# Peaceful chess queen armies

Peaceful chess queen armies
You are encouraged to solve this task according to the task description, using any language you may know.

In chess, a queen attacks positions from where it is, in straight lines up-down and left-right as well as on both its diagonals. It attacks only pieces not of its own colour.

 ⇖ ⇑ ⇗ ⇐ ⇐ ♛ ⇒ ⇒ ⇙ ⇓ ⇘ ⇙ ⇓ ⇘ ⇓

The goal of Peaceful chess queen armies is to arrange `m` black queens and `m` white queens on an `n-by-n` square grid, (the board), so that no queen attacks another of a different colour.

1. Create a routine to represent two-colour queens on a 2-D board. (Alternating black/white background colours, Unicode chess pieces and other embellishments are not necessary, but may be used at your discretion).
2. Create a routine to generate at least one solution to placing `m` equal numbers of black and white queens on an `n` square board.
3. Display here results for the `m=4, n=5` case.

References

## 11l

Translation of: D
```T.enum Piece
EMPTY
BLACK
WHITE

F isAttacking(queen, pos)
R queen.x == pos.x
| queen.y == pos.y
| abs(queen.x - pos.x) == abs(queen.y - pos.y)

F place(m, n, &pBlackQueens, &pWhiteQueens)
I m == 0
R 1B

V placingBlack = 1B
L(i) 0 .< n
L(j) 0 .< n
V pos = (i, j)
L(queen) pBlackQueens
I queen == pos | (!placingBlack & isAttacking(queen, pos))
L.break
L.was_no_break
L(queen) pWhiteQueens
I queen == pos | (placingBlack & isAttacking(queen, pos))
L.break
L.was_no_break
I placingBlack
pBlackQueens [+]= pos
placingBlack = 0B
E
pWhiteQueens [+]= pos
I place(m - 1, n, &pBlackQueens, &pWhiteQueens)
R 1B
pBlackQueens.pop()
pWhiteQueens.pop()
placingBlack = 1B

I !placingBlack
pBlackQueens.pop()
R 0B

F printBoard(n, blackQueens, whiteQueens)
V board = [Piece.EMPTY] * (n * n)

L(queen) blackQueens
board[queen.x * n + queen.y] = Piece.BLACK

L(queen) whiteQueens
board[queen.x * n + queen.y] = Piece.WHITE

L(b) board
V i = L.index
I i != 0 & i % n == 0
print()
I b == BLACK
print(‘B ’, end' ‘’)
E I b == WHITE
print(‘W ’, end' ‘’)
E
V j = i I/ n
V k = i - j * n
I j % 2 == k % 2
print(‘x ’, end' ‘’)
E
print(‘o ’, end' ‘’)
print("\n")

V nms = [
(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6),
(7, 1), (7, 2), (7, 3), (7, 4), (7, 5), (7, 6), (7, 7)
]

L(nm) nms
print(‘#. black and #. white queens on a #. x #. board:’.format(nm[1], nm[1], nm[0], nm[0]))
[(Int, Int)] blackQueens, whiteQueens
I place(nm[1], nm[0], &blackQueens, &whiteQueens)
printBoard(nm[0], blackQueens, whiteQueens)
E
print("No solution exists.\n")```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B o x
o x W
x o x

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B o x o
o x W x
x o x o
o x o x

2 black and 2 white queens on a 4 x 4 board:
B o x o
o x W x
B o x o
o x W x

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B o x o x
o x W x o
x o x o x
o x o x o
x o x o x

2 black and 2 white queens on a 5 x 5 board:
B o x o B
o x W x o
x W x o x
o x o x o
x o x o x

3 black and 3 white queens on a 5 x 5 board:
B o x o B
o x W x o
x W x o x
o x o B o
x W x o x

4 black and 4 white queens on a 5 x 5 board:
x B x B x
o x o x B
W o W o x
o x o x B
W o W o x

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B o x o x o
o x W x o x
x o x o x o
o x o x o x
x o x o x o
o x o x o x

2 black and 2 white queens on a 6 x 6 board:
B o x o B o
o x W x o x
x W x o x o
o x o x o x
x o x o x o
o x o x o x

3 black and 3 white queens on a 6 x 6 board:
B o x o B B
o x W x o x
x W x o x o
o x o x o x
x o W o x o
o x o x o x

4 black and 4 white queens on a 6 x 6 board:
B o x o B B
o x W x o x
x W x o x o
o x o x o B
x o W W x o
o x o x o x

5 black and 5 white queens on a 6 x 6 board:
x B x o B o
o x o B o B
W o x o x o
W x W x o x
x o x o x B
W x W x o x

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B o x o x o x
o x W x o x o
x o x o x o x
o x o x o x o
x o x o x o x
o x o x o x o
x o x o x o x

2 black and 2 white queens on a 7 x 7 board:
B o x o B o x
o x W x o x W
x o x o x o x
o x o x o x o
x o x o x o x
o x o x o x o
x o x o x o x

3 black and 3 white queens on a 7 x 7 board:
B o x o B o x
o x W x o x W
B o x o x o x
o x W x o x o
x o x o x o x
o x o x o x o
x o x o x o x

4 black and 4 white queens on a 7 x 7 board:
B o x o B o x
o x W x o x W
B o x o B o x
o x W x o x W
x o x o x o x
o x o x o x o
x o x o x o x

5 black and 5 white queens on a 7 x 7 board:
B o x o B o x
o x W x o x W
B o x o B o x
o x W x o x W
B o x o x o x
o x W x o x o
x o x o x o x

6 black and 6 white queens on a 7 x 7 board:
B o x o B o x
o x W x o x W
B o x o B o x
o x W x o x W
B o x o B o x
o x W x o x W
x o x o x o x

7 black and 7 white queens on a 7 x 7 board:
x B x o x B x
o B o x B x o
x B x o x B x
o x o x B x o
W o W o x o W
o x o W o x o
W o W W x o x

```

## ATS

Translation of: Scheme

The program can print either all solutions or all solutions that are ‘inequivalent’, in the sense of https://oeis.org/A260680

The program also can stop after printing a specified number of solutions, although the default is to print all solutions.

(Commentary by the author: this program suffers similarly of slowness, in eliminating rotational equivalents, as does its Scheme ancestor. Some reasons: it uses backtracking and that is slow; it uses essentially the same inefficient storage format for solutions [one could for instance use integers], and it does not precompute rotational equivalents. However, it does satisfy the task requirements, and might be regarded as a good start. And it can solve the m=5, n=6 case in practical time on a fast machine. m=7, n=7 is a more annoying case.)

```(********************************************************************)

#include "share/atspre_define.hats"

#define NIL list_vt_nil ()
#define :: list_vt_cons

#ifndef NDEBUG #then
(* Safety is relatively unimportant in this program.
Therefore I have made it so you can put ‘-DATS NDEBUG=1’ on
your patscc command line, to skip some assertloc tests. *)
#define NDEBUG 0
#endif

(********************************************************************)

#define EMPTY 0
#define BLACK 1
#define WHITE ~1

stadef is_color (c : int) : bool = (~1 <= c && c <= 1)
stadef reverse_color (c : int) : int = ~c

typedef color_t (tk : tkind, c : int) =
[is_color c]
g1int (tk, c)
typedef color_t (tk : tkind) =
[c : int | is_color c]
g1int (tk, c)

fn {tk : tkind}
reverse_color {c : int | is_color c}
(c : g1int (tk, c)) :<>
[c_rev : int | is_color c_rev;
c_rev == reverse_color c]
g1int (tk, c_rev) =
(* This template is a fancy way to say ‘minus’. *)
~c

(********************************************************************)

(* Matrix indices will run from 0..n-1 rather than 1..n. *)

#define SIDE_MAX 16             (* The maximum side size. For
power of two. *)
#define BOARD_SIZE 256          (* The storage size for a board. *)

prval _ = prop_verify {SIDE_MAX * SIDE_MAX == BOARD_SIZE} ()

fn {tk : tkind}
row_index {k : int | 0 <= k; k < BOARD_SIZE}
(k : g1int (tk, k)) :<>
[i : int | 0 <= i; i < SIDE_MAX]
g1int (tk, i) =
(* Let the C compiler convert this to bitmasking. *)
g1int_nmod (k, g1i2i SIDE_MAX)

fn {tk : tkind}
column_index {k : int | 0 <= k; k < BOARD_SIZE}
(k : g1int (tk, k)) :<>
[i : int | 0 <= i; i < SIDE_MAX]
g1int (tk, i) =
(* Let the C compiler convert this to a shift. *)
k / g1i2i SIDE_MAX

fn {tk : tkind}
storage_index {i, j : int | 0 <= i; i < SIDE_MAX;
0 <= j; j < SIDE_MAX}
(i : g1int (tk, i),
j : g1int (tk, j)) :<>
[k : int | 0 <= k; k < BOARD_SIZE]
g1int (tk, k) =
(* Let the C compiler convert this to a shift and add. *)
i + (j * g1i2i SIDE_MAX)

(********************************************************************)

extern fn {tk_index : tkind}
test_equiv\$reindex_i
{i, j : int | 0 <= i; 0 <= j}
{n : int | 0 <= n; n <= SIDE_MAX;
i < n; j < n}
(i : g1int (tk_index, i),
j : g1int (tk_index, j),
n : g1int (tk_index, n)) :<>
[i1 : int | 0 <= i1; i1 < SIDE_MAX]
g1int (tk_index, i1)

extern fn {tk_index : tkind}
test_equiv\$reindex_j
{i, j : int | 0 <= i; 0 <= j}
{n : int | 0 <= n; n <= SIDE_MAX;
i < n; j < n}
(i : g1int (tk_index, i),
j : g1int (tk_index, j),
n : g1int (tk_index, n)) :<>
[j1 : int | 0 <= j1; j1 < SIDE_MAX]
g1int (tk_index, j1)

extern fn {tk_color : tkind}
test_equiv\$recolor
{c : int | is_color c}
(c : g1int (tk_color, c)) :<>
[c1 : int | is_color c1]
g1int (tk_color, c1)

fn {tk_index, tk_color : tkind}
test_equiv {n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
macdef reindex_i = test_equiv\$reindex_i<tk_index>
macdef reindex_j = test_equiv\$reindex_j<tk_index>
macdef recolor = test_equiv\$recolor<tk_color>

fun
loopj {j : int | ~1 <= j; j < n} .<j + 1>.
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n),
j : g1int (tk_index, j)) :
bool =
if j < g1i2i 0 then
true
else
let
fun loopi {i : int | ~1 <= i; i < n} .<i + 1>.
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n),
j : g1int (tk_index, j),
i : g1int (tk_index, i)) :
bool =
if i < g1i2i 0 then
true
else
let
val ka = storage_index<tk_index> (i, j)
val color_a = a[ka]

val i1 = test_equiv\$reindex_i<tk_index> (i, j, n)
val j1 = test_equiv\$reindex_j<tk_index> (i, j, n)
val kb = storage_index<tk_index> (i1, j1)
val color_b = recolor b[kb]
in
if color_a = color_b then
loopi (a, b, n, j, pred i)
else
false
end
in
if loopi (a, b, n, j, g1i2i (pred n)) then
loopj (a, b, n, pred j)
else
false
end
in
loopj (a, b, n, g1i2i (pred n))
end

fn {tk_index, tk_color : tkind}
test_equiv_rotate0
{n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
(* No rotations or reflections. *)
implement
test_equiv\$reindex_i<tk_index> (i, j, n) = i
implement
test_equiv\$reindex_j<tk_index> (i, j, n) = j
in
test_equiv<tk_index, tk_color> (a, b, n)
end

fn {tk_index, tk_color : tkind}
test_equiv_rotate90
{n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
(* Matrix rotation counterclockwise by 90 degrees. *)
implement
test_equiv\$reindex_i<tk_index> {i, j} {n} (i, j, n) = pred n - j
implement
test_equiv\$reindex_j<tk_index> (i, j, n) = i
in
test_equiv<tk_index, tk_color> (a, b, n)
end

fn {tk_index, tk_color : tkind}
test_equiv_rotate180
{n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
(* Matrix rotation by 180 degrees. *)
implement
test_equiv\$reindex_i<tk_index> {i, j} {n} (i, j, n) = pred n - i
implement
test_equiv\$reindex_j<tk_index> {i, j} {n} (i, j, n) = pred n - j
in
test_equiv<tk_index, tk_color> (a, b, n)
end

fn {tk_index, tk_color : tkind}
test_equiv_rotate270
{n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
(* Matrix rotation counterclockwise by 270 degrees. *)
implement
test_equiv\$reindex_i<tk_index> (i, j, n) = j
implement
test_equiv\$reindex_j<tk_index> {i, j} {n} (i, j, n) = pred n - i
in
test_equiv<tk_index, tk_color> (a, b, n)
end

fn {tk_index, tk_color : tkind}
test_equiv_reflecti
{n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
(* Reverse the order of the rows. *)
implement
test_equiv\$reindex_i<tk_index> {i, j} {n} (i, j, n) = pred n - i
implement
test_equiv\$reindex_j<tk_index> (i, j, n) = j
in
test_equiv<tk_index, tk_color> (a, b, n)
end

fn {tk_index, tk_color : tkind}
test_equiv_reflectj
{n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
(* Reverse the order of the columns. *)
implement
test_equiv\$reindex_i<tk_index> (i, j, n) = i
implement
test_equiv\$reindex_j<tk_index> {i, j} {n} (i, j, n) = pred n - j
in
test_equiv<tk_index, tk_color> (a, b, n)
end

fn {tk_index, tk_color : tkind}
test_equiv_reflect_diag_down
{n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
(* Transpose the matrix around its main diagonal. *)
implement
test_equiv\$reindex_i<tk_index> (i, j, n) = j
implement
test_equiv\$reindex_j<tk_index> (i, j, n) = i
in
test_equiv<tk_index, tk_color> (a, b, n)
end

fn {tk_index, tk_color : tkind}
test_equiv_reflect_diag_up
{n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
bool =
let
(* Transpose the matrix around its main skew diagonal. *)
implement
test_equiv\$reindex_i<tk_index> {i, j} {n} (i, j, n) = pred n - j
implement
test_equiv\$reindex_j<tk_index> {i, j} {n} (i, j, n) = pred n - i
in
test_equiv<tk_index, tk_color> (a, b, n)
end

fn {tk_index, tk_color : tkind}
board_equiv {n : int | 0 <= n; n <= SIDE_MAX}
(a : &(@[color_t tk_color][BOARD_SIZE]),
b : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n),
rotation_equiv_classes : bool) :
bool =
let
(* Leave the colors unchanged. *)
implement test_equiv\$recolor<tk_color> (c) = c

(* Test without rotations or reflections. *)
val equiv = test_equiv_rotate0<tk_index, tk_color> (a, b, n)
in
if ~rotation_equiv_classes then
equiv
else
let
(* Leave the colors unchanged. *)
implement test_equiv\$recolor<tk_color> (c) = c

val equiv =
(equiv ||
test_equiv_rotate90<tk_index, tk_color> (a, b, n) ||
test_equiv_rotate180<tk_index, tk_color> (a, b, n) ||
test_equiv_rotate270<tk_index, tk_color> (a, b, n) ||
test_equiv_reflecti<tk_index, tk_color> (a, b, n) ||
test_equiv_reflectj<tk_index, tk_color> (a, b, n) ||
test_equiv_reflect_diag_down<tk_index, tk_color> (a, b, n) ||
test_equiv_reflect_diag_up<tk_index, tk_color> (a, b, n))

(* Reverse the colors of b in each test. *)
implement test_equiv\$recolor<tk_color> (c) = reverse_color c

val equiv =
(equiv ||
test_equiv_rotate0<tk_index, tk_color> (a, b, n) ||
test_equiv_rotate90<tk_index, tk_color> (a, b, n) ||
test_equiv_rotate180<tk_index, tk_color> (a, b, n) ||
test_equiv_rotate270<tk_index, tk_color> (a, b, n) ||
test_equiv_reflecti<tk_index, tk_color> (a, b, n) ||
test_equiv_reflectj<tk_index, tk_color> (a, b, n) ||
test_equiv_reflect_diag_down<tk_index, tk_color> (a, b, n) ||
test_equiv_reflect_diag_up<tk_index, tk_color> (a, b, n))
in
equiv
end
end

(********************************************************************)

fn {tk_index : tkind}
fprint_rule {n : int | 0 <= n; n <= SIDE_MAX}
(f : FILEref,
n : g1int (tk_index, n)) :
void =
let
fun
loop {j : int | 0 <= j; j <= n} .<n - j>.
(j : g1int (tk_index, j)) :
void =
if j <> n then
begin
fileref_puts (f, "----+");
loop (succ j)
end
in
fileref_puts (f, "+");
loop (g1i2i 0)
end

fn {tk_index, tk_color : tkind}
fprint_board {n : int | 0 <= n; n <= SIDE_MAX}
(f : FILEref,
a : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n)) :
void =
if n <> 0 then
let
fun
loopi {i : int | ~1 <= i; i < n} .<i + 1>.
(f : FILEref,
a : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n),
i : g1int (tk_index, i)) :
void =
if i <> ~1 then
let
fun
loopj {j : int | 0 <= j; j <= n} .<n - j>.
(f : FILEref,
a : &(@[color_t tk_color][BOARD_SIZE]),
n : g1int (tk_index, n),
i : g1int (tk_index, i),
j : g1int (tk_index, j)) :
void =
if j <> n then
let
val k = storage_index<tk_index> (i, j)
val color = a[k]
val representation =
if color = g1i2i BLACK then
"|  B "
else if color = g1i2i WHITE then
"|  W "
else
"|    "
in
fileref_puts (f, representation);
loopj (f, a, n, i, succ j)
end
in
fileref_puts (f, "\n");
loopj (f, a, n, i, g1i2i 0);
fileref_puts (f, "|\n");
fprint_rule (f, n);
loopi (f, a, n, pred i)
end
in
fprint_rule (f, n);
loopi (f, a, n, pred n)
end

(********************************************************************)

(* M2_MAX equals the maximum number of queens of either color.
Thus it is the maximum of 2*m, where m is the number of queens
in an army. *)
#define M2_MAX BOARD_SIZE

(* The even-index queens are BLACK, the odd-index queens are WHITE. *)

vtypedef board_record_vt (tk_color : tkind,
@{
pf = @[color_t tk_color][BOARD_SIZE] @ p,
pfgc = mfree_gc_v p |
p = ptr p
}
vtypedef board_record_vt (tk_color : tkind) =
[p : addr | null < p]
board_record_vt (tk_color, p)

vtypedef board_record_list_vt (tk_color : tkind,
n : int) =
list_vt (board_record_vt tk_color, n)
vtypedef board_record_list_vt (tk_color : tkind) =
[n : int]
board_record_list_vt (tk_color, n)

fn
board_record_vt_free
{tk_color : tkind}
(record   : board_record_vt (tk_color, p)) :
void =
let
val @{
pf = pf,
pfgc = pfgc |
p = p
} = record
in
array_ptr_free (pf, pfgc | p)
end

fn
board_record_list_vt_free
{tk_color : tkind}
{n        : int}
(lst      : board_record_list_vt (tk_color, n)) :
void =
let
fun
loop {n   : int | 0 <= n} .<n>.
(lst : board_record_list_vt (tk_color, n)) :
void =
case+ lst of
| ~ NIL => ()
| ~ head :: tail =>
begin
loop tail
end

prval _ = lemma_list_vt_param lst
in
loop lst
end

fn {tk_index, tk_color : tkind}
any_board_equiv {n     : int | 0 <= n; n <= SIDE_MAX}
(board : &(@[color_t tk_color][BOARD_SIZE]),
lst   : !board_record_list_vt tk_color,
n     : g1int (tk_index, n),
rotation_equiv_classes : bool) :
bool =
let
macdef board_equiv = board_equiv<tk_index, tk_color>

fun
loop {k : int | 0 <= k} .<k>.
(board : &(@[color_t tk_color][BOARD_SIZE]),
lst   : !board_record_list_vt (tk_color, k),
n     : g1int (tk_index, n)) :
bool =
case+ lst of
| NIL => false
rotation_equiv_classes) then
true
else
loop (board, tail, n)

prval _ = lemma_list_vt_param lst
in
loop (board, lst, n)
end

fn {tk_index, tk_color : tkind}
queens_to_board
{count  : int | 0 <= count; count <= M2_MAX}
(queens : &(@[g1int tk_index][M2_MAX]),
count  : int count) :
[p : addr | null < p]
board_record_vt (tk_color, p) =
let
typedef color_t = color_t tk_color

fun
loop {k : int | ~1 <= k; k < count} .<k + 1>.
(queens : &(@[g1int tk_index][M2_MAX]),
board  : &(@[color_t tk_color][BOARD_SIZE]),
k      : int k) :
void =
if 0 <= k then
let
val [coords : int] coords = queens[k]
#if NDEBUG <> 0 #then
prval _ = \$UN.prop_assert {0 <= coords} ()
prval _ = \$UN.prop_assert {coords < BOARD_SIZE} ()
#else
val _ = assertloc (g1i2i 0 <= coords)
val _ = assertloc (coords < g1i2i BOARD_SIZE)
#endif
in
if g1int_nmod (k, 2) = 0 then
board[coords] := g1i2i BLACK
else
board[coords] := g1i2i WHITE;
loop (queens, board, pred k)
end

val @(pf, pfgc | p) = array_ptr_alloc<color_t> (i2sz BOARD_SIZE)
val _ = array_initize_elt<color_t> (!p, i2sz BOARD_SIZE,
g1i2i EMPTY)
val _ = loop (queens, !p, pred count)
in
@{
pf = pf,
pfgc = pfgc |
p = p
}
end

fn {tk : tkind}
queen_would_fit_in
{count  : int | 0 <= count; count <= M2_MAX}
{i, j   : int | 0 <= i; i < SIDE_MAX;
0 <= j; j < SIDE_MAX}
(queens : &(@[g1int tk][M2_MAX]),
count  : int count,
i      : g1int (tk, i),
j      : g1int (tk, j)) :
bool =
(* Would a new queen at (i,j) be feasible? *)
if count = 0 then
true
else
let
fun
loop {k : int | ~1 <= k; k < count}
(queens : &(@[g1int tk][M2_MAX]),
k      : int k) :
bool =
if k < 0 then
true
else
let
val [coords : int] coords = queens[k]
#if NDEBUG <> 0 #then
prval _ = \$UN.prop_assert {0 <= coords} ()
prval _ = \$UN.prop_assert {coords < BOARD_SIZE} ()
#else
val _ = assertloc (g1i2i 0 <= coords)
val _ = assertloc (coords < g1i2i BOARD_SIZE)
#endif

val i1 = row_index<tk> coords
val j1 = column_index<tk> coords
in
if g1int_nmod (k, 2) = g1int_nmod (count, 2) then
(* The two queens are of the same color. They may not
share the same square. *)
begin
if i <> i1 || j <> j1 then
loop (queens, pred k)
else
false
end
else
(* The two queens are of different colors. They may not
share the same square nor attack each other. *)
begin
if (i <> i1 &&
j <> j1 &&
i + j <> i1 + j1 &&
i - j <> i1 - j1) then
loop (queens, pred k)
else
false
end
end
in
loop (queens, pred count)
end

fn {tk : tkind}
latest_queen_fits_in
{count  : int | 1 <= count; count <= M2_MAX}
(queens : &(@[g1int tk][M2_MAX]),
count  : int count) :
bool =
let
val [coords : int] coords = queens[pred count]
#if NDEBUG <> 0 #then
prval _ = \$UN.prop_assert {0 <= coords} ()
prval _ = \$UN.prop_assert {coords < BOARD_SIZE} ()
#else
val _ = assertloc (g1i2i 0 <= coords)
val _ = assertloc (coords < g1i2i BOARD_SIZE)
#endif

val i = row_index<tk> coords
val j = column_index<tk> coords
in
queen_would_fit_in<tk> (queens, pred count, i, j)
end

fn {tk_index, tk_color : tkind}
find_solutions
{m : int | 0 <= m; 2 * m <= M2_MAX}
{n : int | 0 <= n; n <= SIDE_MAX}
{max_solutions : int | 0 <= max_solutions}
(f : FILEref,
m : int m,
n : g1int (tk_index, n),
rotation_equiv_classes : bool,
max_solutions : int max_solutions) :
[num_solutions : int | 0 <= num_solutions;
num_solutions <= max_solutions]
@(int num_solutions,
board_record_list_vt (tk_color, num_solutions)) =
(* This template function both prints the solutions and returns
them as a linked list. *)
if m = 0 then
@(0, NIL)
else if max_solutions = 0 then
@(0, NIL)
else
let
macdef latest_queen_fits_in = latest_queen_fits_in<tk_index>
macdef queens_to_board = queens_to_board<tk_index, tk_color>
macdef fprint_board = fprint_board<tk_index, tk_color>
macdef any_board_equiv = any_board_equiv<tk_index, tk_color>
macdef row_index = row_index<tk_index>
macdef column_index = column_index<tk_index>
macdef storage_index = storage_index<tk_index>

fnx
loop {num_solutions : int | 0 <= num_solutions;
num_solutions <= max_solutions}
{num_queens    : int | 0 <= num_queens;
num_queens <= 2 * m}
(solutions     : board_record_list_vt (tk_color,
num_solutions),
num_solutions : int num_solutions,
queens        : &(@[g1int tk_index][M2_MAX]),
num_queens    : int num_queens) :
[num_solutions1 : int | 0 <= num_solutions1;
num_solutions1 <= max_solutions]
@(int num_solutions1,
board_record_list_vt (tk_color, num_solutions1)) =
if num_queens = 0 then
@(num_solutions, solutions)
else if num_solutions = max_solutions then
@(num_solutions, solutions)
else if latest_queen_fits_in (queens, num_queens) then
begin
if num_queens = 2 * m then
let
val board = queens_to_board (queens, num_queens)
val equiv_solution =
any_board_equiv (!(board.p), solutions, n,
rotation_equiv_classes)
in
if ~equiv_solution then
begin
fprintln! (f, "Solution ",
succ num_solutions);
fprint_board (f, !(board.p), n);
fileref_puts (f, "\n\n");
move_a_queen (board :: solutions,
succ num_solutions,
queens, num_queens)
end
else
begin
free board;
move_a_queen (solutions, num_solutions,
queens, num_queens)
end
end
else
queens, num_queens)
end
else
move_a_queen (solutions, num_solutions,
queens, num_queens)
and
{num_solutions : int |
0 <= num_solutions;
num_solutions <= max_solutions}
{num_queens : int | 0 <= num_queens;
num_queens + 1 <= 2 * m}
(solutions : board_record_list_vt
(tk_color, num_solutions),
num_solutions : int num_solutions,
queens     : &(@[g1int tk_index][M2_MAX]),
num_queens : int num_queens) :
[num_solutions1 : int | 0 <= num_solutions1;
num_solutions1 <= max_solutions]
@(int num_solutions1,
board_record_list_vt (tk_color, num_solutions1)) =
let
val coords = storage_index (g1i2i 0, g1i2i 0)
in
queens[num_queens] := coords;
loop (solutions, num_solutions, queens, succ num_queens)
end
and
move_a_queen {num_solutions : int |
0 <= num_solutions;
num_solutions <= max_solutions}
{num_queens : int | 0 <= num_queens;
num_queens <= 2 * m}
(solutions : board_record_list_vt
(tk_color, num_solutions),
num_solutions : int num_solutions,
queens : &(@[g1int tk_index][M2_MAX]),
num_queens : int num_queens) :
[num_solutions1 : int | 0 <= num_solutions1;
num_solutions1 <= max_solutions]
@(int num_solutions1,
board_record_list_vt (tk_color, num_solutions1)) =
if num_queens = 0 then
loop (solutions, num_solutions, queens, num_queens)
else
let
val [coords : int] coords = queens[pred num_queens]
#if NDEBUG <> 0 #then
prval _ = \$UN.prop_assert {0 <= coords} ()
prval _ = \$UN.prop_assert {coords < BOARD_SIZE} ()
#else
val _ = assertloc (g1i2i 0 <= coords)
val _ = assertloc (coords < g1i2i BOARD_SIZE)
#endif

val [i : int] i = row_index coords
val [j : int] j = column_index coords

prval _ = prop_verify {0 <= i} ()
prval _ = prop_verify {i < SIDE_MAX} ()

prval _ = prop_verify {0 <= j} ()
prval _ = prop_verify {j < SIDE_MAX} ()

#if NDEBUG <> 0 #then
prval _ = \$UN.prop_assert {i < n} ()
prval _ = \$UN.prop_assert {j < n} ()
#else
val _ = \$effmask_exn assertloc (i < n)
val _ = \$effmask_exn assertloc (j < n)
#endif
in
if j = pred n then
begin
if i = pred n then
(* Backtrack. *)
move_a_queen (solutions, num_solutions,
queens, pred num_queens)
else
let
val coords = storage_index (succ i, j)
in
queens[pred num_queens] := coords;
loop (solutions, num_solutions,
queens, num_queens)
end
end
else
let
#if NDEBUG <> 0 #then
prval _ = \$UN.prop_assert {j < n - 1} ()
#else
val _ = \$effmask_exn assertloc (j < pred n)
#endif
in
if i = pred n then
let
val coords = storage_index (g1i2i 0, succ j)
in
queens[pred num_queens] := coords;
loop (solutions, num_solutions,
queens, num_queens)
end
else
let
val coords = storage_index (succ i, j)
in
queens[pred num_queens] := coords;
loop (solutions, num_solutions,
queens, num_queens)
end
end
end

var queens = @[g1int tk_index][M2_MAX] (g1i2i 0)
in
queens[0] := storage_index (g1i2i 0, g1i2i 0);
loop (NIL, 0, queens, 1)
end

(********************************************************************)

%{^
#include <stdlib.h>
#include <limits.h>
%}

implement
main0 (argc, argv) =
let

macdef usage_error (status) =
begin
println! ("Usage: ", argv[0],
" M N IGNORE_EQUIVALENTS [MAX_SOLUTIONS]");
exit (,(status))
end

val max_max_solutions =
\$extval ([i : int | 0 <= i] int i, "INT_MAX")
in
if 4 <= argc then
let
val m = \$extfcall (int, "atoi", argv[1])
val m = g1ofg0 m
val _ = if m < 0 then usage_error (2)
val _ = assertloc (0 <= m)
val _ =
if M2_MAX < 2 * m then
begin
println! (argv[0], ": M cannot be larger than ",
M2_MAX / 2);
usage_error (2)
end
val _ = assertloc (2 * m <= M2_MAX)

val n = \$extfcall (int, "atoi", argv[2])
val n = g1ofg0 n
val _ = if n < 0 then usage_error (2)
val _ = assertloc (0 <= n)
val _ =
if SIDE_MAX < n then
begin
println! (argv[0], ": N cannot be larger than ",
SIDE_MAX);
usage_error (2)
end
val _ = assertloc (n <= SIDE_MAX)

val ignore_equivalents =
if argv[3] = "T" || argv[3] = "t" || argv[3] = "1" then
true
else if argv[3] = "F" || argv[3] = "f" || argv[3] = "0" then
false
else
begin
println! (argv[0],
": select T=t=1 or F=f=0 ",
"for IGNORE_EQUIVALENTS");
usage_error (2);
false
end
in
if argc = 5 then
let
val max_solutions = \$extfcall (int, "atoi", argv[4])
val max_solutions = g1ofg0 max_solutions
val max_solutions = max (0, max_solutions)

val @(num_solutions, solutions) =
find_solutions<tk_index, tk_color>
(stdout_ref, m, n, ignore_equivalents,
max_solutions)
in
board_record_list_vt_free solutions
end
else
let
val @(num_solutions, solutions) =
find_solutions<tk_index, tk_color>
(stdout_ref, m, n, ignore_equivalents,
max_max_solutions)
in
board_record_list_vt_free solutions
end
end
else
usage_error (1)
end

(********************************************************************)```
Output:

\$ patscc -DATS NDEBUG=1 -O3 -fno-stack-protector -march=native -DATS_MEMALLOC_LIBC -o peaceful_queens peaceful_queens.dats && ./peaceful_queens 4 5 T

```Solution 1
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+

Solution 2
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+

Solution 3
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
```

## C

Translation of: C#
```#include <math.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>

enum Piece {
Empty,
Black,
White,
};

typedef struct Position_t {
int x, y;
} Position;

///////////////////////////////////////////////

struct Node_t {
Position pos;
struct Node_t *next;
};

}

typedef struct List_t {
struct Node_t *tail;
size_t length;
} List;

List makeList() {
return (List) { NULL, NULL, 0 };
}

void releaseList(List *lst) {
if (lst == NULL) return;

lst->tail = NULL;
}

void addNode(List *lst, Position pos) {
struct Node_t *newNode;

if (lst == NULL) {
exit(EXIT_FAILURE);
}

newNode = malloc(sizeof(struct Node_t));
if (newNode == NULL) {
exit(EXIT_FAILURE);
}

newNode->next = NULL;
newNode->pos = pos;

} else {
lst->tail->next = newNode;
lst->tail = newNode;
}

lst->length++;
}

void removeAt(List *lst, size_t pos) {
if (lst == NULL) return;

if (pos == 0) {

lst->tail = NULL;
}

temp->next = NULL;

free(temp);
lst->length--;
} else {
struct Node_t *rem;
size_t i = pos;

while (i-- > 1) {
temp = temp->next;
}

rem = temp->next;
if (rem == lst->tail) {
lst->tail = temp;
}

temp->next = rem->next;

rem->next = NULL;
free(rem);

lst->length--;
}
}

///////////////////////////////////////////////

bool isAttacking(Position queen, Position pos) {
return queen.x == pos.x
|| queen.y == pos.y
|| abs(queen.x - pos.x) == abs(queen.y - pos.y);
}

bool place(int m, int n, List *pBlackQueens, List *pWhiteQueens) {
struct Node_t *queenNode;
bool placingBlack = true;
int i, j;

if (pBlackQueens == NULL || pWhiteQueens == NULL) {
exit(EXIT_FAILURE);
}

if (m == 0) return true;
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
Position pos = { i, j };

while (queenNode != NULL) {
if ((queenNode->pos.x == pos.x && queenNode->pos.y == pos.y) || !placingBlack && isAttacking(queenNode->pos, pos)) {
goto inner;
}
queenNode = queenNode->next;
}

while (queenNode != NULL) {
if ((queenNode->pos.x == pos.x && queenNode->pos.y == pos.y) || placingBlack && isAttacking(queenNode->pos, pos)) {
goto inner;
}
queenNode = queenNode->next;
}

if (placingBlack) {
placingBlack = false;
} else {
if (place(m - 1, n, pBlackQueens, pWhiteQueens)) {
return true;
}
removeAt(pBlackQueens, pBlackQueens->length - 1);
removeAt(pWhiteQueens, pWhiteQueens->length - 1);
placingBlack = true;
}

inner: {}
}
}
if (!placingBlack) {
removeAt(pBlackQueens, pBlackQueens->length - 1);
}
return false;
}

void printBoard(int n, List *pBlackQueens, List *pWhiteQueens) {
size_t length = n * n;
struct Node_t *queenNode;
char *board;
size_t i, j, k;

if (pBlackQueens == NULL || pWhiteQueens == NULL) {
exit(EXIT_FAILURE);
}

board = calloc(length, sizeof(char));
if (board == NULL) {
exit(EXIT_FAILURE);
}

while (queenNode != NULL) {
board[queenNode->pos.x * n + queenNode->pos.y] = Black;
queenNode = queenNode->next;
}

while (queenNode != NULL) {
board[queenNode->pos.x * n + queenNode->pos.y] = White;
queenNode = queenNode->next;
}

for (i = 0; i < length; i++) {
if (i != 0 && i % n == 0) {
printf("\n");
}
switch (board[i]) {
case Black:
printf("B ");
break;
case White:
printf("W ");
break;
default:
j = i / n;
k = i - j * n;
if (j % 2 == k % 2) {
printf("  ");
} else {
printf("# ");
}
break;
}
}

printf("\n\n");
}

void test(int n, int q) {
List blackQueens = makeList();
List whiteQueens = makeList();

printf("%d black and %d white queens on a %d x %d board:\n", q, q, n, n);
if (place(q, n, &blackQueens, &whiteQueens)) {
printBoard(n, &blackQueens, &whiteQueens);
} else {
printf("No solution exists.\n\n");
}

releaseList(&blackQueens);
releaseList(&whiteQueens);
}

int main() {
test(2, 1);

test(3, 1);
test(3, 2);

test(4, 1);
test(4, 2);
test(4, 3);

test(5, 1);
test(5, 2);
test(5, 3);
test(5, 4);
test(5, 5);

test(6, 1);
test(6, 2);
test(6, 3);
test(6, 4);
test(6, 5);
test(6, 6);

test(7, 1);
test(7, 2);
test(7, 3);
test(7, 4);
test(7, 5);
test(7, 6);
test(7, 7);

return EXIT_SUCCESS;
}
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B #
#   W
#

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B #   #
#   W
#   #
#   #

2 black and 2 white queens on a 4 x 4 board:
B #   #
#   W
B #   #
#   W

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B #   #
#   W   #
#   #
#   #   #
#   #

2 black and 2 white queens on a 5 x 5 board:
B #   # B
#   W   #
W   #
#   #   #
#   #

3 black and 3 white queens on a 5 x 5 board:
B #   # B
#   W   #
W   #
#   # B #
W   #

4 black and 4 white queens on a 5 x 5 board:
B   B
#   #   B
W # W #
#   #   B
W # W #

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B #   #   #
#   W   #
#   #   #
#   #   #
#   #   #
#   #   #

2 black and 2 white queens on a 6 x 6 board:
B #   # B #
#   W   #
W   #   #
#   #   #
#   #   #
#   #   #

3 black and 3 white queens on a 6 x 6 board:
B #   # B B
#   W   #
W   #   #
#   #   #
# W #   #
#   #   #

4 black and 4 white queens on a 6 x 6 board:
B #   # B B
#   W   #
W   #   #
#   #   # B
# W W   #
#   #   #

5 black and 5 white queens on a 6 x 6 board:
B   # B #
#   # B # B
W #   #   #
W   W   #
#   #   B
W   W   #

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B #   #   #
#   W   #   #
#   #   #
#   #   #   #
#   #   #
#   #   #   #
#   #   #

2 black and 2 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
#   #   #
#   #   #   #
#   #   #
#   #   #   #
#   #   #

3 black and 3 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
B #   #   #
#   W   #   #
#   #   #
#   #   #   #
#   #   #

4 black and 4 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
B #   # B #
#   W   #   W
#   #   #
#   #   #   #
#   #   #

5 black and 5 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
B #   # B #
#   W   #   W
B #   #   #
#   W   #   #
#   #   #

6 black and 6 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
B #   # B #
#   W   #   W
B #   # B #
#   W   #   W
#   #   #

7 black and 7 white queens on a 7 x 7 board:
B   #   B
# B #   B   #
B   #   B
#   #   B   #
W # W #   # W
#   # W #   #
W # W W   #```

## C#

Translation of: D
```using System;
using System.Collections.Generic;

namespace PeacefulChessQueenArmies {
using Position = Tuple<int, int>;

enum Piece {
Empty,
Black,
White
}

class Program {
static bool IsAttacking(Position queen, Position pos) {
return queen.Item1 == pos.Item1
|| queen.Item2 == pos.Item2
|| Math.Abs(queen.Item1 - pos.Item1) == Math.Abs(queen.Item2 - pos.Item2);
}

static bool Place(int m, int n, List<Position> pBlackQueens, List<Position> pWhiteQueens) {
if (m == 0) {
return true;
}
bool placingBlack = true;
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
var pos = new Position(i, j);
foreach (var queen in pBlackQueens) {
if (queen.Equals(pos) || !placingBlack && IsAttacking(queen, pos)) {
goto inner;
}
}
foreach (var queen in pWhiteQueens) {
if (queen.Equals(pos) || placingBlack && IsAttacking(queen, pos)) {
goto inner;
}
}
if (placingBlack) {
placingBlack = false;
} else {
if (Place(m - 1, n, pBlackQueens, pWhiteQueens)) {
return true;
}
pBlackQueens.RemoveAt(pBlackQueens.Count - 1);
pWhiteQueens.RemoveAt(pWhiteQueens.Count - 1);
placingBlack = true;
}
inner: { }
}
}
if (!placingBlack) {
pBlackQueens.RemoveAt(pBlackQueens.Count - 1);
}
return false;
}

static void PrintBoard(int n, List<Position> blackQueens, List<Position> whiteQueens) {
var board = new Piece[n * n];

foreach (var queen in blackQueens) {
board[queen.Item1 * n + queen.Item2] = Piece.Black;
}
foreach (var queen in whiteQueens) {
board[queen.Item1 * n + queen.Item2] = Piece.White;
}

for (int i = 0; i < board.Length; i++) {
if (i != 0 && i % n == 0) {
Console.WriteLine();
}
switch (board[i]) {
case Piece.Black:
Console.Write("B ");
break;
case Piece.White:
Console.Write("W ");
break;
case Piece.Empty:
int j = i / n;
int k = i - j * n;
if (j % 2 == k % 2) {
Console.Write("  ");
} else {
Console.Write("# ");
}
break;
}
}

Console.WriteLine("\n");
}

static void Main() {
var nms = new int[,] {
{2, 1}, {3, 1}, {3, 2}, {4, 1}, {4, 2}, {4, 3},
{5, 1}, {5, 2}, {5, 3}, {5, 4}, {5, 5},
{6, 1}, {6, 2}, {6, 3}, {6, 4}, {6, 5}, {6, 6},
{7, 1}, {7, 2}, {7, 3}, {7, 4}, {7, 5}, {7, 6}, {7, 7},
};
for (int i = 0; i < nms.GetLength(0); i++) {
Console.WriteLine("{0} black and {0} white queens on a {1} x {1} board:", nms[i, 1], nms[i, 0]);
List<Position> blackQueens = new List<Position>();
List<Position> whiteQueens = new List<Position>();
if (Place(nms[i, 1], nms[i, 0], blackQueens, whiteQueens)) {
PrintBoard(nms[i, 0], blackQueens, whiteQueens);
} else {
Console.WriteLine("No solution exists.\n");
}
}
}
}
}
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B #
#   W
#

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B #   #
#   W
#   #
#   #

2 black and 2 white queens on a 4 x 4 board:
B #   #
#   W
B #   #
#   W

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B #   #
#   W   #
#   #
#   #   #
#   #

2 black and 2 white queens on a 5 x 5 board:
B #   # B
#   W   #
W   #
#   #   #
#   #

3 black and 3 white queens on a 5 x 5 board:
B #   # B
#   W   #
W   #
#   # B #
W   #

4 black and 4 white queens on a 5 x 5 board:
B   B
#   #   B
W # W #
#   #   B
W # W #

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B #   #   #
#   W   #
#   #   #
#   #   #
#   #   #
#   #   #

2 black and 2 white queens on a 6 x 6 board:
B #   # B #
#   W   #
W   #   #
#   #   #
#   #   #
#   #   #

3 black and 3 white queens on a 6 x 6 board:
B #   # B B
#   W   #
W   #   #
#   #   #
# W #   #
#   #   #

4 black and 4 white queens on a 6 x 6 board:
B #   # B B
#   W   #
W   #   #
#   #   # B
# W W   #
#   #   #

5 black and 5 white queens on a 6 x 6 board:
B   # B #
#   # B # B
W #   #   #
W   W   #
#   #   B
W   W   #

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B #   #   #
#   W   #   #
#   #   #
#   #   #   #
#   #   #
#   #   #   #
#   #   #

2 black and 2 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
#   #   #
#   #   #   #
#   #   #
#   #   #   #
#   #   #

3 black and 3 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
B #   #   #
#   W   #   #
#   #   #
#   #   #   #
#   #   #

4 black and 4 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
B #   # B #
#   W   #   W
#   #   #
#   #   #   #
#   #   #

5 black and 5 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
B #   # B #
#   W   #   W
B #   #   #
#   W   #   #
#   #   #

6 black and 6 white queens on a 7 x 7 board:
B #   # B #
#   W   #   W
B #   # B #
#   W   #   W
B #   # B #
#   W   #   W
#   #   #

7 black and 7 white queens on a 7 x 7 board:
B   #   B
# B #   B   #
B   #   B
#   #   B   #
W # W #   # W
#   # W #   #
W # W W   #```

## C++

Translation of: D
```#include <iostream>
#include <vector>

enum class Piece {
empty,
black,
white
};

typedef std::pair<int, int> position;

bool isAttacking(const position &queen, const position &pos) {
return queen.first == pos.first
|| queen.second == pos.second
|| abs(queen.first - pos.first) == abs(queen.second - pos.second);
}

bool place(const int m, const int n, std::vector<position> &pBlackQueens, std::vector<position> &pWhiteQueens) {
if (m == 0) {
return true;
}
bool placingBlack = true;
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
auto pos = std::make_pair(i, j);
for (auto queen : pBlackQueens) {
if (queen == pos || !placingBlack && isAttacking(queen, pos)) {
goto inner;
}
}
for (auto queen : pWhiteQueens) {
if (queen == pos || placingBlack && isAttacking(queen, pos)) {
goto inner;
}
}
if (placingBlack) {
pBlackQueens.push_back(pos);
placingBlack = false;
} else {
pWhiteQueens.push_back(pos);
if (place(m - 1, n, pBlackQueens, pWhiteQueens)) {
return true;
}
pBlackQueens.pop_back();
pWhiteQueens.pop_back();
placingBlack = true;
}

inner: {}
}
}
if (!placingBlack) {
pBlackQueens.pop_back();
}
return false;
}

void printBoard(int n, const std::vector<position> &blackQueens, const std::vector<position> &whiteQueens) {
std::vector<Piece> board(n * n);
std::fill(board.begin(), board.end(), Piece::empty);

for (auto &queen : blackQueens) {
board[queen.first * n + queen.second] = Piece::black;
}
for (auto &queen : whiteQueens) {
board[queen.first * n + queen.second] = Piece::white;
}

for (size_t i = 0; i < board.size(); ++i) {
if (i != 0 && i % n == 0) {
std::cout << '\n';
}
switch (board[i]) {
case Piece::black:
std::cout << "B ";
break;
case Piece::white:
std::cout << "W ";
break;
case Piece::empty:
default:
int j = i / n;
int k = i - j * n;
if (j % 2 == k % 2) {
std::cout << "x ";
} else {
std::cout << "* ";
}
break;
}
}

std::cout << "\n\n";
}

int main() {
std::vector<position> nms = {
{2, 1}, {3, 1}, {3, 2}, {4, 1}, {4, 2}, {4, 3},
{5, 1}, {5, 2}, {5, 3}, {5, 4}, {5, 5},
{6, 1}, {6, 2}, {6, 3}, {6, 4}, {6, 5}, {6, 6},
{7, 1}, {7, 2}, {7, 3}, {7, 4}, {7, 5}, {7, 6}, {7, 7},
};

for (auto nm : nms) {
std::cout << nm.second << " black and " << nm.second << " white queens on a " << nm.first << " x " << nm.first << " board:\n";
std::vector<position> blackQueens, whiteQueens;
if (place(nm.second, nm.first, blackQueens, whiteQueens)) {
printBoard(nm.first, blackQueens, whiteQueens);
} else {
std::cout << "No solution exists.\n\n";
}
}

return 0;
}
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B * x
* x W
x * x

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B * x *
* x W x
x * x *
* x * x

2 black and 2 white queens on a 4 x 4 board:
B * x *
* x W x
B * x *
* x W x

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B * x * x
* x W x *
x * x * x
* x * x *
x * x * x

2 black and 2 white queens on a 5 x 5 board:
B * x * B
* x W x *
x W x * x
* x * x *
x * x * x

3 black and 3 white queens on a 5 x 5 board:
B * x * B
* x W x *
x W x * x
* x * B *
x W x * x

4 black and 4 white queens on a 5 x 5 board:
x B x B x
* x * x B
W * W * x
* x * x B
W * W * x

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B * x * x *
* x W x * x
x * x * x *
* x * x * x
x * x * x *
* x * x * x

2 black and 2 white queens on a 6 x 6 board:
B * x * B *
* x W x * x
x W x * x *
* x * x * x
x * x * x *
* x * x * x

3 black and 3 white queens on a 6 x 6 board:
B * x * B B
* x W x * x
x W x * x *
* x * x * x
x * W * x *
* x * x * x

4 black and 4 white queens on a 6 x 6 board:
B * x * B B
* x W x * x
x W x * x *
* x * x * B
x * W W x *
* x * x * x

5 black and 5 white queens on a 6 x 6 board:
x B x * B *
* x * B * B
W * x * x *
W x W x * x
x * x * x B
W x W x * x

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B * x * x * x
* x W x * x *
x * x * x * x
* x * x * x *
x * x * x * x
* x * x * x *
x * x * x * x

2 black and 2 white queens on a 7 x 7 board:
B * x * B * x
* x W x * x W
x * x * x * x
* x * x * x *
x * x * x * x
* x * x * x *
x * x * x * x

3 black and 3 white queens on a 7 x 7 board:
B * x * B * x
* x W x * x W
B * x * x * x
* x W x * x *
x * x * x * x
* x * x * x *
x * x * x * x

4 black and 4 white queens on a 7 x 7 board:
B * x * B * x
* x W x * x W
B * x * B * x
* x W x * x W
x * x * x * x
* x * x * x *
x * x * x * x

5 black and 5 white queens on a 7 x 7 board:
B * x * B * x
* x W x * x W
B * x * B * x
* x W x * x W
B * x * x * x
* x W x * x *
x * x * x * x

6 black and 6 white queens on a 7 x 7 board:
B * x * B * x
* x W x * x W
B * x * B * x
* x W x * x W
B * x * B * x
* x W x * x W
x * x * x * x

7 black and 7 white queens on a 7 x 7 board:
x B x * x B x
* B * x B x *
x B x * x B x
* x * x B x *
W * W * x * W
* x * W * x *
W * W W x * x```

## D

Translation of: Go
```import std.array;
import std.math;
import std.stdio;
import std.typecons;

enum Piece {
empty,
black,
white,
}

alias position = Tuple!(int, "i", int, "j");

bool place(int m, int n, ref position[] pBlackQueens, ref position[] pWhiteQueens) {
if (m == 0) {
return true;
}
bool placingBlack = true;
foreach (i; 0..n) {
inner:
foreach (j; 0..n) {
auto pos = position(i, j);
foreach (queen; pBlackQueens) {
if (queen == pos || !placingBlack && isAttacking(queen, pos)) {
continue inner;
}
}
foreach (queen; pWhiteQueens) {
if (queen == pos || placingBlack && isAttacking(queen, pos)) {
continue inner;
}
}
if (placingBlack) {
pBlackQueens ~= pos;
placingBlack = false;
} else {
pWhiteQueens ~= pos;
if (place(m - 1, n, pBlackQueens, pWhiteQueens)) {
return true;
}
pBlackQueens.length--;
pWhiteQueens.length--;
placingBlack = true;
}
}
}
if (!placingBlack) {
pBlackQueens.length--;
}
return false;
}

bool isAttacking(position queen, position pos) {
return queen.i == pos.i
|| queen.j == pos.j
|| abs(queen.i - pos.i) == abs(queen.j - pos.j);
}

void printBoard(int n, position[] blackQueens, position[] whiteQueens) {
auto board = uninitializedArray!(Piece[])(n * n);
board[] = Piece.empty;

foreach (queen; blackQueens) {
board[queen.i * n + queen.j] = Piece.black;
}
foreach (queen; whiteQueens) {
board[queen.i * n + queen.j] = Piece.white;
}
foreach (i,b; board) {
if (i != 0 && i % n == 0) {
writeln;
}
final switch (b) {
case Piece.black:
write("B ");
break;
case Piece.white:
write("W ");
break;
case Piece.empty:
int j = i / n;
int k = i - j * n;

if (j % 2 == k % 2) {
write("• "w);
} else {
write("◦ "w);
}
break;
}
}
writeln('\n');
}

void main() {
auto nms = [
[2, 1], [3, 1], [3, 2], [4, 1], [4, 2], [4, 3],
[5, 1], [5, 2], [5, 3], [5, 4], [5, 5],
[6, 1], [6, 2], [6, 3], [6, 4], [6, 5], [6, 6],
[7, 1], [7, 2], [7, 3], [7, 4], [7, 5], [7, 6], [7, 7],
];
foreach (nm; nms) {
writefln("%d black and %d white queens on a %d x %d board:", nm[1], nm[1], nm[0], nm[0]);
position[] blackQueens;
position[] whiteQueens;
if (place(nm[1], nm[0], blackQueens, whiteQueens)) {
printBoard(nm[0], blackQueens, whiteQueens);
} else {
writeln("No solution exists.\n");
}
}
}
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B ◦ •
◦ • W
• ◦ •

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B ◦ • ◦
◦ • W •
• ◦ • ◦
◦ • ◦ •

2 black and 2 white queens on a 4 x 4 board:
B ◦ • ◦
◦ • W •
B ◦ • ◦
◦ • W •

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • W • ◦
• ◦ • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

2 black and 2 white queens on a 5 x 5 board:
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

3 black and 3 white queens on a 5 x 5 board:
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ B ◦
• W • ◦ •

4 black and 4 white queens on a 5 x 5 board:
• B • B •
◦ • ◦ • B
W ◦ W ◦ •
◦ • ◦ • B
W ◦ W ◦ •

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

2 black and 2 white queens on a 6 x 6 board:
B ◦ • ◦ B ◦
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

3 black and 3 white queens on a 6 x 6 board:
B ◦ • ◦ B B
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ W ◦ • ◦
◦ • ◦ • ◦ •

4 black and 4 white queens on a 6 x 6 board:
B ◦ • ◦ B B
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ B
• ◦ W W • ◦
◦ • ◦ • ◦ •

5 black and 5 white queens on a 6 x 6 board:
• B • ◦ B ◦
◦ • ◦ B ◦ B
W ◦ • ◦ • ◦
W • W • ◦ •
• ◦ • ◦ • B
W • W • ◦ •

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

2 black and 2 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

3 black and 3 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

4 black and 4 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

5 black and 5 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •

6 black and 6 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •

7 black and 7 white queens on a 7 x 7 board:
• B • ◦ • B •
◦ B ◦ • B • ◦
• B • ◦ • B •
◦ • ◦ • B • ◦
W ◦ W ◦ • ◦ W
◦ • ◦ W ◦ • ◦
W ◦ W W • ◦ • ```

## Fortran

Works with: gfortran version 11.2.1

The example demonstrates modern Fortran’s capabilities for integer bit manipulation, by using large machine integers (and their entire bitrange) as bitmaps to represent queen armies. Complicated (but nevertheless single-statement) expressions of such integers represent such operations as rotating a chessboard and checking for any attacks.

There are two Fortran programs and a driver script. One program generates a Fortran module for basic operations; the other program (which must be linked with the generated module) does the actual work. The driver script is for Unix shell.

For speed, armies are represented by 64-bit or 128-bit integers, depending on the value of n. A 1-bit represets a queen. Rotations and reflections of the board are elemental integer operations on an army. Checking for any attacks is an elemental integer-to-boolean operation on the two armies (though the program detects rook-like attacks by a different mechanism). Equivalence under interchange of the colors can be tested by reversing which army gets which integer value.

Here is the first program, peaceful_queens_elements_generator.f90, which generates code (specialized for given m and n) to deal with the representations of the armies as integers:

```program peaceful_queens_elements_generator
use, intrinsic :: iso_fortran_env, only: int64
use, intrinsic :: iso_fortran_env, only: error_unit

implicit none

! 64-bit integers, for boards up to 8-by-8.
integer, parameter :: kind8x8 = int64

! 128-bit integers, for boards up to 11-by-11.
! This value is correct for gfortran.
integer, parameter :: kind11x11 = 16

integer(kind = kind11x11), parameter :: one = 1
integer(kind = kind11x11), parameter :: two = 2

integer, parameter :: n_max = 11

integer(kind = kind11x11) :: rook1_masks(0 : n_max - 1)
integer(kind = kind11x11) :: rook2_masks(0 : n_max - 1)
integer(kind = kind11x11) :: bishop1_masks(0 : (2 * n_max) - 4)
integer(kind = kind11x11) :: bishop2_masks(0 : (2 * n_max) - 4)

integer(kind = kind11x11) :: rook_masks(0 : (2 * n_max) - 1)

integer(kind = kind11x11) :: bishop_masks(0 : (4 * n_max) - 7)

! Combines rook and bishop masks.
integer(kind = kind11x11) :: queen_masks(0 : (6 * n_max) - 7)

character(len = 16), parameter :: s_kind8x8 = "kind8x8         "
character(len = 16), parameter :: s_kind11x11 = "kind11x11       "

character(200) :: arg
integer :: arg_count

integer :: m, n, max_solutions
integer :: board_kind

arg_count = command_argument_count ()
if (arg_count /= 3) then
call get_command_argument (0, arg)
write (error_unit, '("Usage: ", A, " M N MAX_SOLUTIONS")') trim (arg)
stop 1
end if

call get_command_argument (1, arg)
if (m < 1) then
write (error_unit, '("M must be between 1 or greater.")')
stop 2
end if

call get_command_argument (2, arg)
if (n < 3 .or. 11 < n) then
write (error_unit, '("N must be between 3 and ", I0, ", inclusive.")') n_max
stop 2
end if

call get_command_argument (3, arg)

write (*, '("module peaceful_queens_elements")')
write (*, '()')
write (*, '("  use, intrinsic :: iso_fortran_env, only: int64")')
write (*, '()')
write (*, '("  implicit none")')
write (*, '("  private")')
write (*, '()')
write (*, '("  integer, parameter, public :: m = ", I0)') m
write (*, '("  integer, parameter, public :: n = ", I0)') n
write (*, '("  integer, parameter, public :: max_solutions = ", I0)') max_solutions
write (*, '()')
if (n <= 8) then
write (*, '("  ! 64-bit integers, for boards up to 8-by-8.")')
write (*, '("  integer, parameter, private :: kind8x8 = int64")')
else
write (*, '("  ! 128-bit integers, for boards up to 11-by-11.")')
write (*, '("  integer, parameter, private :: kind11x11 = ", I0)') kind11x11
end if
write (*, '("  integer, parameter, public :: board_kind = ", A)') trim (s_kindnxn (n))
write (*, '()')
write (*, '()')
write (*, '("  public :: rooks1_attack_check")')
write (*, '("  public :: rooks2_attack_check")')
write (*, '("  public :: rooks_attack_check")')
write (*, '("  public :: bishops1_attack_check")')
write (*, '("  public :: bishops2_attack_check")')
write (*, '("  public :: bishops_attack_check")')
write (*, '("  public :: queens_attack_check")')
write (*, '()')
write (*, '("  public :: board_rotate90")')
write (*, '("  public :: board_rotate180")')
write (*, '("  public :: board_rotate270")')
write (*, '("  public :: board_reflect1")')
write (*, '("  public :: board_reflect2")')
write (*, '("  public :: board_reflect3")')
write (*, '("  public :: board_reflect4")')
write (*, '()')

write (*, '("contains")')
write (*, '()')

call write_rooks1_attack_check
call write_rooks2_attack_check
call write_bishops1_attack_check
call write_bishops2_attack_check
call write_rooks_attack_check
call write_bishops_attack_check
call write_queens_attack_check

call write_board_rotate90
call write_board_rotate180
call write_board_rotate270
call write_board_reflect1
call write_board_reflect2
call write_board_reflect3
call write_board_reflect4

call write_insert_zeros
call write_reverse_insert_zeros

write (*, '("end module peaceful_queens_elements")')

contains

integer :: i

do i = 0, n - 1
write (*, '("  integer(kind = ", A, "), parameter :: rook1_mask_",&
& I0, "x", I0, "_", I0, " = int (z''", Z0.32, "'', kind &
&= ", A, ")")') trim (s_kindnxn (n)), n, n, i,&
end do
write (*, '()')

integer :: i

do i = 0, n - 1
write (*, '("  integer(kind = ", A, "), parameter :: rook2_mask_",&
& I0, "x", I0, "_", I0, " = int (z''", Z0.32, "'', kind &
&= ", A, ")")') trim (s_kindnxn (n)), n, n, i,&
end do
write (*, '()')

integer :: i

do i = 0, (2 * n) - 4
write (*, '("  integer(kind = ", A, "), parameter :: bishop1_mask_",&
& I0, "x", I0, "_", I0, " = int (z''", Z0.32, "'', kind &
&= ", A, ")")') trim (s_kindnxn (n)), n, n, i,&
end do
write (*, '()')

integer :: i

do i = 0, (2 * n) - 4
write (*, '("  integer(kind = ", A, "), parameter :: bishop2_mask_",&
& I0, "x", I0, "_", I0, " = int (z''", Z0.32, "'', kind &
&= ", A, ")")') trim (s_kindnxn (n)), n, n, i,&
end do
write (*, '()')

integer :: i

do i = 0, (2 * n) - 1
write (*, '("  integer(kind = ", A, "), parameter :: rook_mask_",&
& I0, "x", I0, "_", I0, " = int (z''", Z0.32, "'', kind &
&= ", A, ")")') trim (s_kindnxn (n)), n, n, i,&
end do
write (*, '()')

integer :: i

do i = 0, (4 * n) - 7
write (*, '("  integer(kind = ", A, "), parameter :: bishop_mask_",&
& I0, "x", I0, "_", I0, " = int (z''", Z0.32, "'', kind &
&= ", A, ")")') trim (s_kindnxn (n)), n, n, i,&
end do
write (*, '()')

integer :: i

do i = 0, (6 * n) - 7
write (*, '("  integer(kind = ", A, "), parameter :: queen_mask_",&
& I0, "x", I0, "_", I0, " = int (z''", Z0.32, "'', kind &
&= ", A, ")")') trim (s_kindnxn (n)), n, n, i,&
end do
write (*, '()')

subroutine write_rooks1_attack_check
integer :: i

write (*, '("  elemental function rooks1_attack_check (army1, army2) result (attacking)")')
write (*, '("    integer(kind = ", A, "), value :: army1, army2")') trim (s_kindnxn (n))
write (*, '("    logical :: attacking")')
write (*, '()')
write (*, '("    attacking = ((iand (army1, rook1_mask_", I0, "x", I0,&
& "_0) /= 0) .and. (iand (army2, rook1_mask_", I0, "x", I0, "_0) /=&
& 0)) .or. &")') n, n, n, n
do i = 1, n - 1
write (*, '("              & ((iand (army1, rook1_mask_", I0, "x",&
& I0, "_", I0, ") /= 0) .and. (iand (army2, rook1_mask_", I0,&
& "x", I0, "_", I0, ") /= 0))")', advance = 'no') n, n, i, n, n, i
if (i /= n - 1) then
write (*, '(" .or. &")')
else
write (*, '()')
end if
end do
write (*, '("  end function rooks1_attack_check")')
write (*, '()')
end subroutine write_rooks1_attack_check

subroutine write_rooks2_attack_check
integer :: i

write (*, '("  elemental function rooks2_attack_check (army1, army2) result (attacking)")')
write (*, '("    integer(kind = ", A, "), value :: army1, army2")') trim (s_kindnxn (n))
write (*, '("    logical :: attacking")')
write (*, '()')
write (*, '("    attacking = ((iand (army1, rook2_mask_", I0, "x", I0,&
& "_0) /= 0) .and. (iand (army2, rook2_mask_", I0, "x", I0, "_0) /=&
& 0)) .or. &")') n, n, n, n
do i = 1, n - 1
write (*, '("              & ((iand (army1, rook2_mask_", I0, "x",&
& I0, "_", I0, ") /= 0) .and. (iand (army2, rook2_mask_", I0,&
& "x", I0, "_", I0, ") /= 0))")', advance = 'no') n, n, i, n, n, i
if (i /= n - 1) then
write (*, '(" .or. &")')
else
write (*, '()')
end if
end do
write (*, '("  end function rooks2_attack_check")')
write (*, '()')
end subroutine write_rooks2_attack_check

subroutine write_bishops1_attack_check
integer :: i

write (*, '("  elemental function bishops1_attack_check (army1, army2) result (attacking)")')
write (*, '("    integer(kind = ", A, "), value :: army1, army2")') trim (s_kindnxn (n))
write (*, '("    logical :: attacking")')
write (*, '()')
write (*, '("    attacking = ((iand (army1, bishop1_mask_", I0, "x", I0,&
& "_0) /= 0) .and. (iand (army2, bishop1_mask_", I0, "x", I0, "_0) /=&
& 0)) .or. &")') n, n, n, n
do i = 1, (2 * n) - 4
write (*, '("              & ((iand (army1, bishop1_mask_", I0, "x",&
& I0, "_", I0, ") /= 0) .and. (iand (army2, bishop1_mask_", I0,&
& "x", I0, "_", I0, ") /= 0))")', advance = 'no') n, n, i, n, n, i
if (i /= (2 * n) - 4) then
write (*, '(" .or. &")')
else
write (*, '()')
end if
end do
write (*, '("  end function bishops1_attack_check")')
write (*, '()')
end subroutine write_bishops1_attack_check

subroutine write_bishops2_attack_check
integer :: i

write (*, '("  elemental function bishops2_attack_check (army1, army2) result (attacking)")')
write (*, '("    integer(kind = ", A, "), value :: army1, army2")') trim (s_kindnxn (n))
write (*, '("    logical :: attacking")')
write (*, '()')
write (*, '("    attacking = ((iand (army1, bishop2_mask_", I0, "x", I0,&
& "_0) /= 0) .and. (iand (army2, bishop2_mask_", I0, "x", I0, "_0) /=&
& 0)) .or. &")') n, n, n, n
do i = 1, (2 * n) - 4
write (*, '("              & ((iand (army1, bishop2_mask_", I0, "x",&
& I0, "_", I0, ") /= 0) .and. (iand (army2, bishop2_mask_", I0,&
& "x", I0, "_", I0, ") /= 0))")', advance = 'no') n, n, i, n, n, i
if (i /= (2 * n) - 4) then
write (*, '(" .or. &")')
else
write (*, '()')
end if
end do
write (*, '("  end function bishops2_attack_check")')
write (*, '()')
end subroutine write_bishops2_attack_check

subroutine write_rooks_attack_check
integer :: i

write (*, '("  elemental function rooks_attack_check (army1, army2) result (attacking)")')
write (*, '("    integer(kind = ", A, "), value :: army1, army2")') trim (s_kindnxn (n))
write (*, '("    logical :: attacking")')
write (*, '()')
write (*, '("    attacking = ((iand (army1, rook_mask_", I0, "x", I0,&
& "_0) /= 0) .and. (iand (army2, rook_mask_", I0, "x", I0, "_0) /=&
& 0)) .or. &")') n, n, n, n
do i = 1, (2 * n) - 1
write (*, '("              & ((iand (army1, rook_mask_", I0, "x",&
& I0, "_", I0, ") /= 0) .and. (iand (army2, rook_mask_", I0,&
& "x", I0, "_", I0, ") /= 0))")', advance = 'no') n, n, i, n, n, i
if (i /= (2 * n) - 1) then
write (*, '(" .or. &")')
else
write (*, '()')
end if
end do
write (*, '("  end function rooks_attack_check")')
write (*, '()')
end subroutine write_rooks_attack_check

subroutine write_bishops_attack_check
integer :: i

write (*, '("  elemental function bishops_attack_check (army1, army2) result (attacking)")')
write (*, '("    integer(kind = ", A, "), value :: army1, army2")') trim (s_kindnxn (n))
write (*, '("    logical :: attacking")')
write (*, '()')
write (*, '("    attacking = ((iand (army1, bishop_mask_", I0, "x", I0,&
& "_0) /= 0) .and. (iand (army2, bishop_mask_", I0, "x", I0, "_0) /=&
& 0)) .or. &")') n, n, n, n
do i = 1, (4 * n) - 7
write (*, '("              & ((iand (army1, bishop_mask_", I0, "x",&
& I0, "_", I0, ") /= 0) .and. (iand (army2, bishop_mask_", I0,&
& "x", I0, "_", I0, ") /= 0))")', advance = 'no') n, n, i, n, n, i
if (i /= (4 * n) - 7) then
write (*, '(" .or. &")')
else
write (*, '()')
end if
end do
write (*, '("  end function bishops_attack_check")')
write (*, '()')
end subroutine write_bishops_attack_check

subroutine write_queens_attack_check
integer :: i

write (*, '("  elemental function queens_attack_check (army1, army2) result (attacking)")')
write (*, '("    integer(kind = ", A, "), value :: army1, army2")') trim (s_kindnxn (n))
write (*, '("    logical :: attacking")')
write (*, '()')
write (*, '("    attacking = ((iand (army1, queen_mask_", I0, "x", I0,&
& "_0) /= 0) .and. (iand (army2, queen_mask_", I0, "x", I0, "_0) /=&
& 0)) .or. &")') n, n, n, n
do i = 1, (6 * n) - 7
write (*, '("              & ((iand (army1, queen_mask_", I0, "x",&
& I0, "_", I0, ") /= 0) .and. (iand (army2, queen_mask_", I0,&
& "x", I0, "_", I0, ") /= 0))")', advance = 'no') n, n, i, n, n, i
if (i /= (6 * n) - 7) then
write (*, '(" .or. &")')
else
write (*, '()')
end if
end do
write (*, '("  end function queens_attack_check")')
write (*, '()')
end subroutine write_queens_attack_check

subroutine write_board_rotate90
integer :: i, j

write (*, '("  elemental function board_rotate90 (a) result (b)")')
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
write (*, '("    ! Rotation 90 degrees in one of the orientations.")')
write (*, '()')
do i = 0, n - 1
if (i == 0) then
write (*, '("    b = ")', advance = 'no')
else
write (*, '("      & ")', advance = 'no')
do j = 1, i
write (*, '("  ")', advance = 'no')
end do
end if
if (i /= n - 1) then
write (*, '("ior (ishft (reverse_insert_zeros_", I0, " (ishft&
& (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ",&
& I0, ")), ", I0, "), &")') n, n, n, i, -i * n, i
else
write (*, '("   ishft (reverse_insert_zeros_", I0, " (ishft&
& (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ",&
& I0, ")), ", I0, ")")', advance = 'no') n, n, n, i, -i * n, i
do j = 1, n - 1
write (*, '(")")', advance = 'no')
end do
write (*, '()')
end if
end do
write (*, '("  end function board_rotate90")')
write (*, '()')
end subroutine write_board_rotate90

subroutine write_board_rotate180
write (*, '("  elemental function board_rotate180 (a) result (b)")')
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
write (*, '("    ! Rotation 180 degrees.")')
write (*, '()')
write (*, '("    b = board_reflect1 (board_reflect2 (a))")')
write (*, '("  end function board_rotate180")')
write (*, '()')
end subroutine write_board_rotate180

subroutine write_board_rotate270
integer :: i, j

write (*, '("  elemental function board_rotate270 (a) result (b)")')
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
write (*, '("    ! Rotation 270 degrees in one of the orientations.")')
write (*, '()')
do i = 0, n - 1
if (i == 0) then
write (*, '("    b = ")', advance = 'no')
else
write (*, '("      & ")', advance = 'no')
do j = 1, i
write (*, '("  ")', advance = 'no')
end do
end if
if (i /= n - 1) then
write (*, '("ior (ishft (insert_zeros_", I0, " (ishft&
& (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ",&
& I0, ")), ", I0, "), &")') n, n, n, i, -i * n, n - 1 - i
else
write (*, '("   ishft (insert_zeros_", I0, " (ishft&
& (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ",&
& I0, ")), ", I0, ")")', advance = 'no') n, n, n, i, -i * n, n - 1 - i
do j = 1, n - 1
write (*, '(")")', advance = 'no')
end do
write (*, '()')
end if
end do
write (*, '("  end function board_rotate270")')
write (*, '()')
end subroutine write_board_rotate270

subroutine write_board_reflect1
integer :: i, j

write (*, '("  elemental function board_reflect1 (a) result (b)")')
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
write (*, '("    ! Reflection of rows or columns.")')
write (*, '()')
do i = 0, n - 1
if (i == 0) then
write (*, '("    b = ")', advance = 'no')
else
write (*, '("      & ")', advance = 'no')
do j = 1, i
write (*, '("     ")', advance = 'no')
end do
end if
if (i /= n - 1) then
write (*, '("ior (ishft (iand (rook2_mask_", I0, "x", I0, "_", I0, ", a), ", I0, "), &")') &
& n, n, i, (n - 1) - (2 * i)
else
write (*, '("ishft (iand (rook2_mask_", I0, "x", I0, "_", I0, ", a), ", I0, ")")', advance = 'no') &
& n, n, i, (n - 1) - (2 * i)
do j = 1, n - 1
write (*, '(")")', advance = 'no')
end do
write (*, '()')
end if
end do
write (*, '("  end function board_reflect1")')
write (*, '()')
end subroutine write_board_reflect1

subroutine write_board_reflect2
integer :: i, j

write (*, '("  elemental function board_reflect2 (a) result (b)")')
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
write (*, '("    ! Reflection of rows or columns.")')
write (*, '()')
do i = 0, n - 1
if (i == 0) then
write (*, '("    b = ")', advance = 'no')
else
write (*, '("      & ")', advance = 'no')
do j = 1, i
write (*, '("     ")', advance = 'no')
end do
end if
if (i /= n - 1) then
write (*, '("ior (ishft (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ", I0, "), &")') &
& n, n, i, n * ((n - 1) - (2 * i))
else
write (*, '("ishft (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ", I0, ")")', advance = 'no') &
& n, n, i, n * ((n - 1) - (2 * i))
do j = 1, n - 1
write (*, '(")")', advance = 'no')
end do
write (*, '()')
end if
end do
write (*, '("  end function board_reflect2")')
write (*, '()')
end subroutine write_board_reflect2

subroutine write_board_reflect3
integer :: i, j

write (*, '("  elemental function board_reflect3 (a) result (b)")')
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
write (*, '("    ! Reflection around one of the two main diagonals.")')
write (*, '()')
do i = 0, n - 1
if (i == 0) then
write (*, '("    b = ")', advance = 'no')
else
write (*, '("      & ")', advance = 'no')
do j = 1, i
write (*, '("  ")', advance = 'no')
end do
end if
if (i /= n - 1) then
write (*, '("ior (ishft (insert_zeros_", I0, " (ishft&
& (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ",&
& I0, ")), ", I0, "), &")') n, n, n, i, -i * n, i
else
write (*, '("   ishft (insert_zeros_", I0, " (ishft&
& (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ",&
& I0, ")), ", I0, ")")', advance = 'no') n, n, n, i, -i * n, i
do j = 1, n - 1
write (*, '(")")', advance = 'no')
end do
write (*, '()')
end if
end do
write (*, '("  end function board_reflect3")')
write (*, '()')
end subroutine write_board_reflect3

subroutine write_board_reflect4
integer :: i, j

write (*, '("  elemental function board_reflect4 (a) result (b)")')
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
write (*, '("    ! Reflection around one of the two main diagonals.")')
write (*, '()')
do i = 0, n - 1
if (i == 0) then
write (*, '("    b = ")', advance = 'no')
else
write (*, '("      & ")', advance = 'no')
do j = 1, i
write (*, '("  ")', advance = 'no')
end do
end if
if (i /= n - 1) then
write (*, '("ior (ishft (reverse_insert_zeros_", I0, " (ishft&
& (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ",&
& I0, ")), ", I0, "), &")') n, n, n, i, -i * n, n - 1 - i
else
write (*, '("   ishft (reverse_insert_zeros_", I0, " (ishft&
& (iand (rook1_mask_", I0, "x", I0, "_", I0, ", a), ",&
& I0, ")), ", I0, ")")', advance = 'no') n, n, n, i, -i * n, n - 1 - i
do j = 1, n - 1
write (*, '(")")', advance = 'no')
end do
write (*, '()')
end if
end do
write (*, '("  end function board_reflect4")')
write (*, '()')
end subroutine write_board_reflect4

subroutine write_insert_zeros
integer :: i, j

write (*, '("  elemental function insert_zeros_", I0, " (a) result (b)")') n
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
do i = 0, n - 1
if (i == 0) then
write (*, '("    b = ")', advance = 'no')
else
write (*, '("      & ")', advance = 'no')
do j = 1, i
write (*, '("     ")', advance = 'no')
end do
end if
if (i /= n - 1) then
write (*, '("ior (ishft (ibits (a, ", I0, ", 1), ", I0, "), &")') i, i * n
else
write (*, '("ishft (ibits (a, ", I0, ", 1), ", I0, ")")', advance = 'no') i, i * n
do j = 1, n - 1
write (*, '(")")', advance = 'no')
end do
write (*, '()')
end if
end do
write (*, '("  end function insert_zeros_", I0)') n
write (*, '()')
end subroutine write_insert_zeros

subroutine write_reverse_insert_zeros
integer :: i, j

write (*, '("  elemental function reverse_insert_zeros_", I0, " (a) result (b)")') n
write (*, '("    integer(kind = ", A, "), value :: a")') trim (s_kindnxn (n))
write (*, '("    integer(kind = ", A, ") :: b")') trim (s_kindnxn (n))
write (*, '()')
do i = 0, n - 1
if (i == 0) then
write (*, '("    b = ")', advance = 'no')
else
write (*, '("      & ")', advance = 'no')
do j = 1, i
write (*, '("     ")', advance = 'no')
end do
end if
if (i /= n - 1) then
write (*, '("ior (ishft (ibits (a, ", I0, ", 1), ", I0, "), &")') n - 1 - i, i * n
else
write (*, '("ishft (ibits (a, ", I0, ", 1), ", I0, ")")', advance = 'no') n - 1 - i, i * n
do j = 1, n - 1
write (*, '(")")', advance = 'no')
end do
write (*, '()')
end if
end do
write (*, '("  end function reverse_insert_zeros_", I0)') n
write (*, '()')
end subroutine write_reverse_insert_zeros

function s_kindnxn (n) result (s)
integer, intent(in) :: n
character(len = 16) :: s

if (n <= 8) then
s = s_kind8x8
else
s = s_kind11x11
end if
end function s_kindnxn

integer, intent(in) :: n

integer, intent(in) :: n

integer :: i

mask = (two ** n) - 1
do i = 0, n - 1
end do

integer, intent(in) :: n

integer :: i

do i = 0, n - 1
end do
do i = 0, n - 1
end do

integer, intent(in) :: n

integer :: i, j, k

! Masks for diagonals. Put them in order from most densely
! populated to least densely populated.

do k = 0, n - 2
do i = k, n - 1
j = i - k
end do
if (k == 0) then
else
end if
end do

integer, intent(in) :: n

integer :: i, j, k
integer :: i1, j1

! Masks for skew diagonals. Put them in order from most densely
! populated to least densely populated.

do k = 0, n - 2
do i = k, n - 1
j = i - k
i1 = n - 1 - i
j1 = n - 1 - j
end do
if (k == 0) then
else
end if
end do

integer, intent(in) :: n

integer, intent(in) :: n

integer :: i

! Put the masks in order from most densely populated to least
! densely populated.

do i = 0, (2 * n) - 4
end do

integer, intent(in) :: n

end program peaceful_queens_elements_generator
```

Here is the second program, peaceful_queens.f90:

```module peaceful_queens_support
use, non_intrinsic :: peaceful_queens_elements

implicit none
private

public :: write_board
public :: write_board_without_spaces
public :: write_board_with_spaces

public :: save_a_solution

interface write_board
module procedure write_board_without_spaces
module procedure write_board_with_spaces
end interface write_board

contains

subroutine write_board_without_spaces (unit, army_b, army_w)
integer, intent(in) :: unit
integer(kind = board_kind), intent(in) :: army_b, army_w

call write_board_with_spaces (unit, army_b, army_w, 0)
end subroutine write_board_without_spaces

subroutine write_board_with_spaces (unit, army_b, army_w, num_spaces)
integer, intent(in) :: unit
integer(kind = board_kind), intent(in) :: army_b, army_w
integer, intent(in) :: num_spaces

integer(kind = board_kind), parameter :: zero = 0
integer(kind = board_kind), parameter :: one = 1

integer :: i, j
integer(kind = board_kind) :: rank_b, rank_w

character(1), allocatable :: queens(:)
character(4), allocatable :: rules(:)
character(1), allocatable :: spaces(:)

allocate (queens(0 : n - 1))
allocate (rules(0 : n - 1))
allocate (spaces(1 : num_spaces))

rules = "----"
if (0 < num_spaces) then
spaces = " "             ! For putting spaces after newlines.
end if

mask = not (ishft (not (zero), n))
write (unit, '("+", 100(A4, "+"))') rules
do i = 0, n - 1
rank_b = iand (mask, ishft (army_b, -i * n))
rank_w = iand (mask, ishft (army_w, -i * n))
do j = 0, n - 1
if (iand (rank_b, ishft (one, j)) /= 0) then
queens(j) = "B"
else if (iand (rank_w, ishft (one, j)) /= 0) then
queens(j) = "W"
else
queens(j) = " "
end if
end do
write (unit, '(100A1)', advance = 'no') spaces
write (unit, '("|", 100(A3, " |"))') queens
write (unit, '(100A1)', advance = 'no') spaces
if (i /= n - 1) then
write (unit, '("+", 100(A4, "+"))') rules
else
write (unit, '("+", 100(A4, "+"))', advance = 'no') rules
end if
end do
end subroutine write_board_with_spaces

subroutine save_a_solution (army1, army2, num_solutions, armies1, armies2)
integer(kind = board_kind), intent(in) :: army1, army2
integer, intent(inout) :: num_solutions
integer(kind = board_kind), intent(inout) :: armies1(1:8, 1:max_solutions)
integer(kind = board_kind), intent(inout) :: armies2(1:8, 1:max_solutions)

! A sanity check.
if (queens_attack_check (army1, army2)) then
error stop
end if

num_solutions = num_solutions + 1

armies1(1, num_solutions) = army1
armies1(2, num_solutions) = board_rotate90 (army1)
armies1(3, num_solutions) = board_rotate180 (army1)
armies1(4, num_solutions) = board_rotate270 (army1)
armies1(5, num_solutions) = board_reflect1 (army1)
armies1(6, num_solutions) = board_reflect2 (army1)
armies1(7, num_solutions) = board_reflect3 (army1)
armies1(8, num_solutions) = board_reflect4 (army1)

armies2(1, num_solutions) = army2
armies2(2, num_solutions) = board_rotate90 (army2)
armies2(3, num_solutions) = board_rotate180 (army2)
armies2(4, num_solutions) = board_rotate270 (army2)
armies2(5, num_solutions) = board_reflect1 (army2)
armies2(6, num_solutions) = board_reflect2 (army2)
armies2(7, num_solutions) = board_reflect3 (army2)
armies2(8, num_solutions) = board_reflect4 (army2)
end subroutine save_a_solution

end module peaceful_queens_support

module peaceful_queens_solver
use, non_intrinsic :: peaceful_queens_elements
use, non_intrinsic :: peaceful_queens_support

implicit none
private

public :: solve_peaceful_queens

integer(kind = board_kind), parameter :: zero = 0_board_kind
integer(kind = board_kind), parameter :: one = 1_board_kind
integer(kind = board_kind), parameter :: two = 2_board_kind

contains

subroutine solve_peaceful_queens (unit, show_equivalents, &
&                            num_solutions, armies1, armies2)
integer, intent(in) :: unit
logical, intent(in) :: show_equivalents
integer, intent(out) :: num_solutions
integer(kind = board_kind), intent(out) :: armies1(1:8, 1:max_solutions)
integer(kind = board_kind), intent(out) :: armies2(1:8, 1:max_solutions)

call solve (zero, 0, 0, zero, 0, 0, 0)

contains

recursive subroutine solve (army1, rooklike11, rooklike12, &
&                      army2, rooklike21, rooklike22, index)
integer(kind = board_kind), value :: army1
integer, value :: rooklike11, rooklike12
integer(kind = board_kind), value :: army2
integer, value :: rooklike21, rooklike22
integer, value :: index

integer :: num_queens1
integer :: num_queens2
integer(kind = board_kind) :: new_army
integer(kind = board_kind) :: new_army_reversed
integer :: bit1, bit2
logical :: skip

num_queens1 = popcnt (army1)
num_queens2 = popcnt (army2)

if (num_queens1 + num_queens2 == 2 * m) then
if (.not. is_a_duplicate (army1, army2, num_solutions, armies1, armies2)) then
call save_a_solution (army1, army2, num_solutions, armies1, armies2)
write (unit, '("Solution ", I0)') num_solutions
call write_board (unit, army1, army2)
write (unit, '()')
write (unit, '()')
call optionally_write_equivalents
end if
else if (num_queens1 - num_queens2 == 0) then
! It is time to add a queen to army1.
do while (num_solutions < max_solutions .and. index /= n**2)
skip = .false.
new_army = ior (army1, ishft (one, index))
if (new_army == army1) then
skip = .true.
else if (index < n) then
new_army_reversed = board_reflect1 (new_army)
if (new_army_reversed < new_army) then
! Skip a bunch of board_reflect1 equivalents.
skip = .true.
end if
end if
if (skip) then
index = index + 1
else
bit1 = ishft (1, index / n)
bit2 = ishft (1, mod (index, n))
if (iand (rooklike21, bit1) /= 0) then
index = round_up_to_multiple (index + 1, n)
else if (iand (rooklike22, bit2) /= 0) then
index = index + 1
else if (bishops_attack_check (new_army, army2)) then
index = index + 1
else
call solve (new_army, &
&      ior (rooklike11, bit1), &
&      ior (rooklike12, bit2), &
&      army2, rooklike21, rooklike22, &
&      n)
index = index + 1
end if
end if
end do
else
! It is time to add a queen to army2.
do while (num_solutions < max_solutions .and. index /= n**2)
new_army = ior (army2, ishft (one, index))
skip = (new_army == army2)
if (skip) then
index = index + 1
else
bit1 = ishft (1, index / n)
bit2 = ishft (1, mod (index, n))
if (iand (rooklike11, bit1) /= 0) then
index = round_up_to_multiple (index + 1, n)
else if (iand (rooklike12, bit2) /= 0) then
index = index + 1
else if (bishops_attack_check (army1, new_army)) then
index = index + 1
else
call solve (army1, rooklike11, rooklike12, &
&      new_army, &
&      ior (rooklike21, bit1), &
&      ior (rooklike22, bit2), &
&      0)
index = index + 1
end if
end if
end do
end if
end subroutine solve

subroutine optionally_write_equivalents
integer :: i

if (show_equivalents) then
write (unit, '(5X)', advance = 'no')
write (unit, '("Equivalents")')

write (unit, '(5X)', advance = 'no')
call write_board (unit, armies2(1, num_solutions), armies1(1, num_solutions), 5)
write (unit, '()')
write (unit, '()')

do i = 2, 5
if (all ((armies1(i, num_solutions) /= armies1(1 : i - 1, num_solutions) .or. &
&    armies2(i, num_solutions) /= armies2(1 : i - 1, num_solutions)) .and. &
&   (armies2(i, num_solutions) /= armies1(1 : i - 1, num_solutions) .or. &
&    armies1(i, num_solutions) /= armies2(1 : i - 1, num_solutions)))) then
write (unit, '(5X)', advance = 'no')
call write_board (unit, armies1(i, num_solutions), armies2(i, num_solutions), 5)
write (unit, '()')
write (unit, '()')
write (unit, '(5X)', advance = 'no')
call write_board (unit, armies2(i, num_solutions), armies1(i, num_solutions), 5)
write (unit, '()')
write (unit, '()')
end if
end do
end if
end subroutine optionally_write_equivalents

end subroutine solve_peaceful_queens

elemental function round_up_to_multiple (x, n) result (y)
integer, value :: x, n
integer :: y

y = x + mod (n - mod (x, n), n)
end function round_up_to_multiple

pure function is_a_duplicate (army1, army2, num_solutions, armies1, armies2) result (is_dup)
integer(kind = board_kind), intent(in) :: army1, army2
integer, intent(in) :: num_solutions
integer(kind = board_kind), intent(in) :: armies1(1:8, 1:max_solutions)
integer(kind = board_kind), intent(in) :: armies2(1:8, 1:max_solutions)
logical :: is_dup

is_dup = any ((army1 == armies1(:, 1:num_solutions) .and. &
&         army2 == armies2(:, 1:num_solutions)) .or. &
&        (army2 == armies1(:, 1:num_solutions) .and. &
&         army1 == armies2(:, 1:num_solutions)))
end function is_a_duplicate

end module peaceful_queens_solver

program peaceful_queens
use, intrinsic :: iso_fortran_env, only: output_unit
use, non_intrinsic :: peaceful_queens_elements
use, non_intrinsic :: peaceful_queens_support
use, non_intrinsic :: peaceful_queens_solver

implicit none

integer :: num_solutions
logical :: show_equivalents
integer(kind = board_kind) :: armies1(1:8, 1:max_solutions)
integer(kind = board_kind) :: armies2(1:8, 1:max_solutions)

integer :: arg_count
character(len = 200) :: arg

show_equivalents = .false.

arg_count = command_argument_count ()
if (1 <= arg_count) then
call get_command_argument (1, arg)
select case (trim (arg))
case ('1', 't', 'T', 'true', 'y', 'Y', 'yes')
show_equivalents = .true.
end select
end if

call solve_peaceful_queens (output_unit, show_equivalents, &
&                      num_solutions, armies1, armies2)

end program peaceful_queens
```

Here is the driver script:

```#!/bin/sh
#
# Driver script for peaceful_queens in Fortran.
#

if test \${ZSH_VERSION+y} && (emulate sh) >/dev/null 2>&1; then
emulate sh
fi

if test \$# -ne 3 && test \$# -ne 4; then
echo "Usage: \$0 M N MAX_SOLUTIONS [SHOW_EQUIVALENTS]"
exit 1
fi

M=\${1}
N=\${2}
MAX_SOLUTIONS=\${3}
SHOW_EQUIVALENTS=\${4}

RM_GENERATED_SRC=yes
CHECK=no

case \${CHECK} in
0 | f | F | false | N | n | no) FCCHECK="" ;;
1 | t | T | true | Y | y | yes) FCCHECK="-fcheck=all" ;;
*) echo 'CHECK is set incorrectly';
exit 1 ;;
esac

FC="gfortran"
FCFLAGS="-std=f2018 -g -O3 -march=native -fno-stack-protector -Wall -Wextra \${FCCHECK}"

# If you have the graphite optimizer, here are some marginally helpful
# flags. They barely make a difference, for me.
FCFLAGS="\${FCFLAGS} -funroll-loops -floop-nest-optimize"

RUN_IT="yes"

\${FC} -o peaceful_queens_elements_generator peaceful_queens_elements_generator.f90 &&
./peaceful_queens_elements_generator \${M} \${N} \${MAX_SOLUTIONS} > peaceful_queens_elements.f90 &&
\${FC} \${FCFLAGS} -c peaceful_queens_elements.f90 &&
if test x"\${RM_GENERATED_SRC}" != xno; then rm -f peaceful_queens_elements.f90; fi &&
\${FC} \${FCFLAGS} -c peaceful_queens.f90 &&
\${FC} \${FCFLAGS} -o peaceful_queens peaceful_queens_elements.o peaceful_queens.o &&
if test x"\${RUN_IT}" = xyes; then time ./peaceful_queens \${SHOW_EQUIVALENTS}; else :; fi
```
Output:

\$ ./peaceful_queens-fortran-driver.sh 4 5 1000 T

```Solution 1
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+

Equivalents
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+

Solution 2
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+

Equivalents
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+

+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+

+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+

+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+

+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+

+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+

+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+

Solution 3
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+

Equivalents
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+

+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+

+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+

+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+

+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+

+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+

+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
```

On my computer, the program can find all the solutions of m=5, n=6, and eliminate any other possibilities, in under 5 seconds. The m=7, n=7 case took about 4.25 hours, mostly eliminating other possibilities. The next thing to try would be m=9, n=8, but probably a faster program is called for, there.

It would be instructive to save and examine the generated peaceful_queens_elements.f90 files. I leave that as an exercise for the reader. :)

## FreeBASIC

```Type posicion
x As Integer
y As Integer
End Type

Type pieza
empty As Integer
black As Integer
white As Integer
End Type

Function isAttacking(q As posicion, posic As posicion) As Integer
Return (q.x = posic.x Or q.y = posic.y Or Abs(q.x - posic.x) = Abs(q.y - posic.y))
End Function

Sub place(m As Integer, n As Integer, blackQueens() As posicion, whiteQueens() As posicion, Byref result As Integer)
If m = 0 Then
result = -1
Exit Sub
End If
Dim As Integer placingBlack = -1
Dim As Integer i, j, k, equalposicion
Dim As Boolean inner

For i = 0 To n-1
For j = 0 To n-1
Dim As posicion posic = Type<posicion>(i, j)
inner = False
For k = Lbound(blackQueens) To Ubound(blackQueens)
equalposicion = (blackQueens(k).x = posic.x And blackQueens(k).y = posic.y)
If equalposicion Or (Not placingBlack And isAttacking(blackQueens(k), posic)) Then
inner = True
Exit For
End If
Next
If Not inner Then
For k = Lbound(whiteQueens) To Ubound(whiteQueens)
equalposicion = (whiteQueens(k).x = posic.x And whiteQueens(k).y = posic.y)
If equalposicion Or (placingBlack And isAttacking(whiteQueens(k), posic)) Then
inner = True
Exit For
End If
Next
If Not inner Then
If placingBlack Then
Redim Preserve blackQueens(Ubound(blackQueens) + 1)
blackQueens(Ubound(blackQueens)) = posic
placingBlack = 0
Else
Redim Preserve whiteQueens(Ubound(whiteQueens) + 1)
whiteQueens(Ubound(whiteQueens)) = posic
place(m-1, n, blackQueens(), whiteQueens(), result)
If result Then Exit Sub
Redim Preserve blackQueens(Ubound(blackQueens) - 1)
Redim Preserve whiteQueens(Ubound(whiteQueens) - 1)
placingBlack = -1
End If
End If
End If
Next
Next
If Not placingBlack Then Redim Preserve blackQueens(Ubound(blackQueens) - 1)
result = 0
End Sub

Sub printBoard(n As Integer, blackQueens() As posicion, whiteQueens() As posicion)
Dim As Integer board(n * n)
Dim As Integer i, j, k

For i = Lbound(blackQueens) To Ubound(blackQueens)
board(blackQueens(i).x * n + blackQueens(i).y) = 1
Next
For i = Lbound(whiteQueens) To Ubound(whiteQueens)
board(whiteQueens(i).x * n + whiteQueens(i).y) = 2
Next
For i = 0 To n*n-1
If i Mod n = 0 And i <> 0 Then Print
Select Case board(i)
Case 1
Print "B ";
Case 2
Print "W ";
Case Else
j = i \ n
k = i - j * n
If j Mod 2 = k Mod 2 Then
Print Chr(253); " ";
Else
Print Chr(252); " ";
End If
End Select
Next i
Print
End Sub

Dim As posicion nms(23) = { _
Type<posicion>(2, 1), Type<posicion>(3, 1), Type<posicion>(3, 2), Type<posicion>(4, 1), Type<posicion>(4, 2), Type<posicion>(4, 3), _
Type<posicion>(5, 1), Type<posicion>(5, 2), Type<posicion>(5, 3), Type<posicion>(5, 4), Type<posicion>(5, 5), _
Type<posicion>(6, 1), Type<posicion>(6, 2), Type<posicion>(6, 3), Type<posicion>(6, 4), Type<posicion>(6, 5), Type<posicion>(6, 6), _
Type<posicion>(7, 1), Type<posicion>(7, 2), Type<posicion>(7, 3), Type<posicion>(7, 4), Type<posicion>(7, 5), Type<posicion>(7, 6), Type<posicion>(7, 7) }

For i As Integer = Lbound(nms) To Ubound(nms)
Print Chr(10); nms(i).y; " black and "; nms(i).y; " white queens on a "; nms(i).x; " x "; nms(i).x; " board:"
Dim As posicion blackQueens(0)
Dim As posicion whiteQueens(0)
Dim As Integer result
place(nms(i).y, nms(i).x, blackQueens(), whiteQueens(), result)
If result Then
printBoard(nms(i).x, blackQueens(), whiteQueens())
Else
Print "No solution exists."
End If
Next i

Sleep
```
Output:
``` 1 black and  1 white queens on a  2 x  2 board:
No solution exists.

1 black and  1 white queens on a  3 x  3 board:
² ³ ²
³ ² B
W ³ ²

2 black and  2 white queens on a  3 x  3 board:
² B ²
³ ² ³
² ³ W

1 black and  1 white queens on a  4 x  4 board:
² ³ ² ³
³ ² B ²
W ³ ² ³
³ ² ³ ²

2 black and  2 white queens on a  4 x  4 board:
² B ² ³
³ ² ³ W
² ³ ² ³
³ ² ³ ²

3 black and  3 white queens on a  4 x  4 board:
B ³ ² ³
³ ² W ²
² ³ ² ³
³ ² ³ ²

1 black and  1 white queens on a  5 x  5 board:
² ³ ² ³ ²
³ ² B ² ³
W ³ ² ³ ²
³ ² ³ ² ³
² ³ ² ³ ²

2 black and  2 white queens on a  5 x  5 board:
² B ² ³ ²
³ ² ³ W ³
² ³ ² ³ ²
³ ² ³ ² ³
² ³ ² ³ ²

3 black and  3 white queens on a  5 x  5 board:
B ³ ² ³ ²
³ ² W ² ³
² ³ ² ³ ²
³ ² ³ ² ³
² ³ ² ³ ²

4 black and  4 white queens on a  5 x  5 board:
² ³ ² ³ B
W ² ³ ² ³
² ³ ² ³ ²
³ ² ³ ² ³
² ³ ² ³ ²

5 black and  5 white queens on a  5 x  5 board:
² ³ B ³ ²
³ ² ³ ² W
² ³ ² ³ ²
³ ² ³ ² ³
² ³ ² ³ ²

1 black and  1 white queens on a  6 x  6 board:
² ³ ² ³ ² ³
³ ² B ² ³ ²
W ³ ² ³ ² ³
³ ² ³ ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²

2 black and  2 white queens on a  6 x  6 board:
² B ² ³ ² ³
³ ² ³ W ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²

3 black and  3 white queens on a  6 x  6 board:
B ³ ² ³ ² ³
³ ² W ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²

4 black and  4 white queens on a  6 x  6 board:
² ³ ² ³ B ³
W ² ³ ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²

5 black and  5 white queens on a  6 x  6 board:
² ³ B ³ ² ³
³ ² ³ ² W ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²

6 black and  6 white queens on a  6 x  6 board:
B ³ ² ³ ² ³
³ ² W ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²
² ³ ² ³ ² ³
³ ² ³ ² ³ ²

1 black and  1 white queens on a  7 x  7 board:
² ³ ² ³ ² ³ ²
³ ² B ² ³ ² ³
W ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²

2 black and  2 white queens on a  7 x  7 board:
² B ² ³ ² ³ ²
³ ² ³ W ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²

3 black and  3 white queens on a  7 x  7 board:
B ³ ² ³ ² ³ ²
³ ² W ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²

4 black and  4 white queens on a  7 x  7 board:
² ³ ² ³ B ³ ²
W ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²

5 black and  5 white queens on a  7 x  7 board:
² ³ B ³ ² ³ ²
³ ² ³ ² W ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²

6 black and  6 white queens on a  7 x  7 board:
B ³ ² ³ ² ³ ²
³ ² W ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²

7 black and  7 white queens on a  7 x  7 board:
² ³ ² ³ B ³ ²
W ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²
³ ² ³ ² ³ ² ³
² ³ ² ³ ² ³ ²```

## Go

This is based on the C# code here.

Textual rather than HTML output. Whilst the unicode symbols for the black and white queens are recognized by the Ubuntu 16.04 terminal, I found it hard to visually distinguish between them so I've used 'B' and 'W' instead.

```package main

import "fmt"

const (
empty = iota
black
white
)

const (
bqueen  = 'B'
wqueen  = 'W'
bbullet = '•'
wbullet = '◦'
)

type position struct{ i, j int }

func iabs(i int) int {
if i < 0 {
return -i
}
return i
}

func place(m, n int, pBlackQueens, pWhiteQueens *[]position) bool {
if m == 0 {
return true
}
placingBlack := true
for i := 0; i < n; i++ {
inner:
for j := 0; j < n; j++ {
pos := position{i, j}
for _, queen := range *pBlackQueens {
if queen == pos || !placingBlack && isAttacking(queen, pos) {
continue inner
}
}
for _, queen := range *pWhiteQueens {
if queen == pos || placingBlack && isAttacking(queen, pos) {
continue inner
}
}
if placingBlack {
*pBlackQueens = append(*pBlackQueens, pos)
placingBlack = false
} else {
*pWhiteQueens = append(*pWhiteQueens, pos)
if place(m-1, n, pBlackQueens, pWhiteQueens) {
return true
}
*pBlackQueens = (*pBlackQueens)[0 : len(*pBlackQueens)-1]
*pWhiteQueens = (*pWhiteQueens)[0 : len(*pWhiteQueens)-1]
placingBlack = true
}
}
}
if !placingBlack {
*pBlackQueens = (*pBlackQueens)[0 : len(*pBlackQueens)-1]
}
return false
}

func isAttacking(queen, pos position) bool {
if queen.i == pos.i {
return true
}
if queen.j == pos.j {
return true
}
if iabs(queen.i-pos.i) == iabs(queen.j-pos.j) {
return true
}
return false
}

func printBoard(n int, blackQueens, whiteQueens []position) {
board := make([]int, n*n)
for _, queen := range blackQueens {
board[queen.i*n+queen.j] = black
}
for _, queen := range whiteQueens {
board[queen.i*n+queen.j] = white
}

for i, b := range board {
if i != 0 && i%n == 0 {
fmt.Println()
}
switch b {
case black:
fmt.Printf("%c ", bqueen)
case white:
fmt.Printf("%c ", wqueen)
case empty:
if i%2 == 0 {
fmt.Printf("%c ", bbullet)
} else {
fmt.Printf("%c ", wbullet)
}
}
}
fmt.Println("\n")
}

func main() {
nms := [][2]int{
{2, 1}, {3, 1}, {3, 2}, {4, 1}, {4, 2}, {4, 3},
{5, 1}, {5, 2}, {5, 3}, {5, 4}, {5, 5},
{6, 1}, {6, 2}, {6, 3}, {6, 4}, {6, 5}, {6, 6},
{7, 1}, {7, 2}, {7, 3}, {7, 4}, {7, 5}, {7, 6}, {7, 7},
}
for _, nm := range nms {
n, m := nm[0], nm[1]
fmt.Printf("%d black and %d white queens on a %d x %d board:\n", m, m, n, n)
var blackQueens, whiteQueens []position
if place(m, n, &blackQueens, &whiteQueens) {
printBoard(n, blackQueens, whiteQueens)
} else {
fmt.Println("No solution exists.\n")
}
}
}
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B ◦ •
◦ • W
• ◦ •

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B ◦ • ◦
• ◦ W ◦
• ◦ • ◦
• ◦ • ◦

2 black and 2 white queens on a 4 x 4 board:
B ◦ • ◦
• ◦ W ◦
B ◦ • ◦
• ◦ W ◦

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • W • ◦
• ◦ • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

2 black and 2 white queens on a 5 x 5 board:
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

3 black and 3 white queens on a 5 x 5 board:
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ B ◦
• W • ◦ •

4 black and 4 white queens on a 5 x 5 board:
• B • B •
◦ • ◦ • B
W ◦ W ◦ •
◦ • ◦ • B
W ◦ W ◦ •

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
• ◦ W ◦ • ◦
• ◦ • ◦ • ◦
• ◦ • ◦ • ◦
• ◦ • ◦ • ◦
• ◦ • ◦ • ◦

2 black and 2 white queens on a 6 x 6 board:
B ◦ • ◦ B ◦
• ◦ W ◦ • ◦
• W • ◦ • ◦
• ◦ • ◦ • ◦
• ◦ • ◦ • ◦
• ◦ • ◦ • ◦

3 black and 3 white queens on a 6 x 6 board:
B ◦ • ◦ B B
• ◦ W ◦ • ◦
• W • ◦ • ◦
• ◦ • ◦ • ◦
• ◦ W ◦ • ◦
• ◦ • ◦ • ◦

4 black and 4 white queens on a 6 x 6 board:
B ◦ • ◦ B B
• ◦ W ◦ • ◦
• W • ◦ • ◦
• ◦ • ◦ • B
• ◦ W W • ◦
• ◦ • ◦ • ◦

5 black and 5 white queens on a 6 x 6 board:
• B • ◦ B ◦
• ◦ • B • B
W ◦ • ◦ • ◦
W ◦ W ◦ • ◦
• ◦ • ◦ • B
W ◦ W ◦ • ◦

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

2 black and 2 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

3 black and 3 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

4 black and 4 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

5 black and 5 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •

6 black and 6 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •

7 black and 7 white queens on a 7 x 7 board:
• B • ◦ • B •
◦ B ◦ • B • ◦
• B • ◦ • B •
◦ • ◦ • B • ◦
W ◦ W ◦ • ◦ W
◦ • ◦ W ◦ • ◦
W ◦ W W • ◦ •
```

## Java

Translation of: Kotlin
```import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

public class Peaceful {
enum Piece {
Empty,
Black,
White,
}

public static class Position {
public int x, y;

public Position(int x, int y) {
this.x = x;
this.y = y;
}

@Override
public boolean equals(Object obj) {
if (obj instanceof Position) {
Position pos = (Position) obj;
return pos.x == x && pos.y == y;
}
return false;
}
}

private static boolean place(int m, int n, List<Position> pBlackQueens, List<Position> pWhiteQueens) {
if (m == 0) {
return true;
}
boolean placingBlack = true;
for (int i = 0; i < n; ++i) {
inner:
for (int j = 0; j < n; ++j) {
Position pos = new Position(i, j);
for (Position queen : pBlackQueens) {
if (pos.equals(queen) || !placingBlack && isAttacking(queen, pos)) {
continue inner;
}
}
for (Position queen : pWhiteQueens) {
if (pos.equals(queen) || placingBlack && isAttacking(queen, pos)) {
continue inner;
}
}
if (placingBlack) {
placingBlack = false;
} else {
if (place(m - 1, n, pBlackQueens, pWhiteQueens)) {
return true;
}
pBlackQueens.remove(pBlackQueens.size() - 1);
pWhiteQueens.remove(pWhiteQueens.size() - 1);
placingBlack = true;
}
}
}
if (!placingBlack) {
pBlackQueens.remove(pBlackQueens.size() - 1);
}
return false;
}

private static boolean isAttacking(Position queen, Position pos) {
return queen.x == pos.x
|| queen.y == pos.y
|| Math.abs(queen.x - pos.x) == Math.abs(queen.y - pos.y);
}

private static void printBoard(int n, List<Position> blackQueens, List<Position> whiteQueens) {
Piece[] board = new Piece[n * n];
Arrays.fill(board, Piece.Empty);

for (Position queen : blackQueens) {
board[queen.x + n * queen.y] = Piece.Black;
}
for (Position queen : whiteQueens) {
board[queen.x + n * queen.y] = Piece.White;
}
for (int i = 0; i < board.length; ++i) {
if ((i != 0) && i % n == 0) {
System.out.println();
}

Piece b = board[i];
if (b == Piece.Black) {
System.out.print("B ");
} else if (b == Piece.White) {
System.out.print("W ");
} else {
int j = i / n;
int k = i - j * n;
if (j % 2 == k % 2) {
System.out.print("• ");
} else {
System.out.print("◦ ");
}
}
}
System.out.println('\n');
}

public static void main(String[] args) {
List<Position> nms = List.of(
new Position(2, 1),
new Position(3, 1),
new Position(3, 2),
new Position(4, 1),
new Position(4, 2),
new Position(4, 3),
new Position(5, 1),
new Position(5, 2),
new Position(5, 3),
new Position(5, 4),
new Position(5, 5),
new Position(6, 1),
new Position(6, 2),
new Position(6, 3),
new Position(6, 4),
new Position(6, 5),
new Position(6, 6),
new Position(7, 1),
new Position(7, 2),
new Position(7, 3),
new Position(7, 4),
new Position(7, 5),
new Position(7, 6),
new Position(7, 7)
);
for (Position nm : nms) {
int m = nm.y;
int n = nm.x;
System.out.printf("%d black and %d white queens on a %d x %d board:\n", m, m, n, n);
List<Position> blackQueens = new ArrayList<>();
List<Position> whiteQueens = new ArrayList<>();
if (place(m, n, blackQueens, whiteQueens)) {
printBoard(n, blackQueens, whiteQueens);
} else {
System.out.println("No solution exists.\n");
}
}
}
}
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B ◦ •
◦ • ◦
• W •

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B ◦ • ◦
◦ • ◦ •
• W • ◦
◦ • ◦ •

2 black and 2 white queens on a 4 x 4 board:
B ◦ B ◦
◦ • ◦ •
• W • W
◦ • ◦ •

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • ◦ • ◦
• W • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

2 black and 2 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • W • ◦
• W • ◦ •
◦ • ◦ • ◦
B ◦ • ◦ •

3 black and 3 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • W • W
• W • ◦ •
◦ • ◦ B ◦
B ◦ • ◦ •

4 black and 4 white queens on a 5 x 5 board:
• ◦ W ◦ W
B • ◦ • ◦
• ◦ W ◦ W
B • ◦ • ◦
• B • B •

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • ◦ • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

2 black and 2 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
B ◦ • ◦ • ◦
◦ • ◦ • ◦ •

3 black and 3 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• W • ◦ W ◦
◦ • ◦ • ◦ •
B ◦ • ◦ • ◦
B • ◦ • ◦ •

4 black and 4 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• W • ◦ W ◦
◦ • ◦ • W •
B ◦ • ◦ • ◦
B • ◦ B ◦ •

5 black and 5 white queens on a 6 x 6 board:
• ◦ W W • W
B • ◦ • ◦ •
• ◦ • W • W
◦ B ◦ • ◦ •
B ◦ • ◦ • ◦
◦ B ◦ • B •

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

2 black and 2 white queens on a 7 x 7 board:
B ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
B ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • ◦ • ◦ •

3 black and 3 white queens on a 7 x 7 board:
B ◦ B ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • W • ◦ •
◦ • ◦ • ◦ • ◦
B ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • ◦ • ◦ •

4 black and 4 white queens on a 7 x 7 board:
B ◦ B ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • W • ◦ •
◦ • ◦ • ◦ • ◦
B ◦ B ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • W • ◦ •

5 black and 5 white queens on a 7 x 7 board:
B ◦ B ◦ B ◦ •
◦ • ◦ • ◦ • ◦
• W • W • W •
◦ • ◦ • ◦ • ◦
B ◦ B ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • W • ◦ •

6 black and 6 white queens on a 7 x 7 board:
B ◦ B ◦ B ◦ •
◦ • ◦ • ◦ • ◦
• W • W • W •
◦ • ◦ • ◦ • ◦
B ◦ B ◦ B ◦ •
◦ • ◦ • ◦ • ◦
• W • W • W •

7 black and 7 white queens on a 7 x 7 board:
• ◦ • ◦ W ◦ W
B B B • ◦ • ◦
• ◦ • ◦ W ◦ W
◦ • ◦ • ◦ W W
• B • B • ◦ •
B • B • ◦ • ◦
• ◦ • ◦ W ◦ • ```

## jq

Works with jq, the C implementation of jq

Works with gojq, the Go implementation of jq

In the following, positions on the chessboard are represented by {x,y} objects.

```# Is the queen at position . attacking the position \$pos ?
def isAttacking(\$pos):
.x == \$pos.x or
.y == \$pos.y or
(((.x - \$pos.x)|length) == ((.y - \$pos.y)|length)); # i.e. abs

# Place \$q black and \$q white queens on an \$n*\$n board,
# where .blackQueens holds the positions of existing black Queens,
# and similarly for .whiteQueens.
# input: {blackQueens, whiteQueens}
# output: updated input on success, otherwise null.
def place(\$queens; \$n):
def place(\$q):
if \$q == 0 then .ok = true
else .placingBlack = true
| first(
foreach range(0; \$n) as \$i (.;
foreach range(0; \$n) as \$j (.;
{x:\$i, y:\$j} as \$pos
| .placingBlack as \$placingBlack
| if any( .blackQueens[], .whiteQueens[];
((.x == \$pos.x) and (.y == \$pos.y)))
then . # failure
elif .placingBlack
then if any( .whiteQueens[]; isAttacking(\$pos) )
then .
else .blackQueens += [\$pos]
| .placingBlack = false
end
elif any( .blackQueens[]; isAttacking(\$pos) )
then .
else .whiteQueens += [\$pos]
| place(\$q-1) as \$place
| if \$place then \$place  # success
else .blackQueens |= .[:-1]
| .whiteQueens |= .[:-1]
| .placingBlack = true
end
end
| if \$i == \$n-1 and \$j == \$n-1 then .ok = false end );
select(.ok) )
) // null
end;
{blackQueens: [], whiteQueens: [] } | place(\$queens);

# Input {blackQueens, whiteQueens}
def printBoard(\$n):
[range(0; \$n) | 0] as \$row
| .board = [range(0; \$n) | \$row]
| reduce .blackQueens[] as \$queen (.; .board[\$queen.x][\$queen.y] = "B ")
| reduce .whiteQueens[] as \$queen (.; .board[\$queen.x][\$queen.y] = "W ")
| foreach range(0; \$n) as \$i (.;
reduce range(0; \$n) as \$j (.row="";
.board[\$i][\$j] as \$b
| .row +=
(if \$b != 0 then \$b
elif \$i%2 == \$j%2
then "• "
else "◦ "
end) ) )
| .row;

# Use an object {squares, queens} to record the task:
# \$squares is the number of squares on each side of the board,
# and \$queens is the number of queens of each color.

];

| "\(\$t.queens) black and \(\$t.queens) white queens on a \(\$t.squares) x \(\$t.squares) board:",
((place(\$t.queens; \$t.squares)
| select(.)
| printBoard(\$t.squares))
// "No solution exists."),
""```
Output:

See Wren.

## Julia

GUI version, uses the Gtk library. The place! function is condensed from the C# example.

```using Gtk

struct Position
row::Int
col::Int
end

function place!(numeach, bsize, bqueens, wqueens)
isattack(q, pos) = (q.row == pos.row || q.col == pos.col ||
abs(q.row - pos.row) == abs(q.col - pos.col))
noattack(qs, pos) = !any(x -> isattack(x, pos), qs)
positionopen(bqs, wqs, p) = !any(x -> x == p, bqs) && !any(x -> x == p, wqs)

placingbqueens = true
if numeach < 1
return true
end
for i in 1:bsize, j in 1:bsize
bpos = Position(i, j)
if positionopen(bqueens, wqueens, bpos)
if placingbqueens && noattack(wqueens, bpos)
push!(bqueens, bpos)
placingbqueens = false
elseif !placingbqueens && noattack(bqueens, bpos)
push!(wqueens, bpos)
if place!(numeach - 1, bsize, bqueens, wqueens)
return true
end
pop!(bqueens)
pop!(wqueens)
placingbqueens = true
end
end
end
if !placingbqueens
pop!(bqueens)
end
false
end

function peacefulqueenapp()
win = GtkWindow("Peaceful Chess Queen Armies", 800, 800) |> (GtkFrame() |> (box = GtkBox(:v)))
boardsize = 5
numqueenseach = 4
hbox = GtkBox(:h)
boardscale = GtkScale(false, 2:16)
set_gtk_property!(boardscale, :hexpand, true)
blabel = GtkLabel("Choose Board Size")
nqueenscale = GtkScale(false, 1:24)
set_gtk_property!(nqueenscale, :hexpand, true)
qlabel = GtkLabel("Choose Number of Queens Per Side")
solveit = GtkButton("Solve")
set_gtk_property!(solveit, :label, "   Solve   ")
solvequeens(wid) = (boardsize = Int(GAccessor.value(boardscale));
numqueenseach = Int(GAccessor.value(nqueenscale)); update!())
signal_connect(solvequeens, solveit, :clicked)
map(w->push!(hbox, w),[blabel, boardscale, qlabel, nqueenscale, solveit])
scrwin = GtkScrolledWindow()
grid = GtkGrid()
push!(scrwin, grid)
map(w -> push!(box, w),[hbox, scrwin])
piece = (white = "\u2655", black = "\u265B", blank = "   ")
stylist = GtkStyleProvider(Gtk.CssProviderLeaf(data="""
label {background-image: image(cornsilk); font-size: 48px;}
button {background-image: image(tan); font-size: 48px;}"""))

function update!()
bqueens, wqueens = Vector{Position}(), Vector{Position}()
place!(numqueenseach, boardsize, bqueens, wqueens)
if length(bqueens) == 0
warn_dialog("No solution for board size \$boardsize and \$numqueenseach queens each.", win)
return
end
empty!(grid)
labels = Array{Gtk.GtkLabelLeaf, 2}(undef, (boardsize, boardsize))
buttons = Array{GtkButtonLeaf, 2}(undef, (boardsize, boardsize))
for i in 1:boardsize, j in 1:boardsize
if isodd(i + j)
grid[i, j] = buttons[i, j] = GtkButton(piece.blank)
set_gtk_property!(buttons[i, j], :expand, true)
push!(Gtk.GAccessor.style_context(buttons[i, j]), stylist, 600)
else
grid[i, j] = labels[i, j] = GtkLabel(piece.blank)
set_gtk_property!(labels[i, j], :expand, true)
push!(Gtk.GAccessor.style_context(labels[i, j]), stylist, 600)
end
pos = Position(i, j)
if pos in bqueens
set_gtk_property!(grid[i, j], :label, piece.black)
elseif pos in wqueens
set_gtk_property!(grid[i, j], :label, piece.white)
end
end
showall(win)
end

update!()
cond = Condition()
endit(w) = notify(cond)
signal_connect(endit, win, :destroy)
showall(win)
wait(cond)
end

peacefulqueenapp()
```

## Kotlin

Translation of: D
```import kotlin.math.abs

enum class Piece {
Empty,
Black,
White,
}

typealias Position = Pair<Int, Int>

fun place(m: Int, n: Int, pBlackQueens: MutableList<Position>, pWhiteQueens: MutableList<Position>): Boolean {
if (m == 0) {
return true
}
var placingBlack = true
for (i in 0 until n) {
inner@
for (j in 0 until n) {
val pos = Position(i, j)
for (queen in pBlackQueens) {
if (queen == pos || !placingBlack && isAttacking(queen, pos)) {
continue@inner
}
}
for (queen in pWhiteQueens) {
if (queen == pos || placingBlack && isAttacking(queen, pos)) {
continue@inner
}
}
placingBlack = if (placingBlack) {
false
} else {
if (place(m - 1, n, pBlackQueens, pWhiteQueens)) {
return true
}
pBlackQueens.removeAt(pBlackQueens.lastIndex)
pWhiteQueens.removeAt(pWhiteQueens.lastIndex)
true
}
}
}
if (!placingBlack) {
pBlackQueens.removeAt(pBlackQueens.lastIndex)
}
return false
}

fun isAttacking(queen: Position, pos: Position): Boolean {
return queen.first == pos.first
|| queen.second == pos.second
|| abs(queen.first - pos.first) == abs(queen.second - pos.second)
}

fun printBoard(n: Int, blackQueens: List<Position>, whiteQueens: List<Position>) {
val board = MutableList(n * n) { Piece.Empty }

for (queen in blackQueens) {
board[queen.first * n + queen.second] = Piece.Black
}
for (queen in whiteQueens) {
board[queen.first * n + queen.second] = Piece.White
}
for ((i, b) in board.withIndex()) {
if (i != 0 && i % n == 0) {
println()
}
if (b == Piece.Black) {
print("B ")
} else if (b == Piece.White) {
print("W ")
} else {
val j = i / n
val k = i - j * n
if (j % 2 == k % 2) {
print("• ")
} else {
print("◦ ")
}
}
}
println('\n')
}

fun main() {
val nms = listOf(
Pair(2, 1), Pair(3, 1), Pair(3, 2), Pair(4, 1), Pair(4, 2), Pair(4, 3),
Pair(5, 1), Pair(5, 2), Pair(5, 3), Pair(5, 4), Pair(5, 5),
Pair(6, 1), Pair(6, 2), Pair(6, 3), Pair(6, 4), Pair(6, 5), Pair(6, 6),
Pair(7, 1), Pair(7, 2), Pair(7, 3), Pair(7, 4), Pair(7, 5), Pair(7, 6), Pair(7, 7)
)
for ((n, m) in nms) {
println("\$m black and \$m white queens on a \$n x \$n board:")
val blackQueens = mutableListOf<Position>()
val whiteQueens = mutableListOf<Position>()
if (place(m, n, blackQueens, whiteQueens)) {
printBoard(n, blackQueens, whiteQueens)
} else {
println("No solution exists.\n")
}
}
}
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B ◦ •
◦ • W
• ◦ •

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B ◦ • ◦
◦ • W •
• ◦ • ◦
◦ • ◦ •

2 black and 2 white queens on a 4 x 4 board:
B ◦ • ◦
◦ • W •
B ◦ • ◦
◦ • W •

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • W • ◦
• ◦ • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

2 black and 2 white queens on a 5 x 5 board:
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

3 black and 3 white queens on a 5 x 5 board:
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ B ◦
• W • ◦ •

4 black and 4 white queens on a 5 x 5 board:
• B • B •
◦ • ◦ • B
W ◦ W ◦ •
◦ • ◦ • B
W ◦ W ◦ •

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

2 black and 2 white queens on a 6 x 6 board:
B ◦ • ◦ B ◦
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

3 black and 3 white queens on a 6 x 6 board:
B ◦ • ◦ B B
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ W ◦ • ◦
◦ • ◦ • ◦ •

4 black and 4 white queens on a 6 x 6 board:
B ◦ • ◦ B B
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ B
• ◦ W W • ◦
◦ • ◦ • ◦ •

5 black and 5 white queens on a 6 x 6 board:
• B • ◦ B ◦
◦ • ◦ B ◦ B
W ◦ • ◦ • ◦
W • W • ◦ •
• ◦ • ◦ • B
W • W • ◦ •

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

2 black and 2 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

3 black and 3 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

4 black and 4 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

5 black and 5 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •

6 black and 6 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •

7 black and 7 white queens on a 7 x 7 board:
• B • ◦ • B •
◦ B ◦ • B • ◦
• B • ◦ • B •
◦ • ◦ • B • ◦
W ◦ W ◦ • ◦ W
◦ • ◦ W ◦ • ◦
W ◦ W W • ◦ • ```

## Mathematica /Wolfram Language

```ClearAll[ValidSpots, VisibleByQueen, SolveQueen, GetSolution]
VisualizeState[state_] := Module[{q, cells},
q = MapIndexed[If[#["q"] == -1, {}, Text[Style[#["q"], 24], #2]] &, state, {2}];
cells = MapIndexed[{If[OddQ[Total[#2]], FaceForm[],
FaceForm[GrayLevel[0.8]]], EdgeForm[Black],
Rectangle[#2 - 0.5, #2 + 0.5]} &, state, {2}];
Graphics[{cells, q}]
]
ValidSpots[state_, tp_Integer] := Module[{vals},
vals = Catenate@MapIndexed[If[#1["q"] == -1 \[And] DeleteCases[#1["v"], tp] == {}, #2, Missing[]] &, state, {2}];
DeleteMissing[vals]
]
VisibleByQueen[{i_, j_}, {a_, b_}] := i == a \[Or] j == b \[Or] i + j == a + b \[Or] i - j == a - b
PlaceQueen[state_, pos : {i_Integer, j_Integer}, tp_Integer] := Module[{vals, out},
out = state;
out[[i, j]] = Association[out[[i, j]], "q" -> tp];
out = MapIndexed[If[VisibleByQueen[{i, j}, #2], <|#1, "v" -> Append[#1["v"], tp]|>, #1] &, out, {2}];
out
]
SolveQueen[state_, toplace_List] :=
Module[{len = Length[toplace], next, valid, newstate},
If[len == 0,
Print[VisualizeState@state];
Print[StringRiffle[StringJoin /@ Map[ToString, state[[All, All, "q"]] /. -1 -> ".", {2}], "\n"]];
Abort[];
,
next = First[toplace];
valid = ValidSpots[state, next];
Do[
newstate = PlaceQueen[state, v, next];
SolveQueen[newstate, Rest[toplace]]
,
{v, valid}
]
]
]
GetSolution[n_Integer?Positive, m_Integer?Positive, numcol_ : 2] :=
Module[{state, tp},
state = ConstantArray[<|"q" -> -1, "v" -> {}|>, {n, n}];
tp = Flatten[Transpose[ConstantArray[#, m] & /@ Range[numcol]]];
SolveQueen[state, tp]
]
GetSolution[8, 4, 3](* Solves placing 3 armies of each 4 queens on an 8*8 board*)
GetSolution[5, 4, 2](* Solves placing 2 armies of each 4 queens on an 5*5 board*)
```
Output:
```[Graphical object]
1....1..
..2....2
....3...
.3....3.
...1....
1.......
..2....2
....3...

[Graphical object]
1...1
..2..
.2.2.
..2..
1...1```

## Nim

Translation of: Kotlin

Almost a direct translation except for "printBoard" where we have chosen to use a sequence of sequences to simplify the code.

```import sequtils, strformat

type

Piece {.pure.} = enum Empty, Black, White
Position = tuple[x, y: int]

func isAttacking(queen, pos: Position): bool =
queen.x == pos.x or queen.y == pos.y or abs(queen.x - pos.x) == abs(queen.y - pos.y)

func place(m, n: int; blackQueens, whiteQueens: var seq[Position]): bool =

if m == 0: return true

var placingBlack = true
for i in 0..<n:
for j in 0..<n:

block inner:
let pos: Position = (i, j)
for queen in blackQueens:
if queen == pos or not placingBlack and queen.isAttacking(pos):
break inner
for queen in whiteQueens:
if queen == pos or placingBlack and queen.isAttacking(pos):
break inner

if placingBlack:
else:
if place(m - 1, n, blackQueens, whiteQueens): return true
placingBlack = not placingBlack

if not placingBlack:

proc printBoard(n: int; blackQueens, whiteQueens: seq[Position]) =

var board = newSeqWith(n, newSeq[Piece](n))   # Initialized to Empty.

for queen in blackQueens:
board[queen.x][queen.y] = Black
for queen in whiteQueens:
board[queen.x][queen.y] = White

for i in 0..<n:
for j in 0..<n:
stdout.write case board[i][j]
of Black: "B "
of White: "W "
of Empty: (if (i and 1) == (j and 1): "• " else: "◦ ")
stdout.write '\n'

echo ""

const Nms = [(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6),
(7, 1), (7, 2), (7, 3), (7, 4), (7, 5), (7, 6), (7, 7)]

for (n, m) in Nms:
echo &"{m} black and {m} white queens on a {n} x {n} board:"
var blackQueens, whiteQueens: seq[Position]
if place(m, n, blackQueens, whiteQueens):
printBoard(n, blackQueens, whiteQueens)
else:
echo "No solution exists.\n"
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B ◦ •
◦ • W
• ◦ •

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B ◦ • ◦
◦ • W •
• ◦ • ◦
◦ • ◦ •

2 black and 2 white queens on a 4 x 4 board:
B ◦ • ◦
◦ • W •
B ◦ • ◦
◦ • W •

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • W • ◦
• ◦ • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

2 black and 2 white queens on a 5 x 5 board:
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

3 black and 3 white queens on a 5 x 5 board:
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ B ◦
• W • ◦ •

4 black and 4 white queens on a 5 x 5 board:
• B • B •
◦ • ◦ • B
W ◦ W ◦ •
◦ • ◦ • B
W ◦ W ◦ •

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

2 black and 2 white queens on a 6 x 6 board:
B ◦ • ◦ B ◦
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

3 black and 3 white queens on a 6 x 6 board:
B ◦ • ◦ B B
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ W ◦ • ◦
◦ • ◦ • ◦ •

4 black and 4 white queens on a 6 x 6 board:
B ◦ • ◦ B B
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ B
• ◦ W W • ◦
◦ • ◦ • ◦ •

5 black and 5 white queens on a 6 x 6 board:
• B • ◦ B ◦
◦ • ◦ B ◦ B
W ◦ • ◦ • ◦
W • W • ◦ •
• ◦ • ◦ • B
W • W • ◦ •

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

2 black and 2 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

3 black and 3 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

4 black and 4 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

5 black and 5 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •

6 black and 6 white queens on a 7 x 7 board:
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •

7 black and 7 white queens on a 7 x 7 board:
• B • ◦ • B •
◦ B ◦ • B • ◦
• B • ◦ • B •
◦ • ◦ • B • ◦
W ◦ W ◦ • ◦ W
◦ • ◦ W ◦ • ◦
W ◦ W W • ◦ •
```

## Perl

### Terse

```use strict;
use warnings;

my \$m = shift // 4;
my \$n = shift // 5;
my %seen;
my \$gaps = join '|', qr/-*/, map qr/.{\$_}(?:-.{\$_})*/s, \$n-1, \$n, \$n+1;
my \$attack = qr/(\w)(?:\$gaps)(?!\1)\w/;

place( scalar ('-' x \$n . "\n") x \$n );
print "No solution to \$m \$n\n";

sub place
{
local \$_ = shift;
\$seen{\$_}++ || /\$attack/ and return; # previously or attack
(my \$have = tr/WB//) < \$m * 2 or exit !print "Solution to \$m \$n\n\n\$_";
place( s/-\G/ qw(W B)[\$have % 2] /er ) while /-/g; # place next queen
}
```
Output:
```Solution to 4 5

W---W
--B--
-B-B-
--B--
W---W```

### Verbose

A refactored version of the same code, with fancier output.

```use strict;
use warnings;
use feature 'say';
use feature 'state';
use utf8;
binmode(STDOUT, ':utf8');

# recursively place the next queen
sub place {
my(\$board, \$n, \$m, \$empty_square) = @_;
state(%seen,\$attack,\$solution);

# logic of 'attack' regex: queen ( ... paths between queens containing only empty squares ... ) queen of other color
unless (\$attack) {
\$attack =
'([WB])' .      # 1st queen
'(?:' .
join('|',
"[\$empty_square]*",
map {
"(?^s:.{\$_}(?:[\$empty_square].{\$_})*)"
} \$n-1, \$n, \$n+1
) .
')' .
'(?!\1)[WB]';   # 2nd queen
}

# pass first result found back up the stack (omit this line to get last result found)
return \$solution if \$solution;

# bail out if seen this configuration previously, or attack detected
return if \$seen{\$board}++ or \$board =~ /\$attack/;

# success if queen count is m×2
\$solution = \$board and return if \$m * 2 == (my \$have = \$board =~ tr/WB//);

# place the next queen (alternating colors each time)
place(   \$board =~ s/[\$empty_square]\G/ qw<W B>[\$have % 2] /er, \$n, \$m, \$empty_square )
while \$board =~  /[\$empty_square]/g;

return \$solution
}

my(\$m, \$n) = \$#ARGV == 1 ? @ARGV : (4, 5);
my \$empty_square = '◦•';
my \$board = join "\n", map { substr \$empty_square x \$n, \$_%2, \$n } 1..\$n;

my \$solution = place \$board, \$n, \$m, \$empty_square;

say \$solution
? sprintf "Solution to \$m \$n\n\n%s", map { s/(.)/\$1 /gm; s/B /♛/gm; s/W /♕/gmr } \$solution
: "No solution to \$m \$n";
```
Output:
```Solution to 4 5

♕◦ • ◦ ♕
◦ • ♛• ◦
• ♛• ♛•
◦ • ♛• ◦
♕◦ • ◦ ♕```

## Phix

Translation of: Go
Translation of: Python

You can run this online here.

```--
-- demo\rosetta\Queen_Armies.exw
-- =============================
--
with javascript_semantics
requires("1.0.2") -- (puts(fn,x,false) for p2js.js)
string html = ""
constant as_html = true
constant queens = {``,
`♛`,
`<font color="green">♕</font>`,
`<span style="color:red">?</span>`}

procedure showboard(integer n, sequence blackqueens, whitequeens)
sequence board = repeat(repeat('-',n),n)
for i=1 to length(blackqueens) do
integer {qi,qj} = blackqueens[i]
board[qi,qj] = 'B'
{qi,qj} = whitequeens[i]
board[qi,qj] = 'W'
end for
if as_html then
string out = sprintf("<br><b>## %d black and %d white queens on a %d-by-%d board</b><br>\n",
{length(blackqueens),length(whitequeens),n,n}),
tbl = ""
out &= "<table style=\"font-weight:bold\">\n  "
for x=1 to n do
for y=1 to n do
if y=1 then tbl &= "  </tr>\n  <tr valign=\"middle\" align=\"center\">\n" end if
integer xw = find({x,y},blackqueens)!=0,
xb = find({x,y},whitequeens)!=0,
dx = xw+xb*2+1
string ch = queens[dx],
bg = iff(mod(x+y,2)?"":` bgcolor="silver"`)
tbl &= sprintf("    <td style=\"width:14pt; height:14pt;\"%s>%s</td>\n",{bg,ch})
end for
end for
out &= tbl[11..\$]
out &= "  </tr>\n</table>\n<br>\n"
html &= out
else
integer b = length(blackqueens),
w = length(whitequeens)
printf(1,"%d black and %d white queens on a %d x %d board:\n", {b, w, n, n})
puts(1,join(board,"\n")&"\n")
--      ?{n,blackqueens, whitequeens}
end if
end procedure

function isAttacking(sequence queen, pos)
integer {qi,qj} = queen, {pi,pj} = pos
return qi=pi or qj=pj or abs(qi-pi)=abs(qj-pj)
end function

function place(integer m, n, sequence blackqueens = {}, whitequeens = {})
if m == 0 then showboard(n,blackqueens,whitequeens) return true end if
bool placingBlack := true
for i=1 to n do
for j=1 to n do
sequence pos := {i, j}
for q=1 to length(blackqueens) do
sequence queen := blackqueens[q]
if queen == pos or ((not placingBlack) and isAttacking(queen, pos)) then
pos = {}
exit
end if
end for
if pos!={} then
for q=1 to length(whitequeens) do
sequence queen := whitequeens[q]
if queen == pos or (placingBlack and isAttacking(queen, pos)) then
pos = {}
exit
end if
end for
if pos!={} then
if placingBlack then
blackqueens = append(deep_copy(blackqueens), pos)
placingBlack = false
else
whitequeens = append(deep_copy(whitequeens), pos)
if place(m-1, n, blackqueens, whitequeens) then return true end if
blackqueens = blackqueens[1..\$-1]
whitequeens = whitequeens[1..\$-1]
placingBlack = true
end if
end if
end if
end for
end for
return false
end function

for n=2 to 7 do
for m=1 to n-(n<5) do
if not place(m,n) then
string no = sprintf("Cannot place %d+ queen armies on a %d-by-%d board",{m,n,n})
if as_html then
html &= sprintf("<b># %s</b><br><br>\n\n",{no})
else
printf(1,"%s.\n", {no})
end if
end if
end for
end for

<!DOCTYPE html>
<html lang="en">
<meta charset="utf-8" />
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<title>Queen Armies</title>
<body>
<h2>queen armies</h2>
""", -- or <div style="overflow:scroll; height:250px;">
html_footer = """
</body>
</html>
""" -- or </div>

if as_html then
if platform()=JS then
puts(1,html,false)
else
integer fn = open("queen_armies.html","w")
puts(fn,html)
puts(fn,html_footer)
close(fn)
printf(1,"See queen_armies.html\n")
end if
end if

?"done"
{} = wait_key()
```
Output:

with as_html = false

```Cannot place 1+ queen armies on a 2-by-2 board.
1 black and 1 white queens on a 3 x 3 board:
B--
--W
---
Cannot place 2+ queen armies on a 3-by-3 board.
<snip>
7 black and 7 white queens on a 7 x 7 board:
-B---B-
-B--B--
-B---B-
----B--
W-W---W
---W---
W-WW---
```
Output:

with as_html = true

# Cannot place 1+ queen armies on a 2-by-2 board

## 1 black and 1 white queens on a 3-by-3 board

 ♛ ♕

# Cannot place 2+ queen armies on a 3-by-3 board

<snip>

## 7 black and 7 white queens on a 7-by-7 board

 ♛ ♛ ♛ ♛ ♛ ♛ ♛ ♕ ♕ ♕ ♕ ♕ ♕ ♕

## Python

### Python: Textual output

```from itertools import combinations, product, count
from functools import lru_cache, reduce

_bbullet, _wbullet = '\u2022\u25E6'
_or = set.__or__

def place(m, n):
"Place m black and white queens, peacefully, on an n-by-n board"
board = set(product(range(n), repeat=2))  # (x, y) tuples
placements = {frozenset(c) for c in combinations(board, m)}
for blacks in placements:
black_attacks = reduce(_or,
(queen_attacks_from(pos, n) for pos in blacks),
set())
for whites in {frozenset(c)     # Never on blsck attacking squares
for c in combinations(board - black_attacks, m)}:
if not black_attacks & whites:
return blacks, whites
return set(), set()

@lru_cache(maxsize=None)
def queen_attacks_from(pos, n):
x0, y0 = pos
a = set([pos])    # Its position
a.update((x, y0) for x in range(n))    # Its row
a.update((x0, y) for y in range(n))    # Its column
# Diagonals
for x1 in range(n):
# l-to-r diag
y1 = y0 -x0 +x1
if 0 <= y1 < n:
# r-to-l diag
y1 = y0 +x0 -x1
if 0 <= y1 < n:
return a

def pboard(black_white, n):
"Print board"
if black_white is None:
blk, wht = set(), set()
else:
blk, wht = black_white
print(f"## {len(blk)} black and {len(wht)} white queens "
f"on a {n}-by-{n} board:", end='')
for x, y in product(range(n), repeat=2):
if y == 0:
print()
xy = (x, y)
ch = ('?' if xy in blk and xy in wht
else 'B' if xy in blk
else 'W' if xy in wht
else _bbullet if (x + y)%2 else _wbullet)
print('%s' % ch, end='')
print()

if __name__ == '__main__':
n=2
for n in range(2, 7):
print()
for m in count(1):
ans = place(m, n)
if ans[0]:
pboard(ans, n)
else:
print (f"# Can't place {m} queens on a {n}-by-{n} board")
break
#
print('\n')
m, n = 5, 7
ans = place(m, n)
pboard(ans, n)
```
Output:
```# Can't place 1 queens on a 2-by-2 board

## 1 black and 1 white queens on a 3-by-3 board:
◦•◦
B◦•
◦•W
# Can't place 2 queens on a 3-by-3 board

## 1 black and 1 white queens on a 4-by-4 board:
◦•W•
B◦•◦
◦•◦•
•◦•◦
## 2 black and 2 white queens on a 4-by-4 board:
◦B◦•
•B•◦
◦•◦•
W◦W◦
# Can't place 3 queens on a 4-by-4 board

## 1 black and 1 white queens on a 5-by-5 board:
◦•◦•◦
W◦•◦•
◦•◦•◦
•◦•◦B
◦•◦•◦
## 2 black and 2 white queens on a 5-by-5 board:
◦•◦•W
•◦B◦•
◦•◦•◦
•◦•B•
◦W◦•◦
## 3 black and 3 white queens on a 5-by-5 board:
◦W◦•◦
•◦•◦W
B•B•◦
B◦•◦•
◦•◦W◦
## 4 black and 4 white queens on a 5-by-5 board:
◦•B•B
W◦•◦•
◦W◦W◦
W◦•◦•
◦•B•B
# Can't place 5 queens on a 5-by-5 board

## 1 black and 1 white queens on a 6-by-6 board:
◦•◦•◦•
W◦•◦•◦
◦•◦•◦•
•◦•◦B◦
◦•◦•◦•
•◦•◦•◦
## 2 black and 2 white queens on a 6-by-6 board:
◦•◦•◦•
•◦B◦•◦
◦•◦•◦•
•◦•B•◦
◦•◦•◦•
W◦•◦W◦
## 3 black and 3 white queens on a 6-by-6 board:
◦•B•◦•
•B•◦•◦
◦•◦W◦W
•◦•◦•◦
W•◦•◦•
•◦•◦B◦
## 4 black and 4 white queens on a 6-by-6 board:
WW◦•W•
•W•◦•◦
◦•◦•◦B
•◦B◦•◦
◦•◦B◦•
•◦•B•◦
## 5 black and 5 white queens on a 6-by-6 board:
◦•W•W•
B◦•◦•◦
◦•W•◦W
B◦•◦•◦
◦•◦•◦W
BB•B•◦
# Can't place 6 queens on a 6-by-6 board

## 5 black and 5 white queens on a 7-by-7 board:
◦•◦•B•◦
•W•◦•◦W
◦•◦•B•◦
B◦•◦•◦•
◦•B•◦•◦
•◦•B•◦•
◦W◦•◦WW```

### Python: HTML output

Uses the solver function `place` from the above textual output case.

```from peaceful_queen_armies_simpler import place
from itertools import product, count

_bqueenh, _wqueenh = '&#x265b;', '<font color="green">&#x2655;</font>'

def hboard(black_white, n):
"HTML board generator"
if black_white is None:
blk, wht = set(), set()
else:
blk, wht = black_white
out = (f"<br><b>## {len(blk)} black and {len(wht)} white queens "
f"on a {n}-by-{n} board</b><br>\n")
out += '<table style="font-weight:bold">\n  '
tbl = ''
for x, y in product(range(n), repeat=2):
if y == 0:
tbl += '  </tr>\n  <tr valign="middle" align="center">\n'
xy = (x, y)
ch = ('<span style="color:red">?</span>' if xy in blk and xy in wht
else _bqueenh if xy in blk
else _wqueenh if xy in wht
else "")
bg = "" if (x + y)%2 else ' bgcolor="silver"'
tbl += f'    <td style="width:14pt; height:14pt;"{bg}>{ch}</td>\n'
out += tbl[7:]
out += '  </tr>\n</table>\n<br>\n'
return out

if __name__ == '__main__':
n=2
html = ''
for n in range(2, 7):
print()
for m in count(1):
ans = place(m, n)
if ans[0]:
html += hboard(ans, n)
else:
html += (f"<b># Can't place {m} queen armies on a "
f"{n}-by-{n} board</b><br><br>\n\n" )
break
#
html += '<br>\n'
m, n = 6, 7
ans = place(m, n)
html += hboard(ans, n)
with open('peaceful_queen_armies.htm', 'w') as f:
f.write(html)
```
Output:

# Can't place 1 queen armies on a 2-by-2 board

## 1 black and 1 white queens on a 3-by-3 board

 ♛ ♕

# Can't place 2 queen armies on a 3-by-3 board

## 1 black and 1 white queens on a 4-by-4 board

 ♕ ♛

## 2 black and 2 white queens on a 4-by-4 board

 ♛ ♛ ♕ ♕

# Can't place 3 queen armies on a 4-by-4 board

## 1 black and 1 white queens on a 5-by-5 board

 ♕ ♛

## 2 black and 2 white queens on a 5-by-5 board

 ♕ ♛ ♛ ♕

## 3 black and 3 white queens on a 5-by-5 board

 ♕ ♕ ♛ ♛ ♛ ♕

## 4 black and 4 white queens on a 5-by-5 board

 ♛ ♛ ♕ ♕ ♕ ♕ ♛ ♛

# Can't place 5 queen armies on a 5-by-5 board

## 1 black and 1 white queens on a 6-by-6 board

 ♕ ♛

## 2 black and 2 white queens on a 6-by-6 board

 ♛ ♛ ♕ ♕

## 3 black and 3 white queens on a 6-by-6 board

 ♛ ♛ ♕ ♕ ♕ ♛

## 4 black and 4 white queens on a 6-by-6 board

 ♕ ♕ ♕ ♕ ♛ ♛ ♛ ♛

## 5 black and 5 white queens on a 6-by-6 board

 ♕ ♕ ♛ ♕ ♕ ♛ ♕ ♛ ♛ ♛

# Can't place 6 queen armies on a 6-by-6 board

## 6 black and 6 white queens on a 7-by-7 board

 ♛ ♛ ♕ ♕ ♕ ♕ ♕ ♛ ♛ ♕ ♛ ♛

## Raku

(formerly Perl 6)

Translation of: Perl
```# recursively place the next queen
sub place (\$board, \$n, \$m, \$empty-square) {
my \$cnt;
state (%seen,\$attack);
state \$solution = False;

# logic of regex: queen ( ... paths between queens containing only empty squares ... ) queen of other color
once {
my %Q = 'WBBW'.comb; # return the queen of alternate color
my \$re =
'(<[WB]>)' ~                # 1st queen
'[' ~
join(' |',
qq/<[\$empty-square]>*/,
map {
qq/ . ** {\$_}[<[\$empty-square]> . ** {\$_}]*/
}, \$n-1, \$n, \$n+1
) ~
']' ~
'<{%Q{\$0}}>';               # 2nd queen
\$attack = "rx/\$re/".EVAL;
}

# return first result found (omit this line to get last result found)
return \$solution if \$solution;

# bail out if seen this configuration previously, or attack detected
return if %seen{\$board}++ or \$board ~~ \$attack;

# success if queen count is m×2, set state variable and return from recursion
\$solution = \$board and return if \$m * 2 == my \$queens = \$board.comb.Bag{<W B>}.sum;

# place the next queen (alternating colors each time)
place( \$board.subst( /<[◦•]>/, {<W B>[\$queens % 2]}, :nth(\$cnt) ), \$n, \$m, \$empty-square )
while \$board ~~ m:nth(++\$cnt)/<[◦•]>/;

return \$solution
}

my (\$m, \$n) = @*ARGS == 2 ?? @*ARGS !! (4, 5);
my \$empty-square = '◦•';
my \$board = (\$empty-square x \$n**2).comb.rotor(\$n)>>.join[^\$n].join: "\n";

my \$solution = place \$board, \$n, \$m, \$empty-square;

say \$solution
?? "Solution to \$m \$n\n\n{S:g/(\N)/\$0 / with \$solution}"
!! "No solution to \$m \$n";
```
Output:
```W • ◦ • W
• ◦ B ◦ •
◦ B ◦ B ◦
• ◦ B ◦ •
W • ◦ • W```

## Ruby

Translation of: Java
```class Position

def initialize(x, y)
@x = x
@y = y
end

def ==(other)
self.x == other.x &&
self.y == other.y
end

def to_s
'(%d, %d)' % [@x, @y]
end

def to_str
to_s
end
end

def isAttacking(queen, pos)
return queen.x == pos.x ||
queen.y == pos.y ||
(queen.x - pos.x).abs() == (queen.y - pos.y).abs()
end

def place(m, n, blackQueens, whiteQueens)
if m == 0 then
return true
end
placingBlack = true
for i in 0 .. n-1
for j in 0 .. n-1
catch :inner do
pos = Position.new(i, j)
for queen in blackQueens
if pos == queen || !placingBlack && isAttacking(queen, pos) then
throw :inner
end
end
for queen in whiteQueens
if pos == queen || placingBlack && isAttacking(queen, pos) then
throw :inner
end
end
if placingBlack then
blackQueens << pos
placingBlack = false
else
whiteQueens << pos
if place(m - 1, n, blackQueens, whiteQueens) then
return true
end
blackQueens.pop
whiteQueens.pop
placingBlack = true
end
end
end
end
if !placingBlack then
blackQueens.pop
end
return false
end

def printBoard(n, blackQueens, whiteQueens)
# initialize the board
board = Array.new(n) { Array.new(n) { ' ' } }
for i in 0 .. n-1
for j in 0 .. n-1
if i % 2 == j % 2 then
board[i][j] = '•'
else
board[i][j] = '◦'
end
end
end

# insert the queens
for queen in blackQueens
board[queen.y][queen.x] = 'B'
end
for queen in whiteQueens
board[queen.y][queen.x] = 'W'
end

# print the board
for row in board
for cell in row
print cell, ' '
end
print "\n"
end
print "\n"
end

nms = [
[2, 1],
[3, 1], [3, 2],
[4, 1], [4, 2], [4, 3],
[5, 1], [5, 2], [5, 3], [5, 4], [5, 5],
[6, 1], [6, 2], [6, 3], [6, 4], [6, 5], [6, 6],
[7, 1], [7, 2], [7, 3], [7, 4], [7, 5], [7, 6], [7, 7]
]
for nm in nms
m = nm[1]
n = nm[0]
print "%d black and %d white queens on a %d x %d board:\n" % [m, m, n, n]

blackQueens = []
whiteQueens = []
if place(m, n, blackQueens, whiteQueens) then
printBoard(n, blackQueens, whiteQueens)
else
print "No solution exists.\n\n"
end
end
```
Output:
```1 black and 1 white queens on a 2 x 2 board:
No solution exists.

1 black and 1 white queens on a 3 x 3 board:
B ◦ •
◦ • ◦
• W •

2 black and 2 white queens on a 3 x 3 board:
No solution exists.

1 black and 1 white queens on a 4 x 4 board:
B ◦ • ◦
◦ • ◦ •
• W • ◦
◦ • ◦ •

2 black and 2 white queens on a 4 x 4 board:
B ◦ B ◦
◦ • ◦ •
• W • W
◦ • ◦ •

3 black and 3 white queens on a 4 x 4 board:
No solution exists.

1 black and 1 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • ◦ • ◦
• W • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

2 black and 2 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • W • ◦
• W • ◦ •
◦ • ◦ • ◦
B ◦ • ◦ •

3 black and 3 white queens on a 5 x 5 board:
B ◦ • ◦ •
◦ • W • W
• W • ◦ •
◦ • ◦ B ◦
B ◦ • ◦ •

4 black and 4 white queens on a 5 x 5 board:
• ◦ W ◦ W
B • ◦ • ◦
• ◦ W ◦ W
B • ◦ • ◦
• B • B •

5 black and 5 white queens on a 5 x 5 board:
No solution exists.

1 black and 1 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • ◦ • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

2 black and 2 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
B ◦ • ◦ • ◦
◦ • ◦ • ◦ •

3 black and 3 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• W • ◦ W ◦
◦ • ◦ • ◦ •
B ◦ • ◦ • ◦
B • ◦ • ◦ •

4 black and 4 white queens on a 6 x 6 board:
B ◦ • ◦ • ◦
◦ • W • ◦ •
• W • ◦ W ◦
◦ • ◦ • W •
B ◦ • ◦ • ◦
B • ◦ B ◦ •

5 black and 5 white queens on a 6 x 6 board:
• ◦ W W • W
B • ◦ • ◦ •
• ◦ • W • W
◦ B ◦ • ◦ •
B ◦ • ◦ • ◦
◦ B ◦ • B •

6 black and 6 white queens on a 6 x 6 board:
No solution exists.

1 black and 1 white queens on a 7 x 7 board:
B ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

2 black and 2 white queens on a 7 x 7 board:
B ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
B ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • ◦ • ◦ •

3 black and 3 white queens on a 7 x 7 board:
B ◦ B ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • W • ◦ •
◦ • ◦ • ◦ • ◦
B ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • ◦ • ◦ •

4 black and 4 white queens on a 7 x 7 board:
B ◦ B ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • W • ◦ •
◦ • ◦ • ◦ • ◦
B ◦ B ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • W • ◦ •

5 black and 5 white queens on a 7 x 7 board:
B ◦ B ◦ B ◦ •
◦ • ◦ • ◦ • ◦
• W • W • W •
◦ • ◦ • ◦ • ◦
B ◦ B ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• W • W • ◦ •

6 black and 6 white queens on a 7 x 7 board:
B ◦ B ◦ B ◦ •
◦ • ◦ • ◦ • ◦
• W • W • W •
◦ • ◦ • ◦ • ◦
B ◦ B ◦ B ◦ •
◦ • ◦ • ◦ • ◦
• W • W • W •

7 black and 7 white queens on a 7 x 7 board:
• ◦ • ◦ W ◦ W
B B B • ◦ • ◦
• ◦ • ◦ W ◦ W
◦ • ◦ • ◦ W W
• B • B • ◦ •
B • B • ◦ • ◦
• ◦ • ◦ W ◦ •```

## Scheme

### All solutions

Works with: CHICKEN version 5.3.0
Library: srfi-132
```;;;
;;; Solutions to the Peaceful Chess Queen Armies puzzle, in R7RS
;;; Scheme (using also SRFI-132).
;;;
;;; https://rosettacode.org/wiki/Peaceful_chess_queen_armies
;;;

(cond-expand
(r7rs)
(chicken (import (r7rs))))

(import (scheme process-context))
(import (only (srfi 132) list-sort))

(define-record-type <&fail>
(make-the-one-unique-&fail-that-you-must-not-make-twice)
do-not-use-this:&fail?)

(define &fail
(make-the-one-unique-&fail-that-you-must-not-make-twice))

(define (failure? f)
(eq? f &fail))

(define (success? f)
(not (failure? f)))

(define *suspend*
(make-parameter (lambda (x) x)))

(define (suspend v)
((*suspend*) v))

(define (fail-forever)
(let loop ()
(suspend &fail)
(loop)))

(define (make-generator-procedure thunk)
;;
;; Make a suspendable procedure that takes no arguments. It is a
;; simple generator of values. (One can elaborate on this to have
;; the procedure accept an argument upon resumption, like an Icon
;; co-expression.)
;;
(define (next-run return)
(define (my-suspend v)
(set! return
(call/cc
(lambda (resumption-point)
(set! next-run resumption-point)
(return v)))))
(parameterize ((*suspend* my-suspend))
(suspend (thunk))
(fail-forever)))
(lambda ()
(call/cc next-run)))

(define BLACK 'B)
(define WHITE 'W)

(define (flip-color c)
(if (eq? c BLACK) WHITE BLACK))

(define-record-type <queen>
(make-queen color rank file)
queen?
(color queen-color)
(rank queen-rank)
(file queen-file))

(define (serialize-queen queen)
(string-append (if (eq? (queen-color queen) BLACK) "B" "W")
"(" (number->string (queen-rank queen))
"," (number->string (queen-file queen)) ")"))

(define (serialize-queens queens)
(apply string-append
(list-sort string<? (map serialize-queen queens))))

(define (queens->string n queens)

(define board
(let ((board (make-vector (* n n) #f)))
(do ((q queens (cdr q)))
((null? q))
(let* ((color (queen-color (car q)))
(i (queen-rank (car q)))
(j (queen-file (car q))))
(vector-set! board (ij->index n i j) color)))
board))

(define rule
(let ((str "+"))
(do ((j 1 (+ j 1)))
((= j (+ n 1)))
(set! str (string-append str "----+")))
str))

(define str "")

(when (< 0 n)
(set! str rule)
(do ((i n (- i 1)))
((= i 0))
(set! str (string-append str "\n"))
(do ((j 1 (+ j 1)))
((= j (+ n 1)))
(let* ((color (vector-ref board (ij->index n i j)))
(representation
(cond ((eq? color #f) "    ")
((eq? color BLACK) "  B ")
((eq? color WHITE) "  W ")
(else " ?? "))))
(set! str (string-append str "|" representation))))
(set! str (string-append str "|\n" rule))))
str)

(define (queen-fits-in? queen other-queens)
(or (null? other-queens)
(let ((other (car other-queens)))
(let ((colorq (queen-color queen))
(rankq (queen-rank queen))
(fileq (queen-file queen))
(coloro (queen-color other))
(ranko (queen-rank other))
(fileo (queen-file other)))
(if (eq? colorq coloro)
(and (or (not (= rankq ranko))
(not (= fileq fileo)))
(queen-fits-in? queen (cdr other-queens)))
(and (not (= rankq ranko))
(not (= fileq fileo))
(not (= (+ rankq fileq) (+ ranko fileo)))
(not (= (- rankq fileq) (- ranko fileo)))
(queen-fits-in? queen (cdr other-queens))))))))

(define (latest-queen-fits-in? queens)
(or (null? (cdr queens))
(queen-fits-in? (car queens) (cdr queens))))

(define (make-peaceful-queens-generator m n)
(make-generator-procedure
(lambda ()
(define solutions '())

(let loop ((queens (list (make-queen BLACK 1 1)))
(num-queens 1))

(let ((color (flip-color (queen-color (car queens)))))
(loop (cons (make-queen color 1 1) queens)
(+ num-queens 1))))

(define (move-a-queen)
(let drop-one ((queens queens)
(num-queens num-queens))
(if (zero? num-queens)
(loop '() 0)
(let* ((latest (car queens))
(color (queen-color latest))
(rank (queen-rank latest))
(file (queen-file latest)))
(if (and (= rank n) (= file n))
(drop-one (cdr queens) (- num-queens 1))
(let-values (((rank^ file^)
(loop (cons (make-queen color rank^ file^)
(cdr queens))
num-queens)))))))

(cond ((zero? num-queens)
;; There are no more solutions.
&fail)

((latest-queen-fits-in? queens)
(if (= num-queens (* 2 m))
(let ((str (serialize-queens queens)))
;; The current "queens" is a solution.
(unless (member str solutions)
;; The current "queens" is a *new* solution.
(set! solutions (cons str solutions))
(suspend queens))
(move-a-queen))

(else
(move-a-queen)))))))

(define (ij->index n i j)
(let ((i1 (- i 1))
(j1 (- j 1)))
(+ i1 (* n j1))))

(define (index->ij n index)
(let-values (((q r) (floor/ index n)))
(values (+ r 1) (+ q 1))))

(index->ij n (+ (ij->index n i j) 1)))

(define args (command-line))
(unless (or (= (length args) 3)
(= (length args) 4))
(display "Usage: ")
(display (list-ref args 0))
(display " M N [MAX_SOLUTIONS]")
(newline)
(exit 1))
(define m (string->number (list-ref args 1)))
(define n (string->number (list-ref args 2)))
(define max-solutions
(if (= (length args) 4)
(string->number (list-ref args 3))
+inf.0))

(define generate-peaceful-queens
(make-peaceful-queens-generator m n))

(let loop ((next-solution-number 1))
(when (<= next-solution-number max-solutions)
(let ((solution (generate-peaceful-queens)))
(when (success? solution)
(display "Solution ")
(display next-solution-number)
(newline)
(display (queens->string n solution))
(newline)
(newline)
(loop (+ next-solution-number 1))))))
```
Output:

\$ csc -O3 peaceful_queens.scm && ./peaceful_queens 4 5

```Solution 1
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+

Solution 2
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+

Solution 3
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+

Solution 4
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+

Solution 5
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+

Solution 6
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+

Solution 7
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |    |  W |    |  W |
+----+----+----+----+----+
|  B |    |    |    |    |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+

Solution 8
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+

Solution 9
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+

Solution 10
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+

Solution 11
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+

Solution 12
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+

Solution 13
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+

Solution 14
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+
|    |    |  B |    |    |
+----+----+----+----+----+
|  W |    |    |    |  W |
+----+----+----+----+----+

Solution 15
+----+----+----+----+----+
|    |  B |    |  B |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+
|    |    |    |    |  B |
+----+----+----+----+----+
|  W |    |  W |    |    |
+----+----+----+----+----+

Solution 16
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+

Solution 17
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+

Solution 18
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |    |  B |    |  B |
+----+----+----+----+----+
|  W |    |    |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
```

### All non-equivalent solutions

Works with: CHICKEN version 5.3.0
```;;;
;;; Solutions to the Peaceful Chess Queen Armies puzzle, in R7RS
;;; Scheme. This implementation returns only one of each equivalent
;;; solution. See https://oeis.org/A260680
;;;
;;; I weed out equivalent solutions by comparing them tediously
;;;
;;; (At least when compiled with CHICKEN 5.3.0, this program gets kind
;;; of slow for m=5, n=6, once you get past having found the 35
;;; non-equivalent solutions. There are still other, equivalent
;;; solutions to eliminate.)
;;;
;;; https://rosettacode.org/wiki/Peaceful_chess_queen_armies
;;;

(cond-expand
(r7rs)
(chicken (import (r7rs))))

(import (scheme process-context))

(define-record-type <&fail>
(make-the-one-unique-&fail-that-you-must-not-make-twice)
do-not-use-this:&fail?)

(define &fail
(make-the-one-unique-&fail-that-you-must-not-make-twice))

(define (failure? f)
(eq? f &fail))

(define (success? f)
(not (failure? f)))

(define *suspend*
(make-parameter (lambda (x) x)))

(define (suspend v)
((*suspend*) v))

(define (fail-forever)
(let loop ()
(suspend &fail)
(loop)))

(define (make-generator-procedure thunk)
;;
;; Make a suspendable procedure that takes no arguments. It is a
;; simple generator of values. (One can elaborate on this to have
;; the procedure accept an argument upon resumption, like an Icon
;; co-expression.)
;;
(define (next-run return)
(define (my-suspend v)
(set! return
(call/cc
(lambda (resumption-point)
(set! next-run resumption-point)
(return v)))))
(parameterize ((*suspend* my-suspend))
(suspend (thunk))
(fail-forever)))
(lambda ()
(call/cc next-run)))

(define (isqrt m)
;; Integer Newton’s method. See
;; https://en.wikipedia.org/w/index.php?title=Integer_square_root&oldid=1074473475#Using_only_integer_division
(let ((k (truncate-quotient m 2)))
(if (zero? k)
m
(let loop ((k k)
(k^ (truncate-quotient
(+ k (truncate-quotient m k)) 2)))
(if (< k^ k)
(loop k^ (truncate-quotient
(+ k^ (truncate-quotient m k^)) 2))
k)))))

(define (ij->index n i j)
(let ((i1 (- i 1))
(j1 (- j 1)))
(+ i1 (* n j1))))

(define (index->ij n index)
(let-values (((q r) (floor/ index n)))
(values (+ r 1) (+ q 1))))

(index->ij n (+ (ij->index n i j) 1)))

(define (index-rotate90 n index)
(let-values (((i j) (index->ij n index)))
(ij->index n (- n j -1) i)))

(define (index-rotate180 n index)
(let-values (((i j) (index->ij n index)))
(ij->index n (- n i -1) (- n j -1))))

(define (index-rotate270 n index)
(let-values (((i j) (index->ij n index)))
(ij->index n j (- n i -1))))

(define (index-reflecti n index)
(let-values (((i j) (index->ij n index)))
(ij->index n (- n i -1) j)))

(define (index-reflectj n index)
(let-values (((i j) (index->ij n index)))
(ij->index n i (- n j -1))))

(define (index-reflect-diag-down n index)
(let-values (((i j) (index->ij n index)))
(ij->index n j i)))

(define (index-reflect-diag-up n index)
(let-values (((i j) (index->ij n index)))
(ij->index n (- n j -1) (- n i -1))))

(define BLACK 'B)
(define WHITE 'W)

(define (reverse-color c)
(cond ((eq? c WHITE) BLACK)
((eq? c BLACK) WHITE)
(else c)))

(if (eq? c WHITE)
reverse-color
(lambda (x) x)))

(define-record-type <queen>
(make-queen color rank file)
queen?
(color queen-color)
(rank queen-rank)
(file queen-file))

(define (queens->board queens)
(let ((board (make-vector (* n n) #f)))
(do ((q queens (cdr q)))
((null? q))
(let* ((color (queen-color (car q)))
(i (queen-rank (car q)))
(j (queen-file (car q))))
(vector-set! board (ij->index n i j) color)))
board))

(define-syntax board-partial-equiv?
(syntax-rules ()
((_ board1 board2 n*n n reindex recolor)
(let loop ((i 0))
(or (= i n*n)
(let ((color1 (vector-ref board1 i))
(color2 (recolor (vector-ref board2 (reindex n i)))))
(and (eq? color1 color2)
(loop (+ i 1)))))))))

(define (board-equiv? board1 board2)
(define (identity x) x)
(define (2nd-argument n i) i)
(let ((n*n (vector-length board1)))
(or (board-partial-equiv? board1 board2 n*n #f
2nd-argument identity)
(board-partial-equiv? board1 board2 n*n #f
2nd-argument reverse-color)
(let ((n (isqrt n*n)))
(or (board-partial-equiv? board1 board2 n*n n
index-rotate90
identity)
(board-partial-equiv? board1 board2 n*n n
index-rotate90
reverse-color)
(board-partial-equiv? board1 board2 n*n n
index-rotate180
identity)
(board-partial-equiv? board1 board2 n*n n
index-rotate180
reverse-color)
(board-partial-equiv? board1 board2 n*n n
index-rotate270
identity)
(board-partial-equiv? board1 board2 n*n n
index-rotate270
reverse-color)
(board-partial-equiv? board1 board2 n*n n
index-reflecti
identity)
(board-partial-equiv? board1 board2 n*n n
index-reflecti
reverse-color)
(board-partial-equiv? board1 board2 n*n n
index-reflectj
identity)
(board-partial-equiv? board1 board2 n*n n
index-reflectj
reverse-color)
(board-partial-equiv? board1 board2 n*n n
index-reflect-diag-down
identity)
(board-partial-equiv? board1 board2 n*n n
index-reflect-diag-down
reverse-color)
(board-partial-equiv? board1 board2 n*n n
index-reflect-diag-up
identity)
(board-partial-equiv? board1 board2 n*n n
index-reflect-diag-up
reverse-color) )))))

(define (queens->string n queens)

(define board (queens->board queens))

(define rule
(let ((str "+"))
(do ((j 1 (+ j 1)))
((= j (+ n 1)))
(set! str (string-append str "----+")))
str))

(define str "")

(when (< 0 n)
(set! str rule)
(do ((i n (- i 1)))
((= i 0))
(set! str (string-append str "\n"))
(do ((j 1 (+ j 1)))
((= j (+ n 1)))
(let* ((color (vector-ref board (ij->index n i j)))
(representation
(cond ((eq? color #f) "    ")
((eq? color BLACK) "  B ")
((eq? color WHITE) "  W ")
(else " ?? "))))
(set! str (string-append str "|" representation))))
(set! str (string-append str "|\n" rule))))
str)

(define (queen-fits-in? queen other-queens)
(or (null? other-queens)
(let ((other (car other-queens)))
(let ((colorq (queen-color queen))
(rankq (queen-rank queen))
(fileq (queen-file queen))
(coloro (queen-color other))
(ranko (queen-rank other))
(fileo (queen-file other)))
(if (eq? colorq coloro)
(and (or (not (= rankq ranko))
(not (= fileq fileo)))
(queen-fits-in? queen (cdr other-queens)))
(and (not (= rankq ranko))
(not (= fileq fileo))
(not (= (+ rankq fileq) (+ ranko fileo)))
(not (= (- rankq fileq) (- ranko fileo)))
(queen-fits-in? queen (cdr other-queens))))))))

(define (latest-queen-fits-in? queens)
(or (null? (cdr queens))
(queen-fits-in? (car queens) (cdr queens))))

(define (make-peaceful-queens-generator m n)
(make-generator-procedure
(lambda ()
(define solutions '())

(let loop ((queens (list (make-queen BLACK 1 1)))
(num-queens 1))

(let ((color (reverse-color (queen-color (car queens)))))
(loop (cons (make-queen color 1 1) queens)
(+ num-queens 1))))

(define (move-a-queen)
(let drop-one ((queens queens)
(num-queens num-queens))
(if (zero? num-queens)
(loop '() 0)
(let* ((latest (car queens))
(color (queen-color latest))
(rank (queen-rank latest))
(file (queen-file latest)))
(if (and (= rank n) (= file n))
(drop-one (cdr queens) (- num-queens 1))
(let-values (((rank^ file^)
(loop (cons (make-queen color rank^ file^)
(cdr queens))
num-queens)))))))

(cond ((zero? num-queens)
;; There are no more solutions.
&fail)

((latest-queen-fits-in? queens)
(if (= num-queens (* 2 m))
(let ((board (queens->board queens)))
;; The current "queens" is a solution.
(unless (member board solutions board-equiv?)
;; The current "queens" is a *new* solution.
(set! solutions (cons board solutions))
(suspend queens))
(move-a-queen))

(else
(move-a-queen)))))))

(define args (command-line))
(unless (or (= (length args) 3)
(= (length args) 4))
(display "Usage: ")
(display (list-ref args 0))
(display " M N [MAX_SOLUTIONS]")
(newline)
(exit 1))
(define m (string->number (list-ref args 1)))
(define n (string->number (list-ref args 2)))
(define max-solutions
(if (= (length args) 4)
(string->number (list-ref args 3))
+inf.0))

(define generate-peaceful-queens
(make-peaceful-queens-generator m n))

(let loop ((next-solution-number 1))
(when (<= next-solution-number max-solutions)
(let ((solution (generate-peaceful-queens)))
(when (success? solution)
(display "Solution ")
(display next-solution-number)
(newline)
(display (queens->string n solution))
(newline)
(newline)
(loop (+ next-solution-number 1))))))
```
Output:

\$ csc -O5 peaceful_queens2.scm && ./peaceful_queens2 4 5

```Solution 1
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |  W |    |    |
+----+----+----+----+----+
|  B |    |    |    |  B |
+----+----+----+----+----+

Solution 2
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+

Solution 3
+----+----+----+----+----+
|    |  W |    |  W |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
|    |    |    |    |  W |
+----+----+----+----+----+
|  B |    |  B |    |    |
+----+----+----+----+----+
```

## Swift

Translation of: Kotlin
```enum Piece {
case empty, black, white
}

typealias Position = (Int, Int)

func place(_ m: Int, _ n: Int, pBlackQueens: inout [Position], pWhiteQueens: inout [Position]) -> Bool {
guard m != 0 else {
return true
}

var placingBlack = true

for i in 0..<n {
inner: for j in 0..<n {
let pos = (i, j)

for queen in pBlackQueens where queen == pos || !placingBlack && isAttacking(queen, pos) {
continue inner
}

for queen in pWhiteQueens where queen == pos || placingBlack && isAttacking(queen, pos) {
continue inner
}

if placingBlack {
pBlackQueens.append(pos)
placingBlack = false
} else {
placingBlack = true

pWhiteQueens.append(pos)

if place(m - 1, n, pBlackQueens: &pBlackQueens, pWhiteQueens: &pWhiteQueens) {
return true
} else {
pBlackQueens.removeLast()
pWhiteQueens.removeLast()
}
}
}
}

if !placingBlack {
pBlackQueens.removeLast()
}

return false
}

func isAttacking(_ queen: Position, _ pos: Position) -> Bool {
queen.0 == pos.0 || queen.1 == pos.1 || abs(queen.0 - pos.0) == abs(queen.1 - pos.1)
}

func printBoard(n: Int, pBlackQueens: [Position], pWhiteQueens: [Position]) {
var board = Array(repeating: Piece.empty, count: n * n)

for queen in pBlackQueens {
board[queen.0 * n + queen.1] = .black
}

for queen in pWhiteQueens {
board[queen.0 * n + queen.1] = .white
}

for (i, p) in board.enumerated() {
if i != 0 && i % n == 0 {
print()
}

switch p {
case .black:
print("B ", terminator: "")
case .white:
print("W ", terminator: "")
case .empty:
let j = i / n
let k = i - j * n

if j % 2 == k % 2 {
print("• ", terminator: "")
} else {
print("◦ ", terminator: "")
}
}
}

print("\n")
}

let nms = [
(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6),
(7, 1), (7, 2), (7, 3), (7, 4), (7, 5), (7, 6), (7, 7)
]

for (n, m) in nms {
print("\(m) black and white queens on \(n) x \(n) board")

var blackQueens = [Position]()
var whiteQueens = [Position]()

if place(m, n, pBlackQueens: &blackQueens, pWhiteQueens: &whiteQueens) {
printBoard(n: n, pBlackQueens: blackQueens, pWhiteQueens: whiteQueens)
} else {
print("No solution")
}
}
```
Output:
```1 black and white queens on 2 x 2 board
No solution
1 black and white queens on 3 x 3 board
B ◦ •
◦ • W
• ◦ •

2 black and white queens on 3 x 3 board
No solution
1 black and white queens on 4 x 4 board
B ◦ • ◦
◦ • W •
• ◦ • ◦
◦ • ◦ •

2 black and white queens on 4 x 4 board
B ◦ • ◦
◦ • W •
B ◦ • ◦
◦ • W •

3 black and white queens on 4 x 4 board
No solution
1 black and white queens on 5 x 5 board
B ◦ • ◦ •
◦ • W • ◦
• ◦ • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

2 black and white queens on 5 x 5 board
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ • ◦
• ◦ • ◦ •

3 black and white queens on 5 x 5 board
B ◦ • ◦ B
◦ • W • ◦
• W • ◦ •
◦ • ◦ B ◦
• W • ◦ •

4 black and white queens on 5 x 5 board
• B • B •
◦ • ◦ • B
W ◦ W ◦ •
◦ • ◦ • B
W ◦ W ◦ •

5 black and white queens on 5 x 5 board
No solution
1 black and white queens on 6 x 6 board
B ◦ • ◦ • ◦
◦ • W • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

2 black and white queens on 6 x 6 board
B ◦ • ◦ B ◦
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ • ◦ • ◦
◦ • ◦ • ◦ •

3 black and white queens on 6 x 6 board
B ◦ • ◦ B B
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ •
• ◦ W ◦ • ◦
◦ • ◦ • ◦ •

4 black and white queens on 6 x 6 board
B ◦ • ◦ B B
◦ • W • ◦ •
• W • ◦ • ◦
◦ • ◦ • ◦ B
• ◦ W W • ◦
◦ • ◦ • ◦ •

5 black and white queens on 6 x 6 board
• B • ◦ B ◦
◦ • ◦ B ◦ B
W ◦ • ◦ • ◦
W • W • ◦ •
• ◦ • ◦ • B
W • W • ◦ •

6 black and white queens on 6 x 6 board
No solution
1 black and white queens on 7 x 7 board
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

2 black and white queens on 7 x 7 board
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

3 black and white queens on 7 x 7 board
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

4 black and white queens on 7 x 7 board
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •
◦ • ◦ • ◦ • ◦
• ◦ • ◦ • ◦ •

5 black and white queens on 7 x 7 board
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ • ◦ •
◦ • W • ◦ • ◦
• ◦ • ◦ • ◦ •

6 black and white queens on 7 x 7 board
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
B ◦ • ◦ B ◦ •
◦ • W • ◦ • W
• ◦ • ◦ • ◦ •

7 black and white queens on 7 x 7 board
• B • ◦ • B •
◦ B ◦ • B • ◦
• B • ◦ • B •
◦ • ◦ • B • ◦
W ◦ W ◦ • ◦ W
◦ • ◦ W ◦ • ◦
W ◦ W W • ◦ •```

## Wren

Translation of: Kotlin
Library: Wren-dynamic
```import "./dynamic" for Enum, Tuple

var Piece = Enum.create("Piece", ["empty", "black", "white"])

var Pos = Tuple.create("Pos", ["x", "y"])

var isAttacking = Fn.new { |q, pos|
return q.x == pos.x || q.y == pos.y || (q.x - pos.x).abs == (q.y - pos.y).abs
}

var place // recursive
place = Fn.new { |m, n, blackQueens, whiteQueens|
if (m == 0) return true
var placingBlack = true
for (i in 0...n) {
for (j in 0...n) {
var pos = Pos.new(i, j)
var inner = false
for (queen in blackQueens) {
var equalPos = queen.x == pos.x && queen.y == pos.y
if (equalPos || !placingBlack && isAttacking.call(queen, pos)) {
inner = true
break
}
}
if (!inner) {
for (queen in whiteQueens) {
var equalPos = queen.x == pos.x && queen.y == pos.y
if (equalPos || placingBlack && isAttacking.call(queen, pos)) {
inner = true
break
}
}
if (!inner) {
if (placingBlack) {
placingBlack = false
} else {
if (place.call(m-1, n, blackQueens, whiteQueens)) return true
blackQueens.removeAt(-1)
whiteQueens.removeAt(-1)
placingBlack = true
}
}
}
}
}
if (!placingBlack) blackQueens.removeAt(-1)
return false
}

var```