 1) * n / 2);
}
public static int[][] magicSquareOdd(final int n) {
if (n < 3 || n % 2 == 0)
throw new IllegalArgumentException("base must be odd and > 2");
int value = 0;
int gridSize = n * n;
int c = n / 2, r = 0;
int[][] result = new int[n][n];
while (++value <= gridSize) {
result[r][c] = value;
if (r == 0) {
if (c == n - 1) {
r++;
} else {
r = n - 1;
c++;
}
} else if (c == n - 1) {
r--;
c = 0;
} else if (result[r - 1][c + 1] == 0) {
r--;
c++;
} else {
r++;
}
}
return result;
}
static int[][] magicSquareSinglyEven(final int n) {
if (n < 6 || (n - 2) % 4 != 0)
throw new IllegalArgumentException("base must be a positive "
+ "multiple of 4 plus 2");
int size = n * n;
int halfN = n / 2;
int subSquareSize = size / 4;
int[][] subSquare = magicSquareOdd(halfN);
int[] quadrantFactors = {0, 2, 3, 1};
int[][] result = new int[n][n];
for (int r = 0; r < n; r++) {
for (int c = 0; c < n; c++) {
int quadrant = (r / halfN) * 2 + (c / halfN);
result[r][c] = subSquare[r % halfN][c % halfN];
result[r][c] += quadrantFactors[quadrant] * subSquareSize;
}
}
int nColsLeft = halfN / 2;
int nColsRight = nColsLeft - 1;
for (int r = 0; r < halfN; r++)
for (int c = 0; c < n; c++) {
if (c < nColsLeft || c >= n - nColsRight
|| (c == nColsLeft && r == nColsLeft)) {
if (c == 0 && r == nColsLeft)
continue;
int tmp = result[r][c];
result[r][c] = result[r + halfN][c];
result[r + halfN][c] = tmp;
}
}
return result; }
}</lang>
35 1 6 26 19 24 3 32 7 21 23 25 31 9 2 22 27 20 8 28 33 17 10 15 30 5 34 12 14 16 4 36 29 13 18 11 Magic constant: 111
Perl 6
See Magic squares/Perl 6 for a general magic square generator.
- Output:
With a parameter of 6:
35 1 6 26 19 24 3 32 7 21 23 25 31 9 2 22 27 20 8 28 33 17 10 15 30 5 34 12 14 16 4 36 29 13 18 11 The magic number is 111
With a parameter of 10:
92 99 1 8 15 67 74 51 58 40 98 80 7 14 16 73 55 57 64 41 4 81 88 20 22 54 56 63 70 47 85 87 19 21 3 60 62 69 71 28 86 93 25 2 9 61 68 75 52 34 17 24 76 83 90 42 49 26 33 65 23 5 82 89 91 48 30 32 39 66 79 6 13 95 97 29 31 38 45 72 10 12 94 96 78 35 37 44 46 53 11 18 100 77 84 36 43 50 27 59 The magic number is 505
Ruby
<lang ruby>def odd_magic_square(n)
n.times.map{|i| n.times.map{|j| n*((i+j+1+n/2)%n) + ((i+2*j-5)%n) + 1} }
end
def single_even_magic_square(n)
raise ArgumentError, "must be even, but not divisible by 4." unless (n-2) % 4 == 0 raise ArgumentError, "2x2 magic square not possible." if n == 2
order = (n-2)/4
odd_square = odd_magic_square(n/2)
to_add = (0..3).map{|f| f*n*n/4}
quarts = to_add.map{|f| odd_square.dup.map{|row|row.map{|el| el+f}} }
sq = []
quarts[0].zip(quarts[2]){|d1,d2| sq << [d1,d2].flatten}
quarts[3].zip(quarts[1]){|d1,d2| sq << [d1,d2].flatten}
sq = sq.transpose
order.times{|i| sq[i].rotate!(n/2)}
swap(sq[0][order], sq[0][-order-1])
swap(sq[order][order], sq[order][-order-1])
(order-1).times{|i| sq[-(i+1)].rotate!(n/2)}
randomize(sq)
end
def swap(a,b)
a,b = b,a
end
def randomize(square)
square.shuffle.transpose.shuffle
end
def to_string(square)
n = square.size
fmt = "%#{(n*n).to_s.size + 1}d" * n
square.inject(""){|str,row| str << fmt % row << "\n"}
end
puts to_string(single_even_magic_square(6))</lang>
- Output:
23 7 5 21 30 25 18 29 36 13 4 11 14 34 32 12 3 16 19 6 1 26 35 24 27 2 9 22 31 20 10 33 28 17 8 15
LUX method
wp:Conway's LUX method for magic squares <lang ruby>class Magic_square
attr_reader :square
LUX = { L: [[4, 1], [2, 3]], U: [[1, 4], [2, 3]], X: [[1, 4], [3, 2]] }
def initialize(n)
raise ArgumentError, "must be even, but not divisible by 4." unless (n-2) % 4 == 0
raise ArgumentError, "2x2 magic square not possible." if n == 2
@n = n
oms = odd_magic_square(n/2)
mat = make_lux_matrix(n/2)
@square = synthesize(oms, mat)
puts to_s
end
def odd_magic_square(n) # zero beginning, it is 4 multiples.
n.times.map{|i| n.times.map{|j| (n*((i+j+1+n/2)%n) + ((i+2*j-5)%n)) * 4} }
end
def make_lux_matrix(n)
center = n / 2
lux = [*[:L]*(center+1), :U, *[:X]*(n-center-2)]
matrix = lux.map{|x| Array.new(n, x)}
matrix[center][center] = :U
matrix[center+1][center] = :L
matrix
end
def synthesize(oms, mat)
range = 0...@n/2
range.inject([]) do |matrix,i|
row = [[], []]
range.each do |j|
x = oms[i][j]
LUX[mat[i][j]].each_with_index{|lux,k| row[k] << lux.map{|y| x+y}}
end
matrix << row[0].flatten << row[1].flatten
end
end
def to_s
format = "%#{(@n*@n).to_s.size}d " * @n + "\n"
@square.map{|row| format % row}.join
end
end
sq = Magic_square.new(6).square</lang>
- Output:
32 29 4 1 24 21 30 31 2 3 22 23 12 9 17 20 28 25 10 11 18 19 26 27 13 16 36 33 5 8 14 15 34 35 6 7
zkl
<lang zkl>class MagicSquareSinglyEven{
fcn init(n){ var result=magicSquareSinglyEven(n) }
fcn toString{
sink,n:=Sink(String),result.len(); // num collumns
fmt:="%2s ";
foreach row in (result)
{ sink.write(row.apply('wrap(n){ fmt.fmt(n) }).concat(),"\n") }
sink.write("\nMagic constant: %d".fmt((n*n + 1)*n/2));
sink.close();
}
fcn magicSquareOdd(n){
if (n<3 or n%2==0) throw(Exception.ValueError("base must be odd and > 2"));
value,gridSize,c,r:=0, n*n, n/2, 0;
result:=n.pump(List(),n.pump(List(),0).copy); // array[n,n] of zero
while((value+=1)<=gridSize){
result[r][c]=value; if(r==0){ if(c==n-1) r+=1;
else{ r=n-1; c+=1; }
} else if(c==n-1){ r-=1; c=0; } else if(result[r-1][c+1]==0){ r-=1; c+=1; } else r+=1;
}
result;
}
fcn magicSquareSinglyEven(n){
if (n<6 or (n-2)%4!=0)
throw(Exception.ValueError("base must be a positive multiple of 4 +2"));
size,halfN,subSquareSize:=n*n, n/2, size/4;
subSquare:=magicSquareOdd(halfN);
quadrantFactors:=T(0, 2, 3, 1);
result:=n.pump(List(),n.pump(List(),0).copy); // array[n,n] of zero
foreach r,c in (n,n){
quadrant:=(r/halfN)*2 + (c/halfN);
result[r][c]=subSquare[r%halfN][c%halfN]; result[r][c]+=quadrantFactors[quadrant]*subSquareSize;
}
nColsLeft,nColsRight:=halfN/2, nColsLeft-1;
foreach r,c in (halfN,n){
if ( c<nColsLeft or c>=(n-nColsRight) or
(c==nColsLeft and r==nColsLeft) ){
if(c==0 and r==nColsLeft) continue; tmp:=result[r][c]; result[r][c]=result[r+halfN][c]; result[r+halfN][c]=tmp; }
}
result
}
}</lang> <lang zkl>MagicSquareSinglyEven(6).println();</lang>
- Output:
35 1 6 26 19 24 3 32 7 21 23 25 31 9 2 22 27 20 8 28 33 17 10 15 30 5 34 12 14 16 4 36 29 13 18 11 Magic constant: 111