# N-queens minimum and knights and bishops

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N-queens minimum and knights and bishops is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

N-Queens: you've been there; done that; have the T-shirt. It is time to find the minimum number of Queens, Bishops or Knights that can be placed on an NxN board such that no piece attacks another but every unoccupied square is attacked.

For N=1 to 10 discover the minimum number of Queens, Bishops, and Knights required to fulfill the above requirement. For N=8 print out a possible solution for Queens and Bishops.

## F#

```// Minimum knights to attack all squares not occupied on an NxN chess board. Nigel Galloway: May 12th., 2022
type att={n:uint64; g:uint64}
static member att n g=let g=g|>Seq.fold(fun n g->n ||| (1UL<<<g)) 0UL in {n=n|>Seq.fold(fun n g->n ||| (1UL<<<g)) 0UL; g=g}
static member (+) (n,g)=let x=n.g ||| g.g in {n=n.n ||| g.n; g=x}
let     fN g=let fG n g=[n-g-g-1;n-g-g+1;n-g+2;n-g-2;n+g+g-1;n+g+g+1;n+g-2;n+g+2]|>List.filter(fun x->0<=x && x<g*g && abs(x%g-n%g)+abs(x/g-n/g)=3)|>List.distinct|>List.map(fun n->n/2)
let n,g=Array.init(g*g)(fun n->att.att [n/2] (fG n g)), Array.init(g*g)(fun n->att.att (fG n g) [n/2]) in (fun g->n.[g]),(fun n->g.[n])
type cand={att:att; n:int; g:int}
type Solver={n:cand seq; i:int[]; g:(int -> att) * (int -> att); e:att; l:int[]}
member this.test()=let rec test n i g e l=match g with 0UL->(if i=this.e then Some(n,e) else None)|g when g%2UL=1UL->test n (i+((snd this.g)(this.i.[l])))(g/2UL)(e+1)(l+1) |_->test n i (g/2UL) e (l+1)
let n=this.n|>Seq.choose(fun n->test n n.att (this.e.g^^^n.att.g) 0 0) in if Seq.isEmpty n then None else Some(n|>Seq.minBy snd)
member this.xP()  ={this with n=this.n|>Seq.collect(fun n->[for g in n.n..n.g do let att=n.att+((fst this.g)(this.l.[g])) in yield {n with att=att; n=g}])}
let rec slvK (n:Solver) i g l = match n.test() with Some(r,ta)->match min l (g+ta) with t when t>2*(g+1) || l<t->slvK (n.xP()) (if t<l then Some(r,ta) else i) (g+1) (min t l) |t->Some(min t l,r)
|_->slvK (n.xP()) i (g+1) l
let tC bw s (att:att)=let n=Array2D.init s s (fun n g->if (n+g)%2=bw then (if att.n &&& pown 2UL ((n*s+g)/2) > 0UL then "X" else ".") else (if att.g &&& pown 2UL ((n*s+g)/2) > 0UL then "~" else "o"))
for g in 0..s-1 do n.[g,0..s-1]|>Seq.iter(fun g->printf "%s" g); printfn ""
let solveK g=printfn "\nSolving for %dx%d board" g g
let bs,ws=[|for n in g..g+g..(g*g-1)/2 do for z in 0..g+1..(g*g-1)/2-n->((n+z)/g,(n+z)%g)|],[|for n in 0..g+g..(g*g-1)/2 do for z in 0..g+1..(g*g-1)/2-n->((n+z)/g,(n+z)%g)|]
let i,l=let n,i=[|for n in 0..g-1 do for g in 0..g-1->(n,g)|]|>Array.partition(fun(n,g)->(n+g)%2=1) in n|>Array.map(fun(n,i)->n*g+i), i|>Array.map(fun(n,i)->n*g+i)
let t,f=System.DateTime.UtcNow,fN g
let bK={l=Array.concat[bs|>Array.map(fun(n,i)->n*g+i);i]|>Array.distinct; i=l; e=att.att [0..i.Length-1] [0..l.Length-1]; n=bs|>Array.mapi(fun l (n,e)->let att=((fst f)(n*g+e)) in {att=att; n=l+1; g=i.Length-1}); g=fN g}
let wK={l=Array.concat[ws|>Array.map(fun(n,i)->n*g+i);l]|>Array.distinct; i=i; e=att.att [0..l.Length-1] [0..i.Length-1]; n=ws|>Array.mapi(fun i (n,e)->let att=((fst f)(n*g+e)) in {att=att; n=i+1; g=l.Length-1}); g=fN g}
let (rn,rb),tc=match g with 1|2->(slvK wK None 1 (g*g/2+g%2)).Value, tC 0 g
|x when x%2=0->(slvK bK None 1 (g*g/2)).Value, tC 1 g
|_->let x,y=(slvK bK None 1 (g*g/2)).Value, (slvK wK None 1 (g*g/2+1)).Value in if (fst x)<(fst y) then x,tC 1 g else y,tC 0 g
printfn "Solution found using %d knights in %A:" rn (System.DateTime.UtcNow-t); tc rb.att
for n in 1..10 do solveK n
```
Output:
```Solving for 1x1 board
Solution found using 1 knights in 00:00:00.0331768:
X

Solving for 2x2 board
Solution found using 2 knights in 00:00:00:
Xo
oX

Solving for 3x3 board
Solution found using 4 knights in 00:00:00.0156191:
Xo.
oX~
.~.

Solving for 4x4 board
Solution found using 4 knights in 00:00:00:
~.~.
XoXo
~.~.
.~.~

Solving for 5x5 board
Solution found using 5 knights in 00:00:00:
.o.~.
~X~.~
.o.~.
~X~.~
.o.~.

Solving for 6x6 board
Solution found using 8 knights in 00:00:00:
~.~.~.
.~.~.~
oX~.oX
Xo.~Xo
~.~.~.
.~.~.~

Solving for 7x7 board
Solution found using 13 knights in 00:00:00.1426817:
X~.~.o.
oX~.~.~
X~.~.o.
~.~.~Xo
.~.~.~.
o.oX~.~
.~.o.~X

Solving for 8x8 board
Solution found using 14 knights in 00:00:00.2655969:
o.~X~.~.
X~.~.~.~
o.~.~Xo.
.~.~.o.~
~.o.~.~.
.oX~.~.o
~.~.~.~X
.~.~X~.o

Solving for 9x9 board
Solution found using 14 knights in 00:00:10.2331055:
.~.~.~.~.
~.o.~.o.~
.~Xo.oX~.
~.~.~.~.~
X~.~.~.~X
~.~.~.~.~
.~Xo.oX~.
~.o.~.o.~
.~.~.~.~.

Solving for 10x10 board
Solution found using 16 knights in 00:04:44.0573668:
~.~.~.~.~.
.~.~.~Xo.~
~Xo.~.oX~.
.oX~.~.~.~
~.~.~.~.~.
.~.~.~.~.~
~.~.~.~Xo.
.~Xo.~.oX~
~.oX~.~.~.
.~.~.~.~.~
```

## Go

This was originally a translation of the Wren entry but was substantially improved by Pete Lomax using suggestions from the talk page and has been improved further since then, resulting in an overall execution time of about 22.4 seconds.

Timing is for an Intel Core i7-8565U machine running Ubuntu 20.04.

```package main

import (
"fmt"
"math"
"strings"
"time"
)

var board [][]bool
var diag1, diag2 [][]int
var diag1Lookup, diag2Lookup []bool
var n, minCount int
var layout string

func isAttacked(piece string, row, col int) bool {
if piece == "Q" {
for i := 0; i < n; i++ {
if board[i][col] || board[row][i] {
return true
}
}
if diag1Lookup[diag1[row][col]] || diag2Lookup[diag2[row][col]] {
return true
}
} else if piece == "B" {
if diag1Lookup[diag1[row][col]] || diag2Lookup[diag2[row][col]] {
return true
}
} else { // piece == "K"
if board[row][col] {
return true
}
if row+2 < n && col-1 >= 0 && board[row+2][col-1] {
return true
}
if row-2 >= 0 && col-1 >= 0 && board[row-2][col-1] {
return true
}
if row+2 < n && col+1 < n && board[row+2][col+1] {
return true
}
if row-2 >= 0 && col+1 < n && board[row-2][col+1] {
return true
}
if row+1 < n && col+2 < n && board[row+1][col+2] {
return true
}
if row-1 >= 0 && col+2 < n && board[row-1][col+2] {
return true
}
if row+1 < n && col-2 >= 0 && board[row+1][col-2] {
return true
}
if row-1 >= 0 && col-2 >= 0 && board[row-1][col-2] {
return true
}
}
return false
}

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

func attacks(piece string, row, col, trow, tcol int) bool {
if piece == "Q" {
return row == trow || col == tcol || abs(row-trow) == abs(col-tcol)
} else if piece == "B" {
return abs(row-trow) == abs(col-tcol)
} else { // piece == "K"
rd := abs(trow - row)
cd := abs(tcol - col)
return (rd == 1 && cd == 2) || (rd == 2 && cd == 1)
}
}

func storeLayout(piece string) {
var sb strings.Builder
for _, row := range board {
for _, cell := range row {
if cell {
sb.WriteString(piece + " ")
} else {
sb.WriteString(". ")
}
}
sb.WriteString("\n")
}
layout = sb.String()
}

func placePiece(piece string, countSoFar, maxCount int) {
if countSoFar >= minCount {
return
}
allAttacked := true
ti := 0
tj := 0
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
if !isAttacked(piece, i, j) {
allAttacked = false
ti = i
tj = j
break
}
}
if !allAttacked {
break
}
}
if allAttacked {
minCount = countSoFar
storeLayout(piece)
return
}
if countSoFar <= maxCount {
si := ti
sj := tj
if piece == "K" {
si = si - 2
sj = sj - 2
if si < 0 {
si = 0
}
if sj < 0 {
sj = 0
}
}
for i := si; i < n; i++ {
for j := sj; j < n; j++ {
if !isAttacked(piece, i, j) {
if (i == ti && j == tj) || attacks(piece, i, j, ti, tj) {
board[i][j] = true
if piece != "K" {
diag1Lookup[diag1[i][j]] = true
diag2Lookup[diag2[i][j]] = true
}
placePiece(piece, countSoFar+1, maxCount)
board[i][j] = false
if piece != "K" {
diag1Lookup[diag1[i][j]] = false
diag2Lookup[diag2[i][j]] = false
}
}
}
}
}
}
}

func main() {
start := time.Now()
pieces := []string{"Q", "B", "K"}
limits := map[string]int{"Q": 10, "B": 10, "K": 10}
names := map[string]string{"Q": "Queens", "B": "Bishops", "K": "Knights"}
for _, piece := range pieces {
fmt.Println(names[piece])
fmt.Println("=======\n")

for n = 1; ; n++ {
board = make([][]bool, n)
for i := 0; i < n; i++ {
board[i] = make([]bool, n)
}
if piece != "K" {
diag1 = make([][]int, n)
for i := 0; i < n; i++ {
diag1[i] = make([]int, n)
for j := 0; j < n; j++ {
diag1[i][j] = i + j
}
}
diag2 = make([][]int, n)
for i := 0; i < n; i++ {
diag2[i] = make([]int, n)
for j := 0; j < n; j++ {
diag2[i][j] = i - j + n - 1
}
}
diag1Lookup = make([]bool, 2*n-1)
diag2Lookup = make([]bool, 2*n-1)
}
minCount = math.MaxInt32
layout = ""
for maxCount := 1; maxCount <= n*n; maxCount++ {
placePiece(piece, 0, maxCount)
if minCount <= n*n {
break
}
}
fmt.Printf("%2d x %-2d : %d\n", n, n, minCount)
if n == limits[piece] {
fmt.Printf("\n%s on a %d x %d board:\n", names[piece], n, n)
fmt.Println("\n" + layout)
break
}
}
}
elapsed := time.Now().Sub(start)
fmt.Printf("Took %s\n", elapsed)
}
```
Output:
```Queens
=======

1 x 1  : 1
2 x 2  : 1
3 x 3  : 1
4 x 4  : 3
5 x 5  : 3
6 x 6  : 4
7 x 7  : 4
8 x 8  : 5
9 x 9  : 5
10 x 10 : 5

Queens on a 10 x 10 board:

. . Q . . . . . . .
. . . . . . . . . .
. . . . . . . . Q .
. . . . . . . . . .
. . . . Q . . . . .
. . . . . . . . . .
Q . . . . . . . . .
. . . . . . . . . .
. . . . . . Q . . .
. . . . . . . . . .

Bishops
=======

1 x 1  : 1
2 x 2  : 2
3 x 3  : 3
4 x 4  : 4
5 x 5  : 5
6 x 6  : 6
7 x 7  : 7
8 x 8  : 8
9 x 9  : 9
10 x 10 : 10

Bishops on a 10 x 10 board:

. . . . . . . . . B
. . . . . . . . . .
. . . B . B . . . .
. . . B . B . B . .
B . . . . . . . . .
. . . . . . . . . .
. . . . . B . . . .
. . . . . B . . . .
. . . . . B . . . .
. . . . . . . . . .

Knights
=======

1 x 1  : 1
2 x 2  : 4
3 x 3  : 4
4 x 4  : 4
5 x 5  : 5
6 x 6  : 8
7 x 7  : 13
8 x 8  : 14
9 x 9  : 14
10 x 10 : 16

Knights on a 10 x 10 board:

. . . . . . . . . .
. . K K . . . . . .
. . K K . . . K K .
. . . . . . . K K .
. . . . . . . . . .
. . . . . . . . . .
. K K . . . . . . .
. K K . . . K K . .
. . . . . . K K . .
. . . . . . . . . .

Took 22.383253365s
```

## J

This is a crude attempt -- brute force search with some minor state space pruning. I am not patient enough to run this for boards larger than 7x7 for knights:

```genboard=: {{
safelen=:2*len=: {.y
shape=: 2\$len
board=: shape\$0
safeshape=: ,~safelen
c=:,coords=: safeshape#.shape#:i.shape
qrow=. i:{.shape-1
qcol=. qrow*safelen
qdiag1=. qrow+qcol
qdiag2=. qrow-qcol
queen=: ~.qrow,qcol,qdiag1,qdiag2
k1=. ,(1 _1*safelen)+/2 _2
k2=. ,(2 _2*safelen)+/1 _1
knight=: 0,k1,k2
bishop=: ~.qdiag1,qdiag2
row=. i.len
first=: ~.,coords#"1~(row<:>.<:len%2) * >:/~ row
EMPTY
}}

placebishops=: {{coords #&,~ 1 (<.-:len)} board}}

placequeens=: {{
N=. 0
while. N=. N+1 do.
assert. N<:#c
for_seq. first do.
board=. coords e.queen+seq
if. 0 e.,board do.
if. 1<N do.
seq=. board queen place1 N seq
if. #seq do.
assert. N-:#seq
assert. */c e.,queen+/seq
seq return.
end.
end.
else.
seq return.
end.
end.
end.
EMPTY
}}

placeknights=:{{
N=. 0
while. N=. N+1 do.
assert. N<:#c
for_seq. c do.
board=. coords e.knight+seq
if. 0 e.,board do.
if. 1<N do.
seq=. board knight place1 N seq
if. #seq do.
assert. N-:#seq
assert. */c e.,knight+/seq
seq return.
end.
end.
else.
seq return.
end.
end.
end.
EMPTY
}}

NB. x: board with currently attacked locations marked
NB. m: move targets
NB. n: best sequence length so far
NB. y: coords of placed pieces
place1=: {{
for_seq. y,"1 0(#~ (>./y) < ])(,0=x)#c do.
board=. x>.coords e.,m+/seq
if. 0 e.,board do.          NB. further work needed?
if.n>#seq do.
seq=. board m place1 n seq
if.#seq do.seq return.end.
end.
else. seq return.
end.
end.
EMPTY
}}

B=:Q=:K=:i.0
for_order.1+i.y do.
genboard order
B=: 1j1#"1'.B'{~coords e.b=. placebishops ''
Q=: 1j1#"1'.Q'{~coords e.q=. placequeens ''
if. 8>order do.
K=: 1j1#"1'.K'{~coords e.k=. placeknights ''
echo (":B;Q;K),&.|:>(' B: ',":#b);(' Q: ',":#q);' K: ',":#k
else.
echo (":B;Q),&.|:>(' B: ',":#b);' Q: ',":#q
end.
end.
}}
```

```   task 10
+--+--+--+ B: 1
|B |Q |K | Q: 1
+--+--+--+ K: 1
+----+----+----+ B: 2
|. . |Q . |K K | Q: 1
|B B |. . |K K | K: 4
+----+----+----+
+------+------+------+ B: 3
|. . . |. . . |K K K | Q: 1
|B B B |. Q . |. K . | K: 4
|. . . |. . . |. . . |
+------+------+------+
+--------+--------+--------+ B: 4
|. . . . |Q . . . |. K K . | Q: 3
|. . . . |. . Q . |. K K . | K: 4
|B B B B |. . . . |. . . . |
|. . . . |. Q . . |. . . . |
+--------+--------+--------+
+----------+----------+----------+ B: 5
|. . . . . |Q . . . . |K . . . . | Q: 3
|. . . . . |. . . Q . |. . . . . | K: 5
|B B B B B |. . . . . |. . K K . |
|. . . . . |. . Q . . |. . K K . |
|. . . . . |. . . . . |. . . . . |
+----------+----------+----------+
+------------+------------+------------+ B: 6
|. . . . . . |Q . . . . . |K . . . . K | Q: 4
|. . . . . . |. . Q . . . |. . . . . . | K: 8
|. . . . . . |. . . . . Q |. . K K . . |
|B B B B B B |. . . . . . |. . K K . . |
|. . . . . . |. . . . . . |. . . . . . |
|. . . . . . |. Q . . . . |K . . . . K |
+------------+------------+------------+
+--------------+--------------+--------------+ B: 7
|. . . . . . . |. . . . . . . |K K K K . K . | Q: 4
|. . . . . . . |. Q . . . . . |. . . . . . . | K: 13
|. . . . . . . |. . . . Q . . |. . . . . K . |
|B B B B B B B |. . . . . . . |K . . . . K . |
|. . . . . . . |. . . . . . . |. . . . . . . |
|. . . . . . . |Q . . . . . . |K . K K . K . |
|. . . . . . . |. . . Q . . . |. . . . . . K |
+--------------+--------------+--------------+
+----------------+----------------+ B: 8
|. . . . . . . . |Q . . . . . . . | Q: 5
|. . . . . . . . |. . Q . . . . . |
|. . . . . . . . |. . . . Q . . . |
|. . . . . . . . |. Q . . . . . . |
|B B B B B B B B |. . . Q . . . . |
|. . . . . . . . |. . . . . . . . |
|. . . . . . . . |. . . . . . . . |
|. . . . . . . . |. . . . . . . . |
+----------------+----------------+
+------------------+------------------+ B: 9
|. . . . . . . . . |Q . . . . . . . . | Q: 5
|. . . . . . . . . |. . Q . . . . . . |
|. . . . . . . . . |. . . . . . Q . . |
|. . . . . . . . . |. . . . . . . . . |
|B B B B B B B B B |. . . . . . . . . |
|. . . . . . . . . |. Q . . . . . . . |
|. . . . . . . . . |. . . . . Q . . . |
|. . . . . . . . . |. . . . . . . . . |
|. . . . . . . . . |. . . . . . . . . |
+------------------+------------------+
+--------------------+--------------------+ B: 10
|. . . . . . . . . . |Q . . . . . . . . . | Q: 6
|. . . . . . . . . . |. . Q . . . . . . . |
|. . . . . . . . . . |. . . . . . . . Q . |
|. . . . . . . . . . |. . . . . . . . . . |
|. . . . . . . . . . |. . . . . . . . . . |
|B B B B B B B B B B |. . . Q . . . . . . |
|. . . . . . . . . . |. . . . . . . . . Q |
|. . . . . . . . . . |. . . . . . . . . . |
|. . . . . . . . . . |. . . . . Q . . . . |
|. . . . . . . . . . |. . . . . . . . . . |
+--------------------+--------------------+
```

## Julia

Uses the Cbc optimizer (github.com/coin-or/Cbc) for its complementarity support.

```""" Rosetta code task N-queens_minimum_and_knights_and_bishops """

import Cbc
using JuMP
using LinearAlgebra

""" Make a printable string from a matrix of zeros and ones using a char ch for 1, a period for !=1. """
function smatrix(x, n, ch)
s = ""
for i in 1:n, j in 1:n
s *= lpad(x[i, j] == 1 ? "\$ch" : ".", 2) * (j == n ? "\n" : "")
end
return s
end

""" N-queens minimum, oeis.org/A075458 """
function queensminimum(N, char = "Q")
if  N < 4
a = zeros(Int, N, N)
a[N ÷ 2 + 1, N ÷ 2 + 1] = 1
return 1, smatrix(a, N, char)
end
model = Model(Cbc.Optimizer)
@variable(model, x[1:N, 1:N], Bin)

for i in 1:N
@constraint(model, sum(x[i, :]) <= 1)
@constraint(model, sum(x[:, i]) <= 1)
end
for i in -(N - 1):(N-1)
@constraint(model, sum(diag(x, i)) <= 1)
@constraint(model, sum(diag(reverse(x, dims = 1), i)) <= 1)
end
for i in 1:N, j in 1:N
@constraint(model, sum(x[i, :]) + sum(x[:, j]) +  sum(diag(x, j - i)) +
sum(diag(rotl90(x), N - j - i + 1)) >= 1)
end

set_silent(model)
@objective(model, Min, sum(x))
optimize!(model)

solution = round.(Int, value.(x))
minresult = sum(solution)
return minresult, smatrix(solution, N, char)
end

""" N-bishops minimum, same as N """
function bishopsminimum(N, char = "B")
N == 1 && return 1, "\$char"
N == 2 && return 2, "\$char .\n\$char .\n"

model = Model(Cbc.Optimizer)
@variable(model, x[1:N, 1:N], Bin)

for i in 1:N, j in 1:N
@constraint(model, sum(diag(x, j - i)) + sum(diag(rotl90(x), N - j - i + 1)) >= 1)
end

set_silent(model)
@objective(model, Min, sum(x))
optimize!(model)

solution = round.(Int, value.(x))
minresult = sum(solution)
return minresult, smatrix(solution, N, char)
end

""" N-knights minimum, oeis.org/A342576 """
function knightsminimum(N, char = "N")
N < 2 && return 1, "\$char"

knightdeltas = [(1, 2), (2, 1), (2, -1), (1, -2), (-1, -2), (-2, -1), (-2, 1), (-1, 2)]

model = Model(Cbc.Optimizer)

# to simplify the constraints, embed the board of size N inside a board of size n + 4
@variable(model, x[1:N+4, 1:N+4], Bin)

@constraint(model, x[:, 1:2] .== 0)
@constraint(model, x[1:2, :] .== 0)
@constraint(model, x[:, N+3:N+4] .== 0)
@constraint(model, x[N+3:N+4, :] .== 0)

for i in 3:N+2, j in 3:N+2
@constraint(model, x[i, j] + sum(x[i + d[1], j + d[2]] for d in knightdeltas) >= 1)
@constraint(model, x[i, j] => {sum(x[i + d[1], j + d[2]] for d in knightdeltas) == 0})
end

set_silent(model)
@objective(model, Min, sum(x))
optimize!(model)

solution = round.(Int, value.(x))
minresult = sum(solution)
return minresult, smatrix(solution[3:N+2, 3:N+2], N, char)
end

const examples = fill("", 3)
println("   Squares    Queens   Bishops   Knights")
@time for N in 1:10
i, examples[1] = queensminimum(N)
i, examples[2] = bishopsminimum(N)
i, examples[3] = knightsminimum(N)
end

println("\nExamples for N = 10:\n\nQueens:\n", examples[1], "\nBishops:\n", examples[2],
"\nKnights:\n", examples[3])
```
Output:
```   Squares    Queens   Bishops   Knights
1         1         1         1
4         1         2         4
9         1         3         4
16         3         4         4
25         3         5         5
36         4         6         8
49         4         7        13
64         5         8        14
81         5         9        14
100         5        10        16
49.922920 seconds (30.87 M allocations: 1.656 GiB, 2.39% gc time, 65.49% compilation time)

Examples for N = 10:

Queens:
. . . . . . . . . .
. . Q . . . . . . .
. . . . . . . . . .
. . . . . . . . Q .
. . . . . . . . . .
. . . . Q . . . . .
. . . . . . . . . .
Q . . . . . . . . .
. . . . . . . . . .
. . . . . . Q . . .

Bishops:
. . . . . . . . . .
. . . . . B . . . .
. . . . . B . . . .
. B . . . . . . . .
. B . . . . B . . .
. . . . . . B . . .
. . B . . . B . . .
. . . . . . B . . .
. B . . . . . . . .
. . . . . . . . . .

Knights:
. . . . . . . . . .
. . N N . . . . . .
. . N N . . . N N .
. . . . . . . N N .
. . . . . . . . . .
. . . . . . . . . .
. N N . . . . . . .
. N N . . . N N . .
. . . . . . N N . .
. . . . . . . . . .

```

## Nim

Translation of: Go
```import std/[monotimes, sequtils, strformat]

type

Piece {.pure.} = enum Queen, Bishop, Knight
Solver = object
n: Natural
board: seq[seq[bool]]
diag1, diag2: seq[seq[int]]
diag1Lookup, diag2Lookup: seq[bool]
minCount: int
layout: string

func isAttacked(s: Solver; piece: Piece; row, col: Natural): bool =
case piece
of Queen:
for i in 0..<s.n:
if s.board[i][col] or s.board[row][i]:
return true
result = s.diag1Lookup[s.diag1[row][col]] or s.diag2Lookup[s.diag2[row][col]]
of Bishop:
result = s.diag1Lookup[s.diag1[row][col]] or s.diag2Lookup[s.diag2[row][col]]:
of Knight:
result = s.board[row][col] or
row + 2 < s.n and col - 1 >= 0  and s.board[row + 2][col - 1] or
row - 2 >= 0  and col - 1 >= 0  and s.board[row - 2][col - 1] or
row + 2 < s.n and col + 1 < s.n and s.board[row + 2][col + 1] or
row - 2 >= 0  and col + 1 < s.n and s.board[row - 2][col + 1] or
row + 1 < s.n and col + 2 < s.n and s.board[row + 1][col + 2] or
row - 1 >= 0  and col + 2 < s.n and s.board[row - 1][col + 2] or
row + 1 < s.n and col - 2 >= 0  and s.board[row + 1][col - 2] or
row - 1 >= 0  and col - 2 >= 0  and s.board[row - 1][col - 2]

func attacks(piece: Piece; row, col, trow, tcol: int): bool =
case piece
of Queen:
result = row == trow or col == tcol or abs(row - trow) == abs(col - tcol)
of Bishop:
result = abs(row - trow) == abs(col - tcol)
of Knight:
let rd = abs(trow - row)
let cd = abs(tcol - col)
result = (rd == 1 and cd == 2) or (rd == 2 and cd == 1)

func storeLayout(s: var Solver; piece: Piece) =
for row in s.board:
for cell in row:
s.layout.add if cell: (\$piece)[0] & ' ' else: ". "

func placePiece(s: var Solver; piece: Piece; countSoFar, maxCount: int) =
if countSoFar >= s.minCount: return

var allAttacked = true
var ti, tj = 0
block CheckAttacked:
for i in 0..<s.n:
for j in 0..<s.n:
if not s.isAttacked(piece, i, j):
allAttacked = false
ti = i
tj = j
break CheckAttacked

if allAttacked:
s.minCount = countSoFar
s.storeLayout(piece)
return

if countSoFar <= maxCount:
var si = ti
var sj = tj
if piece == Knight:
dec si, 2
dec sj, 2
if si < 0: si = 0
if sj < 0: sj = 0

for i in si..<s.n:
for j in sj..<s.n:
if not s.isAttacked(piece, i, j):
if (i == ti and j == tj) or attacks(piece, i, j, ti, tj):
s.board[i][j] = true
if piece != Knight:
s.diag1Lookup[s.diag1[i][j]] = true
s.diag2Lookup[s.diag2[i][j]] = true
s.placePiece(piece, countSoFar + 1, maxCount)
s.board[i][j] = false
if piece != Knight:
s.diag1Lookup[s.diag1[i][j]] = false
s.diag2Lookup[s.diag2[i][j]] = false

func initSolver(n: Positive; piece: Piece): Solver =
result.n = n
result.board = newSeqWith(n, newSeq[bool](n))
if piece != Knight:
result.diag1 = newSeqWith(n, newSeq[int](n))
result.diag2 = newSeqWith(n, newSeq[int](n))
for i in 0..<n:
for j in 0..<n:
result.diag1[i][j] = i + j
result.diag2[i][j] = i - j + n - 1
result.diag1Lookup = newSeq[bool](2 * n - 1)
result.diag2Lookup = newSeq[bool](2 * n - 1)
result.minCount = int32.high

proc main() =
let start = getMonoTime()
const Limits = [Queen: 10, Bishop: 10, Knight: 10]
for piece in Piece.low..Piece.high:
echo \$piece & 's'
echo "=======\n"
var n = 0
while true:
inc n
var solver = initSolver(n , piece)
for maxCount in 1..(n * n):
solver.placePiece(piece, 0, maxCount)
if solver.minCount <= n * n:
break
echo &"{n:>2} × {n:<2} : {solver.minCount}"
if n == Limits[piece]:
echo &"\n{\$piece}s on a {n} × {n} board:"
echo '\n' & solver.layout
break
let elapsed = getMonoTime() - start
echo "Took: ", elapsed

main()
```
Output:
```Queens
=======

1 × 1  : 1
2 × 2  : 1
3 × 3  : 1
4 × 4  : 3
5 × 5  : 3
6 × 6  : 4
7 × 7  : 4
8 × 8  : 5
9 × 9  : 5
10 × 10 : 5

Queens on a 10 × 10 board:

. . Q . . . . . . .
. . . . . . . . . .
. . . . . . . . Q .
. . . . . . . . . .
. . . . Q . . . . .
. . . . . . . . . .
Q . . . . . . . . .
. . . . . . . . . .
. . . . . . Q . . .
. . . . . . . . . .

Bishops
=======

1 × 1  : 1
2 × 2  : 2
3 × 3  : 3
4 × 4  : 4
5 × 5  : 5
6 × 6  : 6
7 × 7  : 7
8 × 8  : 8
9 × 9  : 9
10 × 10 : 10

Bishops on a 10 × 10 board:

. . . . . . . . . B
. . . . . . . . . .
. . . B . B . . . .
. . . B . B . B . .
B . . . . . . . . .
. . . . . . . . . .
. . . . . B . . . .
. . . . . B . . . .
. . . . . B . . . .
. . . . . . . . . .

Knights
=======

1 × 1  : 1
2 × 2  : 4
3 × 3  : 4
4 × 4  : 4
5 × 5  : 5
6 × 6  : 8
7 × 7  : 13
8 × 8  : 14
9 × 9  : 14
10 × 10 : 16

Knights on a 10 × 10 board:

. . . . . . . . . .
. . K K . . . . . .
. . K K . . . K K .
. . . . . . . K K .
. . . . . . . . . .
. . . . . . . . . .
. K K . . . . . . .
. K K . . . K K . .
. . . . . . K K . .
. . . . . . . . . .

Took: (seconds: 19, nanosecond: 942476699)
```

## Pascal

### Free Pascal

The first Q,B in the first row is only placed lmt..mid because of symmetry reasons.
14 Queens takes 2 min @home ~2.5x faster than TIO.RUN

```program TestMinimalQueen;
{\$MODE DELPHI}{\$OPTIMIZATION ON,ALL}

uses
sysutils;
type
tDeltaKoor = packed record
dRow,
dCol : Int8;
end;
const
cKnightAttacks : array[0..7] of tDeltaKoor =
((dRow:-2;dCol:-1),(dRow:-2;dCol:+1),
(dRow:-1;dCol:-2),(dRow:-1;dCol:+2),
(dRow:+1;dCol:-2),(dRow:+1;dCol:+2),
(dRow:+2;dCol:-1),(dRow:+2;dCol:+1));

KoorCOUNT = 16;

type
tLimit = 0..KoorCOUNT-1;

tPlayGround = array[tLimit,tLimit] of byte;
tCheckPG = array[0..2*KoorCOUNT] of tplayGround;
tpPlayGround = ^tPlayGround;

var
{\$ALIGN 32}
CPG :tCheckPG;
Qsol,BSol,KSol :tPlayGround;
pgIdx,minIdx : nativeInt;

procedure pG_Out(pSol:tpPlayGround;ConvChar : string;lmt: NativeInt);
var
row,col: NativeInt;
begin
iF length(ConvChar)<>3 then
EXIT;
for row := lmt downto 0 do
Begin
for col := 0 to lmt do
write(ConvChar[1+pSol^[row,col]],' ');
writeln;
end;
writeln;
end;

procedure LeftAscDia(row,col,lmt: NativeInt);
var
pPG :tpPlayGround;
j: NativeInt;
begin
pPG := @CPG[pgIdx];
if row >= col then
begin
j := row-col;
col := lmt-j;
row := lmt;
repeat
pPG^[row,col] := 1;
dec(col);
dec(row);
until col < 0;
end
else
begin
j := col-row;
row := lmt-j;
col := lmt;
repeat
pPG^[row,col] := 1;
dec(row);
dec(col);
until row < 0;
end;
end;

procedure RightAscDia(row,col,lmt: NativeInt);
var
pPG :tpPlayGround;
j: NativeInt;
begin
pPG := @CPG[pgIdx];
j := row+col;
if j <= lmt then
begin
col := j;
row := 0;
repeat
pPG^[row,col] := 1;
dec(col);
inc(row);
until col < 0;
end
else
begin
col := lmt;
row := j-lmt;
repeat
pPG^[row,col] := 1;
inc(row);
dec(col);
until row > lmt;
end;
end;

function check(lmt:nativeInt):boolean;
//check, if all fields are attacked
var
pPG :tpPlayGround;
pRow : pByte;
row,col: NativeInt;
Begin
pPG := @CPG[pgIdx];
For row := lmt downto 0 do
begin
pRow := @pPG^[row,0];
For col := lmt downto 0 do
if pRow[col] = 0 then
EXIT(false);
end;
exit(true);
end;

procedure SetQueen(row,lmt: NativeInt);
var
pPG :tpPlayGround;
i,col,t: NativeInt;
begin
t := pgIdx+1;
if t = minIDX then
EXIT;
pgIdx:= t;
//use state before
//  CPG[pgIdx]:=CPG[pgIdx-1];
move(CPG[t-1],CPG[t],SizeOf(tPlayGround));
col := lmt;
//first row only check one half -> symmetry
if row = 0 then
col := col shr 1;

//check every column
For col := col downto 0 do
begin
pPG := @CPG[pgIdx];
if pPG^[row,col] <> 0 then
continue;
//set diagonals
RightAscDia(row,col,lmt);
LeftAscDia(row,col,lmt);
//set row and column as attacked
For i := 0 to lmt do
Begin
pPG^[row,i] := 1;
pPG^[i,col] := 1;
end;
//now set position of queen
pPG^[row,col] := 2;

if check(lmt) then
begin
if minIdx> pgIdx then
begin
minIdx := pgIdx;
Qsol := pPG^;
end;
end
else
if row > lmt then
BREAK
else
//check next rows
For t := row+1 to lmt do
SetQueen(t,lmt);
//copy last state
t := pgIdx;
move(CPG[t-1],CPG[t],SizeOf(tPlayGround));
//    CPG[pgIdx]:=CPG[pgIdx-1];
end;
dec(pgIdx);
end;

procedure SetBishop(row,lmt: NativeInt);
var
pPG :tpPlayGround;
col,t: NativeInt;
begin
if pgIdx = minIDX then
EXIT;
inc(pgIdx);
move(CPG[pgIdx-1],CPG[pgIdx],SizeOf(tPlayGround));
col := lmt;
if row = 0 then
col := col shr 1;
For col := col downto 0 do
begin
pPG := @CPG[pgIdx];
if pPG^[row,col] <> 0 then
continue;

RightAscDia(row,col,lmt);
LeftAscDia(row,col,lmt);

//set position of bishop
pPG^[row,col] := 2;

if check(lmt) then
begin
if minIdx> pgIdx then
begin
minIdx := pgIdx;
Bsol := pPG^;
end;
end
else
if row > lmt then
BREAK
else
begin
//check same row
SetBishop(row,lmt);
//check next row
t := row+1;
if (t <= lmt) then
SetBishop(t,lmt);
end;
move(CPG[pgIdx-1],CPG[pgIdx],SizeOf(tPlayGround));
end;
dec(pgIdx);
end;

var
lmt,max : NativeInt;

BEGIN
max := 10;
write(' nxn  n=:');
For lmt := 1 to max do
write(lmt:3);
writeln;

write(' Queens :');
For lmt := 0 to max-1 do
begin
pgIdx := 0;
minIdx := lmt;
setQueen(0,lmt);
write(minIDX:3);
end;
writeln;

write(' Bishop :');
For lmt := 0 to max-1 do
begin
pgIdx := 0;
minIdx := 2*lmt+1;
setBishop(0,lmt);
write(minIDX:3);
end;
writeln;

pG_Out(@Qsol,'_.Q',max-1);
writeln;

pG_Out(@Bsol,'_.B',max-1);
END.
```
@TIO.RUN:
``` nxn  n=:  1  2  3  4  5  6  7  8  9 10
Queens :  1  1  2  3  3  4  4  5  5  5
Bishop :  1  2  3  4  5  6  7  8  9 10
. . . . . . . . . .
. . . . . . . Q . .
. . . . . . . . . .
. Q . . . . . . . .
. . . . . . . . . .
. . . . . Q . . . .
. . . . . . . . . .
. . . . . . . . . Q
. . . . . . . . . .
. . . Q . . . . . .

. . . . . . . . . .
. . . . B . . . . .
. . . . . . B . . .
. . . . B . . . . .
. . . . . . B . . .
. B . . . . . . . .
. . . B . B . . . .
. . . . . B . . . .
. . . . . . . . B .
. . . . B . . . . .

Real time: 25.935 s CPU share: 99.27 % //@home AMD 5600G real	0m10,784s
```

## Perl

Shows how the latest release of Perl can now use booleans.

Translation of: Raku
```use v5.36;
use builtin 'true', 'false';
no warnings 'experimental::for_list', 'experimental::builtin';

my(@B, @D1, @D2, @D1x, @D2x, \$N, \$Min, \$Layout);

sub X (\$a,\$b)       { my @c; for my \$aa (0..\$a-1) { for my \$bb (0..\$b-1) { push @c, \$aa, \$bb } } @c }
sub Xr(\$a,\$b,\$c,\$d) { my @c; for my \$ab (\$a..\$b)  { for my \$cd (\$c..\$d)  { push @c, \$ab, \$cd } } @c }

sub is_attacked(\$piece, \$r, \$c) {
if (\$piece eq 'Q') {
for (0..\$N-1) { return true if \$B[\$_][\$c] or \$B[\$r][\$_] }
return true if \$D1x[ \$D1[\$r][\$c] ] or
\$D2x[ \$D2[\$r][\$c] ]
} elsif (\$piece eq 'B') {
return true if \$D1x[ \$D1[\$r][\$c] ] or \$D2x[ \$D2[\$r][\$c] ]
} else { # 'K'
return true if (
\$B[\$r][\$c] or
\$r+2 < \$N and \$c-1 >= 0 and \$B[\$r+2][\$c-1] or
\$r-2 >= 0 and \$c-1 >= 0 and \$B[\$r-2][\$c-1] or
\$r+2 < \$N and \$c+1 < \$N and \$B[\$r+2][\$c+1] or
\$r-2 >= 0 and \$c+1 < \$N and \$B[\$r-2][\$c+1] or
\$r+1 < \$N and \$c+2 < \$N and \$B[\$r+1][\$c+2] or
\$r-1 >= 0 and \$c+2 < \$N and \$B[\$r-1][\$c+2] or
\$r+1 < \$N and \$c-2 >= 0 and \$B[\$r+1][\$c-2] or
\$r-1 >= 0 and \$c-2 >= 0 and \$B[\$r-1][\$c-2]
)
}
false
}

sub attacks(\$piece, \$r, \$c, \$tr, \$tc) {
if    (\$piece eq 'Q') { \$r==\$tr or \$c==\$tc or abs(\$r - \$tr)==abs(\$c - \$tc) }
elsif (\$piece eq 'B') { abs(\$r - \$tr) == abs(\$c - \$tc) }
else                  {
my (\$rd, \$cd) = (abs(\$tr - \$r), abs(\$tc - \$c));
(\$rd == 1 and \$cd == 2) or (\$rd == 2 and \$cd == 1)
}
}

sub store_layout(\$piece) {
\$Layout = '';
for (@B) {
map { \$Layout .= \$_ ?  \$piece.' ' : '. ' } @\$_;
\$Layout .=  "\n";
}
}

sub place_piece(\$piece, \$so_far, \$max) {
return if \$so_far >= \$Min;
my (\$all_attacked,\$ti,\$tj) = (true,0,0);
for my(\$i,\$j) (X \$N, \$N) {
unless (is_attacked(\$piece, \$i, \$j)) {
(\$all_attacked,\$ti,\$tj) = (false,\$i,\$j) and last
}
last unless \$all_attacked
}
if (\$all_attacked) {
\$Min = \$so_far;
store_layout(\$piece);
} elsif (\$so_far <= \$max) {
my(\$si,\$sj) = (\$ti,\$tj);
if (\$piece eq 'K') {
\$si -= 2; \$si = 0 if \$si < 0;
\$sj -= 2; \$sj = 0 if \$sj < 0;
}
for my (\$i,\$j) (Xr \$si, \$N-1, \$sj, \$N-1) {
unless (is_attacked(\$piece, \$i, \$j)) {
if ((\$i == \$ti and \$j == \$tj) or attacks(\$piece, \$i, \$j, \$ti, \$tj)) {
\$B[\$i][\$j] = true;
unless (\$piece eq 'K') {
(\$D1x[ \$D1[\$i][\$j] ], \$D2x[ \$D2[\$i][\$j] ]) = (true,true);
};
place_piece(\$piece, \$so_far+1, \$max);
\$B[\$i][\$j] = false;
unless (\$piece eq 'K') {
(\$D1x[ \$D1[\$i][\$j] ], \$D2x[ \$D2[\$i][\$j] ]) = (false,false);
}
}
}
}
}
}

my @Pieces = <Q B K>;
my %Limits = ( 'Q' =>   10,     'B' =>    10,     'K' =>    10   );
my %Names  = ( 'Q' => 'Queens', 'B' => 'Bishops', 'K' =>'Knights');

for my \$piece (@Pieces) {
say \$Names{\$piece} . "\n=======\n";
for (\$N = 1 ; ; \$N++) {
@B = map { [ (false) x \$N ] } 1..\$N;
unless (\$piece eq 'K') {
@D2 = reverse @D1 = map { [\$_ .. \$N+\$_-1] } 0..\$N-1;
@D2x = @D1x = (false) x ((2*\$N)-1);
}
\$Min = 2**31 - 1;
my \$nSQ   = \$N**2;
for my \$max (1..\$nSQ) {
place_piece(\$piece, 0, \$max);
last if \$Min <= \$nSQ
}
printf("%2d x %-2d : %d\n", \$N, \$N, \$Min);
if (\$N == \$Limits{\$piece}) {
printf "\n%s on a %d x %d board:\n", \$Names{\$piece}, \$N, \$N;
say \$Layout and last
}
}
}
```
Output:
```Queens
=======

1 x 1  : 1
2 x 2  : 1
3 x 3  : 1
4 x 4  : 3
5 x 5  : 3
6 x 6  : 4
7 x 7  : 4
8 x 8  : 5
9 x 9  : 5
10 x 10 : 5

Queens on a 10 x 10 board:
. . Q . . . . . . .
. . . . . . . . . .
. . . . . . . . Q .
. . . . . . . . . .
. . . . Q . . . . .
. . . . . . . . . .
Q . . . . . . . . .
. . . . . . . . . .
. . . . . . Q . . .
. . . . . . . . . .

Bishops
=======

1 x 1  : 1
2 x 2  : 2
3 x 3  : 3
4 x 4  : 4
5 x 5  : 5
6 x 6  : 6
7 x 7  : 7
8 x 8  : 8
9 x 9  : 9
10 x 10 : 10

Bishops on a 10 x 10 board:
. . . . . . . . . B
. . . . . . . . . .
. . . B . B . . . .
. . . B . B . B . .
B . . . . . . . . .
. . . . . . . . . .
. . . . . B . . . .
. . . . . B . . . .
. . . . . B . . . .
. . . . . . . . . .

Knights
=======

1 x 1  : 1
2 x 2  : 4
3 x 3  : 4
4 x 4  : 4
5 x 5  : 5
6 x 6  : 8
7 x 7  : 13
8 x 8  : 14
9 x 9  : 14
10 x 10 : 16

Knights on a 10 x 10 board:
. . . . . . . . . .
. . K K . . . . . .
. . K K . . . K K .
. . . . . . . K K .
. . . . . . . . . .
. . . . . . . . . .
. K K . . . . . . .
. K K . . . K K . .
. . . . . . K K . .
. . . . . . . . . .```

## Phix

Library: Phix/pGUI
Library: Phix/online

You can run this online here, with cheat mode on so it should all be done in 22s (8.3 on the desktop). Cheat mode drops the final tricky 10N search from 8,163,658 positions to just 183,937. Without cheat mode it takes 1 min 20s to completely finish on the desktop (would be ~7mins in a browser), but it always remains fully interactive throughout.

```--
-- demo\rosetta\minQBN.exw
-- =======================
--
with javascript_semantics
atom t0 = time()
include pGUI.e
constant title = "Minimum QBN",
help_text = """
Finds the minimum number of Queens, Bishops, or Knights that can be placed on an NxN board
such that no piece attacks another but every unoccupied square is attacked.

The legend on the right shows the current search status: a red number means that is still
being or yet to be tried, and a green number means it has found a solution.

Click on the legend, or press Q/B/N and 1..9/T and/or +/-, to show the answer or maniaical
workings of a particular sub-task. Use space to toggle the timer on and off.

The window title shows additional information for the selected item.
"""
constant maxq = 10, -- 1.0s (13 is 3minutes 58s, with maxn=1)
maxb = 10, -- 1s (100 is 10s, with maxq=1 and maxn=1)
maxn = 10, -- 1mins 10s (9: 11.8s, 8: 8.1s, with maxq=1)
maxqbn = max({maxq,maxb,maxn})

bool cheat = true   -- eg find 16N on a 10x10 w/o disproving 15N first.
-- the total time drops to 8.3s (21.9s under p2js).

sequence board

constant bm = {{-1,-1},{-1,+1},
{+1,-1},{+1,+1}},
rm = {{-1,0},{0,-1},
{+1,0},{0,+1}},
qm = rm&bm,
nm = {{-1,-2},{-2,-1},{-2,+1},{-1,+2},
{+1,-2},{+2,-1},{+2,+1},{+1,+2}}

function get_mm(integer piece)
switch piece do
case 'Q': return qm
case 'B': return bm
case 'N': return nm
end switch
crash("uh?")
end function

function mm_baby(sequence mm, integer piece, row, col, n)
sequence res = {}
for i=1 to length(mm) do
integer {dx,dy} = mm[i],
mx = row,
my = col
while true do
mx += dx
my += dy
if mx<1 or mx>n
or my<1 or my>n then
exit
end if
res &= {{mx,my}}
if piece='N' then exit end if
end while
end for
return res
end function

function get_moves(integer piece, n, row, col)
-- either occupy or attack [row][col]
-- we only have to collect right and down
sequence res = {{row,col}}, -- (occupy)
mm = get_mm(piece)
mm = iff(piece='Q'?extract(mm,{3,4,7,8})
:mm[length(mm)/2+1..\$])
mm = mm_baby(mm,piece,row,col,n)
for i=1 to length(mm) do
integer {mx,my} = mm[i]
if board[mx,my]='0' then
res &= {{mx,my}}
end if
end for
integer m = ceil(n/2)
if piece='B' then
-- As pointed out on the talk page, *every*
-- valid bishop solution can be transformed
-- into all in column m so search only that
for i=1 to length(res) do
if res[i][2]=m then
res = res[i..i]
exit
end if
end for
assert(length(res)=1)
elsif row=1 and col=1 and n>1 then
if piece='Q' then
-- first queen on first half of top row
-- or first half of diagonal (not cols)
assert(length(res)=3*n-2)
res = res[1..m]&res[2*n..2*n+m-2]
elsif piece='N' and n>2 then
-- first knight, was occupy+2, but by
-- symmetry we only need it to be 1+1
assert(length(res)=3)
res = res[1..2]
end if
end if
-- this cheeky little fella cuts more than half off the
-- last phase of 10N... (a circumstantial optimisation)
res = reverse(res)
return res
end function

procedure move(sequence rc, integer piece, n)
integer {row,col} = rc
board[row][col] = piece
sequence mm = mm_baby(get_mm(piece),piece,row,col,n)
for i=1 to length(mm) do
integer {mx,my} = mm[i]
board[mx][my] += 1
end for
end procedure

Ihandle dlg, canvas, timer
cdCanvas cddbuffer, cdcanvas

constant SQBN = " QBN",  -- (or " RBN")
QBNU = utf8_to_utf32("♕♗♘")

integer bn = 8, -- chosen board is nxn (1..10)
cp = 1  -- piece (1,2,3 for QBN)

sequence state

-- eg go straight for 16N on 10x10, avoid disproving 15N (from OEIS)
constant cheats = {{1,1,1,3,3,4,4,5,5,5,5,7,7,8,9,9,9,10,11,11,11,12,13,13,13},
{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22},
{1,4,4,4,5,8,13,14,14,16,22,24,29,33}}
-- some more timings with cheat mode ON:
-- 11Q: 1s, 12Q: 1.2s, 13Q: 1 min 7s, 14Q: 3 min 35s
-- 11N: 1 min 42s, 12N: gave up

procedure reset()
state = {repeat(0,maxq),
repeat(0,maxb),
repeat(0,maxn)}
-- (in case the maxq/b/n settings altered:)
if bn>length(state[cp]) then bn = 1 end if
for p=1 to 3 do
integer piece = SQBN[p+1]
for n=1 to length(state[p]) do
atom scolour = CD_RED
--          integer m = 1
integer m = iff(cheat?cheats[p][n]:1)
board = repeat(repeat('0',n),n)
sequence moves = get_moves(piece,n,1,1)
string undo = join(board,'\n')
state[p][n] = {scolour,m,{moves},{undo},undo,0}
end for
end for
IupSetInt(timer,"RUN",true)
end procedure

procedure iterative_solve()
-- find something not yet done
integer n = 0, p
for ndx=1 to maxqbn do
for pdx=1 to 3 do
if ndx<=length(state[pdx])
and state[pdx][ndx][1]!=CD_DARK_GREEN then
p = pdx
n = ndx
exit
end if
end for
if n!=0 then exit end if
end for
if n=0 then
?{"solved",(elapsed(time()-t0))}
IupSetInt(timer,"RUN",false)
else
-- but work on currently selected first, if needed
if state[cp][bn][1]!=CD_DARK_GREEN then
p = cp
n = bn
end if
atom t1 = time()+0.04, scolour
integer piece = SQBN[p+1], m, count
state[p][n] = 0
if n>1 and state[p][n-1][1]=CD_DARK_GREEN and m<state[p][n-1][2] then
-- if we needed (eg) 7 bishops to solve a 7x7, that means
-- we couldn't solve it with 6, so we are not going to be
-- able to solve an 8x8 with 6 (or less) now are we!
m = state[p][n-1][2]
else
while length(stack) do
sequence rc = stack[\$][1]
stack[\$] = stack[\$][2..\$]
board = split(undo[\$],'\n')
move(rc,piece,n)
count += 1
bool solved = true
for row=1 to n do
for col=1 to n do
if board[row][col]='0' then
if length(stack)<m then
stack &= {get_moves(piece,n,row,col)}
undo &= {join(board,'\n')}
end if
solved = false
exit
end if
end for
if not solved then exit end if
end for
if solved then
scolour = CD_DARK_GREEN
stack = {}
undo = {}
end if
while length(stack) and stack[\$]={} do
stack = stack[1..-2]
undo = undo[1..-2]
if length(undo)=0 then exit end if
end while
if solved or time()>t1 then
return
end if
end while
m += 1
end if
board = repeat(repeat('0',n),n)
stack = {get_moves(piece,n,1,1)}
undo = {join(board,'\n')}
end if
end procedure

atom dx, dy -- (saved for button_cb)

function redraw_cb(Ihandle /*canvas*/)
integer {w, h} = IupGetIntInt(canvas, "DRAWSIZE")
dx = w/40                 -- legend fifth
dy = h/(maxqbn+1)         -- legend delta
atom ly = h-dy/2,         -- legend top
cx = w/2,            -- board centre
cy = h/2,
bs = min(w*.7,h*.9), -- board size
ss = bs/bn           -- square size

cdCanvasActivate(cddbuffer)
cdCanvasClear(cddbuffer)
atom fontsize = min(ceil(dy/6),ceil(dx/2))
cdCanvasFont(cddbuffer, "Helvetica", CD_PLAIN, fontsize)
for i=0 to maxqbn do
atom lx = dx*36
for j=0 to 3 do
if j=0 or i<=length(state[j]) then
string txt = iff(i=0?SQBN[j+1..j+1]:
sprintf("%d",iff(j=0?i:state[j][i][2])))
atom colour = iff(i==0 or j==0?CD_BLACK:state[j][i][1])
cdCanvasSetForeground(cddbuffer, colour)
cdCanvasText(cddbuffer,lx,ly,txt)
end if
lx += dx
end for
ly -= dy
end for
atom x = cx-bs/2,
y = cy+bs/2
cdCanvasSetForeground(cddbuffer, CD_DARK_GREY)
for i=1 to bn do
for j=1+even(i) to bn by 2 do
atom sx = x+j*ss,
sy = y-i*ss
cdCanvasBox(cddbuffer,sx-ss,sx,sy+ss,sy)
end for
end for
cdCanvasRect(cddbuffer,x,x+bs,y,y-bs)
string piece_text = utf32_to_utf8({QBNU[cp]})
integer piece = SQBN[cp+1]
sequence mm = get_mm(piece),
st = state[cp][bn]
bool solved = st[1]=CD_DARK_GREEN
-- show the solution/mt or undo[\$] aka maniaical workings
board = split(iff(solved or st[4]={}?st[5]:st[4][\$]),'\n')
for row=1 to bn do
for col=1 to bn do
if board[row][col]=piece then
atom sx = x+col*ss-ss/2,
sy = y-row*ss+ss/2
cdCanvasSetForeground(cddbuffer, CD_BLACK)
cdCanvasFont(cddbuffer, "Helvetica", CD_PLAIN, fontsize*5)
cdCanvasText(cddbuffer,sx,sy+iff(platform()=JS?0:5),piece_text)
-- and mark all attacked squares
cdCanvasFont(cddbuffer, "Helvetica", CD_PLAIN, fontsize*2)
cdCanvasSetForeground(cddbuffer, CD_AMBER)
sequence mnm = mm_baby(mm,piece,col,row,bn)
for i=1 to length(mnm) do
integer {mx,my} = mnm[i]
string ac = board[my,mx]&""
cdCanvasText(cddbuffer,sx+ss*(mx-col),sy-ss*(my-row),ac)
end for
end if
end for
end for
cdCanvasFlush(cddbuffer)
integer m = st[2], count = st[6]
string pdesc = {"Queens", "Bishops", "Knights"}[cp][1..-1-(m=1)],
status = iff(solved?"solved in":"working:"),
attempt = iff(count=1?"attempt":"attempts")
IupSetStrAttribute(dlg,"TITLE","%s - %d %s on %dx%d %s %,d %s",{title,m,pdesc,bn,bn,status,count,attempt})
return IUP_DEFAULT
end function

function map_cb(Ihandle canvas)
cdcanvas = cdCreateCanvas(CD_IUP, canvas)
cddbuffer = cdCreateCanvas(CD_DBUFFER, cdcanvas)
cdCanvasSetBackground(cddbuffer, CD_PARCHMENT)
cdCanvasSetTextAlignment(cddbuffer,CD_CENTER)
return IUP_DEFAULT
end function

function help()
IupMessage(title,help_text)
return IUP_IGNORE -- (don't open browser help!)
end function

function key_cb(Ihandle ih, atom c)
if c=K_ESC then
IupSetInt(timer,"RUN",false)
return IUP_CLOSE
end if
if c=K_F5 then return IUP_DEFAULT end if -- (let browser reload work)
if c=K_F1 then return help() end if
integer k = find(upper(c),SQBN&"123456789T+-!")
if k then
if k=1 then IupSetInt(timer,"RUN",not IupGetInt(timer,"RUN"))
elsif k<=4 then     cp = k-1
elsif k<=14 then    bn = k-4
elsif c='+' then    bn = min(bn+1,maxqbn)
elsif c='-' then    bn = max(bn-1,1)
end if
bn = min(bn,length(state[cp]))
IupUpdate(ih)
end if
return IUP_IGNORE
end function

function button_cb(Ihandle ih, integer button, pressed, x, y, atom /*pStatus*/)
if button=IUP_BUTTON1 and pressed then      -- (left button pressed)
integer p = floor((x+dx/2)/dx)-36,
n = floor(y/dy)
if p>=1 and p<=3
and n>=1 and n<=length(state[p]) then
{cp,bn} = {p,n}
IupUpdate(ih)
end if
end if
return IUP_CONTINUE
end function

function timer_cb(Ihandln /*ih*/)
iterative_solve()
IupUpdate(canvas)
return IUP_IGNORE
end function

procedure main()
IupOpen()
IupSetGlobal("UTF8MODE","YES")
canvas = IupGLCanvas("RASTERSIZE=640x480")
IupSetCallbacks(canvas, {"ACTION", Icallback("redraw_cb"),
"MAP_CB", Icallback("map_cb"),
"BUTTON_CB", Icallback("button_cb")})
dlg = IupDialog(canvas,`TITLE="%s"`,{title})
IupSetCallback(dlg,"KEY_CB", Icallback("key_cb"))
IupSetAttributeHandle(NULL,"PARENTDIALOG",dlg)
timer = IupTimer(Icallback("timer_cb"), 30)
reset()
IupShow(dlg)
IupSetAttribute(canvas, "RASTERSIZE", NULL)
if platform()!=JS then
IupMainLoop()
IupClose()
end if
end procedure

main()
```

## Python

Translation of: Julia
```""" For Rosetta Code task N-queens_minimum_and_knights_and_bishops """

from mip import Model, BINARY, xsum, minimize

def n_queens_min(N):
""" N-queens minimum problem, oeis.org/A075458 """
if N < 4:
brd = [[0 for i in range(N)] for j in range(N)]
brd[0 if N < 2 else 1][0 if N < 2 else 1] = 1
return 1, brd

model = Model()
board = [[model.add_var(var_type=BINARY) for j in range(N)] for i in range(N)]
for k in range(N):
model += xsum(board[k][j] for j in range(N)) <= 1
model += xsum(board[i][k] for i in range(N)) <= 1

for k in range(1, 2 * N - 2):
model += xsum(board[k - j][j] for j in range(max(0, k - N + 1), min(k + 1, N))) <= 1

for k in range(2 - N, N - 1):
model += xsum(board[k + j][j] for j in range(max(0, -k), min(N - k, N))) <= 1

for i in range(N):
for j in range(N):
model += xsum([xsum(board[i][k] for k in range(N)),
xsum(board[k][j] for k in range(N)),
xsum(board[i + k][j + k] for k in range(-N, N)
if 0 <= i + k < N and 0 <= j + k < N),
xsum(board[i - k][j + k] for k in range(-N, N)
if 0 <= i - k < N and 0 <= j + k < N)]) >= 1

model.objective = minimize(xsum(board[i][j] for i in range(N) for j in range(N)))
model.optimize()
return model.objective_value, [[board[i][j].x for i in range(N)] for j in range(N)]

def n_bishops_min(N):
""" N-Bishops minimum problem (returns N)"""
model = Model()
board = [[model.add_var(var_type=BINARY) for j in range(N)] for i in range(N)]

for i in range(N):
for j in range(N):
model += xsum([
xsum(board[i + k][j + k] for k in range(-N, N)
if 0 <= i + k < N and 0 <= j + k < N),
xsum(board[i - k][j + k] for k in range(-N, N)
if 0 <= i - k < N and 0 <= j + k < N)]) >= 1

model.objective = minimize(xsum(board[i][j] for i in range(N) for j in range(N)))
model.optimize()
return model.objective_value, [[board[i][j].x for i in range(N)] for j in range(N)]

def n_knights_min(N):
""" N-knights minimum problem, oeis.org/A342576 """
if N < 2:
return 1, "N"

knightdeltas = [(1, 2), (2, 1), (2, -1), (1, -2), (-1, -2), (-2, -1), (-2, 1), (-1, 2)]
model = Model()
# to simplify the constraints, embed the board of size N inside a board of size N + 4
board = [[model.add_var(var_type=BINARY) for j in range(N + 4)] for i in range(N + 4)]
for i in range(N + 4):
model += xsum(board[i][j] for j in [0, 1, N + 2, N + 3]) == 0
for j in range(N + 4):
model += xsum(board[i][j] for i in [0, 1, N + 2, N + 3]) == 0

for i in range(2, N + 2):
for j in range(2, N + 2):
model += xsum([board[i][j]] + [board[i + d[0]][j + d[1]]
for d in knightdeltas]) >= 1
model += xsum([board[i + d[0]][j + d[1]]
for d in knightdeltas] + [100 * board[i][j]]) <= 100

model.objective = minimize(xsum(board[i][j] for i in range(2, N + 2) for j in range(2, N + 2)))
model.optimize()
minresult = model.objective_value
return minresult, [[board[i][j].x for i in range(2, N + 2)] for j in range(2, N + 2)]

if __name__ == '__main__':
examples, pieces, chars = [[], [], []], ["Queens", "Bishops", "Knights"], ['Q', 'B', 'N']
print("   Squares    Queens   Bishops   Knights")
for nrows in range(1, 11):
print(str(nrows * nrows).rjust(10), end='')
minval, examples[0] = n_queens_min(nrows)
print(str(int(minval)).rjust(10), end='')
minval, examples[1] = n_bishops_min(nrows)
print(str(int(minval)).rjust(10), end='')
minval, examples[2] = n_knights_min(nrows)
print(str(int(minval)).rjust(10))
if nrows == 10:
print("\nExamples for N = 10:")
for idx, piece in enumerate(chars):
print(f"\n{pieces[idx]}:")
for row in examples[idx]:
for sqr in row:
print(chars[idx] if sqr == 1 else '.', '', end = '')
print()
print()
```
Output:
```   Squares    Queens   Bishops   Knights
1         1         1         1
4         1         2         4
9         1         3         4
16         3         4         4
25         3         5         5
36         4         6         8
49         4         7        13
64         5         8        14
81         5         9        14
100         5        10        16

Examples for N = 10:

Queens:
. . . . . . . . . .
. . Q . . . . . . .
. . . . . . . . . .
. . . . . . . . Q .
. . . . . . . . . .
. . . . Q . . . . .
. . . . . . . . . .
Q . . . . . . . . .
. . . . . . . . . .
. . . . . . Q . . .

Bishops:
. . . . . . . . . .
. . . . B B . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . B . B . . .
. . B . . . B . . .
. . . B . . . . . .
. . . . . . B . . .
. . . . B . B . . .
. . . . . . . . . .

Knights:
. . . . . . . . . .
. . N N . . . . . .
. . N N . . . N N .
. . . . . . . N N .
. . . . . . . . . .
. . . . . . . . . .
. N N . . . . . . .
. N N . . . N N . .
. . . . . . N N . .
. . . . . . . . . .
```

## Raku

Due to the time it's taking only a subset of the task are attempted.

Translation of: Go
```# 20220705 Raku programming solution

my (@board, @diag1, @diag2, @diag1Lookup, @diag2Lookup, \$n, \$minCount, \$layout);

my %limits   = ( my @pieces = <Q B K> ) Z=> 7,7,6; # >>=>>> 10;
my %names    = @pieces Z=> <Queens Bishops Knights>;

sub isAttacked(\piece, \row, \col) {
given piece {
when 'Q' {
(^\$n)>>.&{ return True if @board[\$_;col] || @board[row;\$_] }
return True if @diag1Lookup[@diag1[row;col]] ||
@diag2Lookup[@diag2[row;col]]
}
when 'B' {
return True if @diag1Lookup[@diag1[row;col]] ||
@diag2Lookup[@diag2[row;col]]
}
default { # 'K'
return True if (
@board[row;col] or
row+2 < \$n && col-1 >= 0 && @board[row+2;col-1] or
row-2 >= 0 && col-1 >= 0 && @board[row-2;col-1] or
row+2 < \$n && col+1 < \$n && @board[row+2;col+1] or
row-2 >= 0 && col+1 < \$n && @board[row-2;col+1] or
row+1 < \$n && col+2 < \$n && @board[row+1;col+2] or
row-1 >= 0 && col+2 < \$n && @board[row-1;col+2] or
row+1 < \$n && col-2 >= 0 && @board[row+1;col-2] or
row-1 >= 0 && col-2 >= 0 && @board[row-1;col-2]
)
}
}
return False
}

sub attacks(\piece, \row, \col, \trow, \tcol) {
given piece {
when 'Q' { row==trow || col==tcol || abs(row - trow)==abs(col - tcol) }
when 'B' { abs(row - trow) == abs(col - tcol) }
default  { my (\rd,\cd) = ((trow - row),(tcol - col))>>.abs; # when 'K'
(rd == 1 && cd == 2) || (rd == 2 && cd == 1)               }
}
}

sub storeLayout(\piece) {
\$layout = [~] @board.map: -> @row {
[~] ( @row.map: { \$_ ??  piece~' ' !! '. ' } ) , "\n"
}
}

sub placePiece(\piece, \countSoFar, \maxCount) {
return if countSoFar >= \$minCount;
my (\$allAttacked,\$ti,\$tj) = True,0,0;
for ^\$n X ^\$n -> (\i,\j) {
unless isAttacked(piece, i, j) {
(\$allAttacked,\$ti,\$tj) = False,i,j andthen last
}
last unless \$allAttacked
}
if \$allAttacked {
\$minCount = countSoFar;
storeLayout(piece);
return
}
if countSoFar <= maxCount {
my (\$si,\$sj) = \$ti,\$tj;
if piece eq 'K' {
(\$si,\$sj) >>-=>> 2;
\$si = 0 if \$si < 0;
\$sj = 0 if \$sj < 0;
}
for (\$si..^\$n) X (\$sj..^\$n) -> (\i,\j) {
unless isAttacked(piece, i, j) {
if (i == \$ti && j == \$tj) || attacks(piece, i, j, \$ti, \$tj) {
@board[i][j] = True;
unless piece eq 'K' {
(@diag1Lookup[@diag1[i;j]],@diag2Lookup[@diag2[i;j]])=True xx *
}
placePiece(piece, countSoFar+1, maxCount);
@board[i][j] = False;
unless piece eq 'K' {
(@diag1Lookup[@diag1[i;j]],@diag2Lookup[@diag2[i;j]])=False xx *
}
}
}
}
}
}

for @pieces -> \piece {
say %names{piece}~"\n=======\n";
loop (\$n = 1 ; ; \$n++) {
@board = [ [ False xx \$n ] xx \$n ];
unless piece eq 'K' {
@diag1 = ^\$n .map: { \$_ ... \$n+\$_-1 } ;
@diag2 = ^\$n .map: { \$n+\$_-1 ... \$_ } ;
@diag2Lookup = @diag1Lookup = [ False xx 2*\$n-1 ]
}
\$minCount = 2³¹ - 1; # golang: math.MaxInt32
my \nSQ   = \$n*\$n;
for 1..nSQ -> \maxCount {
placePiece(piece, 0, maxCount);
last if \$minCount <= nSQ
}
printf("%2d x %-2d : %d\n", \$n, \$n, \$minCount);
if \$n == %limits{piece} {
printf "\n%s on a %d x %d board:\n", %names{piece}, \$n, \$n;
say \$layout andthen last
}
}
}
```
Output:
```Queens
=======

1 x 1  : 1
2 x 2  : 1
3 x 3  : 1
4 x 4  : 3
5 x 5  : 3
6 x 6  : 4
7 x 7  : 4

Queens on a 7 x 7 board:
.  Q  .  .  .  .  .
.  .  .  .  .  Q  .
.  .  .  .  .  .  .
Q  .  .  .  .  .  .
.  .  .  .  Q  .  .
.  .  .  .  .  .  .
.  .  .  .  .  .  .

Bishops
=======

1 x 1  : 1
2 x 2  : 2
3 x 3  : 3
4 x 4  : 4
5 x 5  : 5
6 x 6  : 6
7 x 7  : 7

Bishops on a 7 x 7 board:
.  .  .  .  .  B  .
.  .  B  .  .  .  .
.  .  B  .  B  .  .
.  .  .  .  .  .  B
.  .  .  B  .  .  .
.  .  .  B  .  .  .
.  .  .  .  .  .  .

Knights
=======

1 x 1  : 1
2 x 2  : 4
3 x 3  : 4
4 x 4  : 4
5 x 5  : 5
6 x 6  : 8

Knights on a 6 x 6 board:
K  .  .  .  .  K
.  .  .  .  .  .
.  .  K  K  .  .
.  .  K  K  .  .
.  .  .  .  .  .
K  .  .  .  .  K
```

## Wren

### CLI

Library: Wren-fmt

This was originally based on the Java code here which uses a backtracking algorithm and which I extended to deal with bishops and knights as well as queens when translating to Wren. I then used the more efficient way for checking the diagonals described here and have now incorporated the improvements made to the Go version.

Although far quicker than it was originally (it now gets to 7 knights in less than a minute), it struggles after that and needs north of 21 minutes to get to 10.

```import "./fmt" for Fmt

var board
var diag1
var diag2
var diag1Lookup
var diag2Lookup
var n
var minCount
var layout

var isAttacked = Fn.new { |piece, row, col|
if (piece == "Q") {
for (i in 0...n) {
if (board[i][col] || board[row][i]) return true
}
if (diag1Lookup[diag1[row][col]] ||
diag2Lookup[diag2[row][col]]) return true
} else if (piece == "B") {
if (diag1Lookup[diag1[row][col]] ||
diag2Lookup[diag2[row][col]]) return true
} else { // piece == "K"
if (board[row][col]) return true
if (row + 2 < n  && col - 1 >= 0 && board[row + 2][col - 1]) return true
if (row - 2 >= 0 && col - 1 >= 0 && board[row - 2][col - 1]) return true
if (row + 2 < n  && col + 1 < n  && board[row + 2][col + 1]) return true
if (row - 2 >= 0 && col + 1 < n  && board[row - 2][col + 1]) return true
if (row + 1 < n  && col + 2 < n  && board[row + 1][col + 2]) return true
if (row - 1 >= 0 && col + 2 < n  && board[row - 1][col + 2]) return true
if (row + 1 < n  && col - 2 >= 0 && board[row + 1][col - 2]) return true
if (row - 1 >= 0 && col - 2 >= 0 && board[row - 1][col - 2]) return true
}
return false
}

var attacks = Fn.new { |piece, row, col, trow, tcol|
if (piece == "Q") {
return row == trow || col == tcol || (row-trow).abs == (col-tcol).abs
} else if (piece == "B") {
return (row-trow).abs == (col-tcol).abs
} else { // piece == "K"
var rd = (trow - row).abs
var cd = (tcol - col).abs
return (rd == 1 && cd == 2) || (rd == 2 && cd == 1)
}
}

var storeLayout = Fn.new { |piece|
var sb = ""
for (row in board) {
for (cell in row) sb = sb + (cell ? piece + " " : ". ")
sb = sb + "\n"
}
layout = sb
}

var placePiece // recursive function
placePiece = Fn.new { |piece, countSoFar, maxCount|
if (countSoFar >= minCount) return
var allAttacked = true
var ti = 0
var tj = 0
for (i in 0...n) {
for (j in 0...n) {
if (!isAttacked.call(piece, i, j)) {
allAttacked = false
ti = i
tj = j
break
}
}
if (!allAttacked) break
}
if (allAttacked) {
minCount = countSoFar
storeLayout.call(piece)
return
}
if (countSoFar <= maxCount) {
var si = (piece == "K") ? (ti-2).max(0) : ti
var sj = (piece == "K") ? (tj-2).max(0) : tj
for (i in si...n) {
for (j in sj...n) {
if (!isAttacked.call(piece, i, j)) {
if ((i == ti && j == tj) || attacks.call(piece, i, j, ti, tj)) {
board[i][j] = true
if (piece != "K") {
diag1Lookup[diag1[i][j]] = true
diag2Lookup[diag2[i][j]] = true
}
placePiece.call(piece, countSoFar + 1, maxCount)
board[i][j] = false
if (piece != "K") {
diag1Lookup[diag1[i][j]] = false
diag2Lookup[diag2[i][j]] = false
}
}
}
}
}
}
}

var start = System.clock
var pieces = ["Q", "B", "K"]
var limits = {"Q": 10, "B": 10, "K": 10}
var names  = {"Q": "Queens", "B": "Bishops", "K": "Knights"}
for (piece in pieces) {
System.print(names[piece])
System.print("=======\n")
n = 1
while (true) {
board = List.filled(n, null)
for (i in 0...n) board[i] = List.filled(n, false)
if (piece != "K") {
diag1 = List.filled(n, null)
for (i in 0...n) {
diag1[i] = List.filled(n, 0)
for (j in 0...n) diag1[i][j] = i + j
}
diag2 = List.filled(n, null)
for (i in 0...n) {
diag2[i] = List.filled(n, 0)
for (j in 0...n) diag2[i][j] = i - j + n - 1
}
diag1Lookup = List.filled(2*n-1, false)
diag2Lookup = List.filled(2*n-1, false)
}
minCount = Num.maxSafeInteger
layout = ""
for (maxCount in 1..n*n) {
placePiece.call(piece, 0, maxCount)
if (minCount <= n*n) break
}
Fmt.print("\$2d x \$-2d : \$d", n, n, minCount)
if (n == limits[piece]) {
Fmt.print("\n\$s on a \$d x \$d board:", names[piece], n, n)
System.print("\n" + layout)
break
}
n = n + 1
}
}
System.print("Took %(System.clock - start) seconds.")
```
Output:
```Queens
=======

1 x 1  : 1
2 x 2  : 1
3 x 3  : 1
4 x 4  : 3
5 x 5  : 3
6 x 6  : 4
7 x 7  : 4
8 x 8  : 5
9 x 9  : 5
10 x 10 : 5

Queens on a 10 x 10 board:

. . Q . . . . . . .
. . . . . . . . . .
. . . . . . . . Q .
. . . . . . . . . .
. . . . Q . . . . .
. . . . . . . . . .
Q . . . . . . . . .
. . . . . . . . . .
. . . . . . Q . . .
. . . . . . . . . .

Bishops
=======

1 x 1  : 1
2 x 2  : 2
3 x 3  : 3
4 x 4  : 4
5 x 5  : 5
6 x 6  : 6
7 x 7  : 7
8 x 8  : 8
9 x 9  : 9
10 x 10 : 10

Bishops on a 10 x 10 board:

. . . . . . . . . B
. . . . . . . . . .
. . . B . B . . . .
. . . B . B . B . .
B . . . . . . . . .
. . . . . . . . . .
. . . . . B . . . .
. . . . . B . . . .
. . . . . B . . . .
. . . . . . . . . .

Knights
=======

1 x 1  : 1
2 x 2  : 4
3 x 3  : 4
4 x 4  : 4
5 x 5  : 5
6 x 6  : 8
7 x 7  : 13
8 x 8  : 14
9 x 9  : 14
10 x 10 : 16

Knights on a 10 x 10 board:

. . . . . . . . . .
. . K K . . . . . .
. . K K . . . K K .
. . . . . . . K K .
. . . . . . . . . .
. . . . . . . . . .
. K K . . . . . . .
. K K . . . K K . .
. . . . . . K K . .
. . . . . . . . . .

Took 1276.522608 seconds.
```

### Embedded

Library: Wren-linear

This is the first outing for the above module which is a wrapper for GLPK.

As there are quite a lot of variables and constraints in this task, I have used MathProg scripts to solve it rather than calling the basic API routines directly. The script file needs to be changed for each chess piece and each value of 'n' as there appear to be no looping constructs in MathProg itself.

Despite this, the program runs in only 3.25 seconds which is far quicker than I was expecting.

I have borrowed one or two tricks from the Julia/Python versions in formulating the constraints.

```import "./linear" for Prob, Glp, Tran, File
import "./fmt" for Fmt

var start = System.clock

var queenMpl = """
var x{1..n, 1..n}, binary;
s.t. a{i in 1..n}: sum{j in 1..n} x[i,j] <= 1;
s.t. b{j in 1..n}: sum{i in 1..n} x[i,j] <= 1;
s.t. c{k in 2-n..n-2}: sum{i in 1..n, j in 1..n: i-j == k} x[i,j] <= 1;
s.t. d{k in 3..n+n-1}: sum{i in 1..n, j in 1..n: i+j == k} x[i,j] <= 1;
s.t. e{i in 1..n, j in 1..n}:
sum{k in 1..n} x[i,k] +
sum{k in 1..n} x[k,j] +
sum{k in (1-n)..n: 1 <= i + k && i + k <= n && 1 <= j + k && j + k <=n} x[i+k,j+k]  +
sum{k in (1-n)..n: 1 <= i - k && i - k <= n && 1 <= j + k && j + k <=n} x[i-k, k+j] >= 1;

minimize obj: sum{i in 1..n, j in 1..n} x[i,j];
solve;
end;

"""

var bishopMpl = """
var x{1..n, 1..n}, binary;
s.t. a{k in 2-n..n-2}: sum{i in 1..n, j in 1..n: i-j == k} x[i,j] <= 1;
s.t. b{k in 3..n+n-1}: sum{i in 1..n, j in 1..n: i+j == k} x[i,j] <= 1;
s.t. c{i in 1..n, j in 1..n}:
sum{k in (1-n)..n: 1 <= i + k && i + k <= n && 1 <= j + k && j + k <=n} x[i+k,j+k]  +
sum{k in (1-n)..n: 1 <= i - k && i - k <= n && 1 <= j + k && j + k <=n} x[i-k, k+j] >= 1;

minimize obj: sum{i in 1..n, j in 1..n} x[i,j];
solve;
end;

"""

var knightMpl = """
set deltas, dimen 2;
var x{1..n+4, 1..n+4}, binary;
s.t. a{i in 1..n+4}: sum{j in 1..n+4: j < 3 || j > n + 2} x[i,j] = 0;
s.t. b{j in 1..n+4}: sum{i in 1..n+4: i < 3 || i > n + 2} x[i,j] = 0;
s.t. c{i in 3..n+2, j in 3..n+2}: x[i, j] + sum{(k, l) in deltas} x[i + k, j + l] >= 1;
s.t. d{i in 3..n+2, j in 3..n+2}: sum{(k, l) in deltas} x[i + k, j + l] + 100 * x[i, j] <= 100;

minimize obj: sum{i in 3..n+2, j in 3..n+2} x[i,j];
solve;
data;
set deltas := (1,2) (2,1) (2,-1) (1,-2) (-1,-2) (-2,-1) (-2,1) (-1,2);
end;

"""

var mpls   = {"Q": queenMpl, "B": bishopMpl, "K": knightMpl}
var pieces = ["Q", "B", "K"]
var limits = {"Q": 10, "B": 10, "K": 10}
var names  = {"Q": "Queens", "B": "Bishops", "K": "Knights"}
var fname  = "n_pieces.mod"

Glp.termOut(Glp.OFF)
for (piece in pieces) {
System.print(names[piece])
System.print("=======\n")
for (n in 1..limits[piece]) {
var first = "param n, integer, > 0, default %(n);\n"
File.write(fname, first + mpls[piece])
var mip = Prob.create()
var tran = Tran.mplAllocWksp()
if (ret != 0) System.print("Error on translating model.")
if (ret == 0) {
ret = tran.mplGenerate(null)
if (ret != 0) System.print("Error on generating model.")
if (ret == 0) {
tran.mplBuildProb(mip)
mip.simplex(null)
mip.intOpt(null)
Fmt.print("\$2d x \$-2d : \$d", n, n, mip.mipObjVal.round)
if (n == limits[piece]) {
Fmt.print("\n\$s on a \$d x \$d board:\n", names[piece], n, n)
var cols = {}
if (piece != "K") {
for (i in 1..n*n) cols[mip.colName(i)] = mip.mipColVal(i)
for (i in 1..n) {
for (j in 1..n) {
var char = (cols["x[%(i),%(j)]"] == 1) ? "%(piece) " : ". "
System.write(char)
}
System.print()
}
} else {
for (i in 1..(n+4)*(n+4)) cols[mip.colName(i)] = mip.mipColVal(i)
for (i in 3..n+2) {
for (j in 3..n+2) {
var char = (cols["x[%(i),%(j)]"] == 1) ? "%(piece) " : ". "
System.write(char)
}
System.print()
}
}
}
}
}
tran.mplFreeWksp()
mip.delete()
}
System.print()
}
File.remove(fname)
System.print("Took %(System.clock - start) seconds.")
```
Output:
```Queens
=======

1 x 1  : 1
2 x 2  : 1
3 x 3  : 1
4 x 4  : 3
5 x 5  : 3
6 x 6  : 4
7 x 7  : 4
8 x 8  : 5
9 x 9  : 5
10 x 10 : 5

Queens on a 10 x 10 board:

. . . . . . Q . . .
. . . . . . . . . .
Q . . . . . . . . .
. . . . . . . . . .
. . . . Q . . . . .
. . . . . . . . . .
. . . . . . . . Q .
. . . . . . . . . .
. . Q . . . . . . .
. . . . . . . . . .

Bishops
=======

1 x 1  : 1
2 x 2  : 2
3 x 3  : 3
4 x 4  : 4
5 x 5  : 5
6 x 6  : 6
7 x 7  : 7
8 x 8  : 8
9 x 9  : 9
10 x 10 : 10

Bishops on a 10 x 10 board:

B . . . . . . . . .
. . . . B . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . B . B B . B .
. B . B . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . B . . . .
. . B . . . . . . .

Knights
=======

1 x 1  : 1
2 x 2  : 4
3 x 3  : 4
4 x 4  : 4
5 x 5  : 5
6 x 6  : 8
7 x 7  : 13
8 x 8  : 14
9 x 9  : 14
10 x 10 : 16

Knights on a 10 x 10 board:

. . . . . K . . . .
. . K . . . . . . .
. . K K . . . K K .
. . . . . . . K . .
K . . . . . . . . .
. . . . . . . . . K
. . K . . . . . . .
. K K . . . K K . .
. . . . . . . K . .
. . . . K . . . . .

Took 3.244584 seconds.
```