Sudoku

Sudoku
You are encouraged to solve this task according to the task description, using any language you may know.

Solve a partially filled-in normal   9x9   Sudoku grid   and display the result in a human-readable format.

references

11l

Translation of: Kotlin
```T Sudoku
solved = 0B
grid = [0] * 81

F (rows)
assert(rows.len == 9 & all(rows.map(row -> row.len == 9)), ‘Grid must be 9 x 9’)
L(i) 9
L(j) 9
.grid[9 * i + j] = Int(rows[i][j])

F solve()
print("Starting grid:\n\n"(.))
.placeNumber(0)
print(I .solved {"Solution:\n\n"(.)} E ‘Unsolvable!’)

F placeNumber(pos)
I .solved
R
I pos == 81
.solved = 1B
R

I .grid[pos] > 0
.placeNumber(pos + 1)
R

L(n) 1..9
I .checkValidity(n, pos % 9, pos I/ 9)
.grid[pos] = n
.placeNumber(pos + 1)
I .solved
R
.grid[pos] = 0

F checkValidity(v, x, y)
L(i) 9
I .grid[y * 9 + i] == v |
.grid[i * 9 + x] == v
R 0B

V startX = (x I/ 3) * 3
V startY = (y I/ 3) * 3
L(i) startY .< startY + 3
L(j) startX .< startX + 3
I .grid[i * 9 + j] == v
R 0B

R 1B

F String()
V s = ‘’
L(i) 9
L(j) 9
s ‘’= .grid[i * 9 + j]‘ ’
I j C (2, 5)
s ‘’= ‘| ’
s ‘’= "\n"
I i C (2, 5)
s ‘’= "------+-------+------\n"
R s

V rows = [‘850002400’,
‘720000009’,
‘004000000’,
‘000107002’,
‘305000900’,
‘040000000’,
‘000080070’,
‘017000000’,
‘000036040’]

Sudoku(rows).solve()```
Output:
```Starting grid:

8 5 0 | 0 0 2 | 4 0 0
7 2 0 | 0 0 0 | 0 0 9
0 0 4 | 0 0 0 | 0 0 0
------+-------+------
0 0 0 | 1 0 7 | 0 0 2
3 0 5 | 0 0 0 | 9 0 0
0 4 0 | 0 0 0 | 0 0 0
------+-------+------
0 0 0 | 0 8 0 | 0 7 0
0 1 7 | 0 0 0 | 0 0 0
0 0 0 | 0 3 6 | 0 4 0

Solution:

8 5 9 | 6 1 2 | 4 3 7
7 2 3 | 8 5 4 | 1 6 9
1 6 4 | 3 7 9 | 5 2 8
------+-------+------
9 8 6 | 1 4 7 | 3 5 2
3 7 5 | 2 6 8 | 9 1 4
2 4 1 | 5 9 3 | 7 8 6
------+-------+------
4 3 2 | 9 8 1 | 6 7 5
6 1 7 | 4 2 5 | 8 9 3
5 9 8 | 7 3 6 | 2 4 1

```

8th

```\
\  Simple iterative backtracking Sudoku solver for 8th
\
needs array/each-slice

[  00, 00, 00, 03, 03, 03, 06, 06, 06,
00, 00, 00, 03, 03, 03, 06, 06, 06,
00, 00, 00, 03, 03, 03, 06, 06, 06,
27, 27, 27, 30, 30, 30, 33, 33, 33,
27, 27, 27, 30, 30, 30, 33, 33, 33,
27, 27, 27, 30, 30, 30, 33, 33, 33,
54, 54, 54, 57, 57, 57, 60, 60, 60,
54, 54, 54, 57, 57, 57, 60, 60, 60,
54, 54, 54, 57, 57, 57, 60, 60, 60 ] constant top-left-cell

\ Bit number presentations
a:new 2 b:new b:clear a:push ( 2 b:new b:clear swap 1 b:bit! a:push ) 0 8 loop constant posbit

: posbit?  \ n -- s
posbit swap a:@ nip ;

: search  \ b -- n
null swap
( dup -rot b:bit@ if rot drop break else nip then ) 0 8 loop
swap ;

: b-or  \ b b -- b
' n:bor b:op ;

: b-and  \ b b -- b
' n:band b:op ;

: b-xor  \ b b -- b
b:xor
[ xff, x01 ] b:new
b-and ;

: b-not  \ b -- b
xff b:xor
[ xff, x01 ] b:new
b-and ;

: b-any  \ a -- b
' b-or 0 posbit? a:reduce ;

: row \ a row -- a
9 n:* 9 a:slice ;

: col  \ a col -- a
-1 9 a:slice+ ;

\ For testing sub boards
: sub  \ a n -- a
top-left-cell swap a:@ nip over over 3 a:slice
-rot 9 n:+ 2dup 3 a:slice
-rot 9 n:+ 3 a:slice
a:+ a:+ ;

a:new 0 args "Give Sudoku text file as param" thrownull
f:slurp "Cannot read file" thrownull >s "" s:/
' >n a:map ( posbit? a:push ) a:each! drop constant board

: display-board
board ( search nip -1 ?: n:1+ ) a:map
"+-----+-----+-----+\n"
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"+-----+-----+-----+\n" s:+
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"+-----+-----+-----+\n" s:+
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"|%d %d %d|%d %d %d|%d %d %d|\n" s:+
"+-----+-----+-----+\n" s:+
s:strfmt . ;

\ Store move history
a:new constant history

\ Possible numbers for a cell
: candidates?  \ n -- s
dup dup 9 n:/ n:int swap 9 n:mod \ row col
board swap col b-any
board rot row b-any
b-or
board rot sub b-any
b-or
b-not ;

\ If found:     -- n T
: find-free-cell
false
board ( 0 posbit? b:= if nip true break else drop then ) a:each drop ;

: validate
true
board
( dup -rot a:@ swap 2 pick 0 posbit? a:! 2 pick candidates? 2 pick b:= if
-rot a:!
else
2drop drop
false swap
break
then ) 0 80 loop drop ;

: solve
repeat
find-free-cell if
dup candidates?
repeat
search null? if
drop board -rot a:! drop
history a:len 0 n:= if
drop false ;;
then
a:pop nip
a:open
else
n:1+ posbit?
dup
board 4 pick rot a:! drop
b-xor
2 a:close
history swap a:push drop
break
then
again
else
validate
break
then
again ;

: app:main
"Sudoku puzzle:\n" .
display-board cr
solve if
"Sudoku solved:\n" .
display-board
else
"No solution!\n" .
then ;```

Translation of: C++
```with Ada.Text_IO;

procedure Sudoku is
type sudoku_ar_t is array ( integer range 0..80 ) of integer range 0..9;
FINISH_EXCEPTION : exception;

procedure prettyprint(sudoku_ar: sudoku_ar_t);
function checkValidity( val : integer; x : integer; y : integer;  sudoku_ar: in  sudoku_ar_t) return Boolean;
procedure placeNumber(pos: Integer; sudoku_ar: in out sudoku_ar_t);
procedure solve(sudoku_ar: in out sudoku_ar_t);

function checkValidity( val : integer; x : integer; y : integer;  sudoku_ar: in  sudoku_ar_t) return Boolean
is
begin
for i in 0..8 loop

if ( sudoku_ar( y * 9 + i ) = val or sudoku_ar( i * 9 + x ) = val ) then
return False;
end if;
end loop;

declare
startX : constant integer := ( x / 3 ) * 3;
startY : constant integer := ( y / 3 ) * 3;
begin
for i in startY..startY+2 loop
for j in startX..startX+2 loop
if ( sudoku_ar( i * 9 +j ) = val ) then
return False;
end if;
end loop;
end loop;
return True;
end;
end checkValidity;

procedure placeNumber(pos: Integer; sudoku_ar: in out sudoku_ar_t)
is
begin
if ( pos = 81 ) then
raise FINISH_EXCEPTION;
end if;
if (  sudoku_ar(pos) > 0 ) then
placeNumber(pos+1, sudoku_ar);
return;
end if;
for n in 1..9 loop
if( checkValidity( n,  pos mod 9, pos / 9 , sudoku_ar ) ) then
sudoku_ar(pos) := n;
placeNumber(pos + 1, sudoku_ar );
sudoku_ar(pos) := 0;
end if;
end loop;
end placeNumber;

procedure solve(sudoku_ar: in out sudoku_ar_t)
is
begin
placeNumber( 0, sudoku_ar );
exception
when FINISH_EXCEPTION =>
prettyprint(sudoku_ar);
end solve;

procedure prettyprint(sudoku_ar: sudoku_ar_t)
is
line_sep   : constant String  := "------+------+------";
begin
for i in sudoku_ar'Range loop
if (i+1) mod 3 = 0 and not((i+1) mod 9 = 0) then
end if;
if (i+1) mod 9 = 0 then
end if;
if (i+1) mod 27 = 0 then
end if;
end loop;
end prettyprint;

sudoku_ar : sudoku_ar_t :=
(
8,5,0,0,0,2,4,0,0,
7,2,0,0,0,0,0,0,9,
0,0,4,0,0,0,0,0,0,
0,0,0,1,0,7,0,0,2,
3,0,5,0,0,0,9,0,0,
0,4,0,0,0,0,0,0,0,
0,0,0,0,8,0,0,7,0,
0,1,7,0,0,0,0,0,0,
0,0,0,0,3,6,0,4,0
);

begin
solve( sudoku_ar );
end Sudoku;
```
Output:
```Finished !
8 5 9| 6 1 2| 4 3 7
7 2 3| 8 5 4| 1 6 9
1 6 4| 3 7 9| 5 2 8
------+------+------
9 8 6| 1 4 7| 3 5 2
3 7 5| 2 6 8| 9 1 4
2 4 1| 5 9 3| 7 8 6
------+------+------
4 3 2| 9 8 1| 6 7 5
6 1 7| 4 2 5| 8 9 3
5 9 8| 7 3 6| 2 4 1
```

ALGOL 68

Translation of: D
Note: This specimen retains the original D coding style.
Works with: ALGOL 68 version Revision 1 - no extensions to language used.
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny.
```MODE AVAIL = [9]BOOL;
MODE BOX = [3, 3]CHAR;

FORMAT row fmt = \$"|"3(" "3(g" ")"|")l\$;
FORMAT line = \$"+"3(7"-","+")l\$;
FORMAT puzzle fmt = \$f(line)3(3(f(row fmt))f(line))\$;

AVAIL gen full = (TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE, TRUE);

OP REPR = (AVAIL avail)STRING: (
STRING out := "";
FOR i FROM LWB avail TO UPB avail DO
IF avail[i] THEN out +:= REPR(ABS "0" + i) FI
OD;
out
);

CHAR empty = "_";

OP -:= = (REF AVAIL set, CHAR index)VOID: (
set[ABS index - ABS "0"]:=FALSE
);

#  these two functions assume that the number has not already been found #
PROC avail slice = (REF[]CHAR slice, REF AVAIL available)REF AVAIL:(
FOR ele FROM LWB slice TO UPB slice DO
IF slice[ele] /= empty THEN available-:=slice[ele] FI
OD;
available
);

PROC avail box = (INT x, y, REF AVAIL available)REF AVAIL:(
#  x designates row, y designates column #
#  get a base index for the boxes #
INT bx := x - (x-1) MOD 3;
INT by := y - (y-1) MOD 3;
REF BOX box = puzzle[bx:bx+2, by:by+2];
FOR i FROM LWB box TO UPB box DO
FOR j FROM 2 LWB box TO 2 UPB box DO
IF box[i, j] /= empty THEN available-:=box[i, j] FI
OD
OD;
available
);

[9, 9]CHAR puzzle;
PROC solve = ([,]CHAR in puzzle)VOID:(
puzzle := in puzzle;
TO UPB puzzle UP 2 DO
BOOL done := TRUE;
FOR i FROM LWB puzzle TO UPB puzzle DO
FOR j FROM 2 LWB puzzle TO 2 UPB puzzle DO
CHAR ele := puzzle[i, j];
IF ele = empty THEN
#  poke at the elements that are "_" #
AVAIL remaining := avail box(i, j,
avail slice(puzzle[i, ],
avail slice(puzzle[, j],
LOC AVAIL := gen full)));
STRING s = REPR remaining;
IF UPB s = 1 THEN puzzle[i, j] := s[LWB s]
ELSE done := FALSE
FI
FI
OD
OD;
IF done THEN break FI
OD;
break:
#  write out completed puzzle #
printf((\$gl\$, "Completed puzzle:"));
printf((puzzle fmt, puzzle))
);
main:(
solve(("394__267_",
"___3__4__",
"5__69__2_",
"_45___9__",
"6_______7",
"__7___58_",
"_1__67__8",
"__9__8___",
"_264__735"))
CO # note: This codes/algorithm does not [yet] solve: #
solve(("9__2__5__",
"_4__6__3_",
"__3_____6",
"___9__2__",
"____5__8_",
"__7__4__3",
"7_____1__",
"_5__2__4_",
"__1__6__9"))
END CO
)```
Output:
```Completed puzzle:
+-------+-------+-------+
| 3 9 4 | 8 5 2 | 6 7 1 |
| 2 6 8 | 3 7 1 | 4 5 9 |
| 5 7 1 | 6 9 4 | 8 2 3 |
+-------+-------+-------+
| 1 4 5 | 7 8 3 | 9 6 2 |
| 6 8 2 | 9 4 5 | 3 1 7 |
| 9 3 7 | 1 2 6 | 5 8 4 |
+-------+-------+-------+
| 4 1 3 | 5 6 7 | 2 9 8 |
| 7 5 9 | 2 3 8 | 1 4 6 |
| 8 2 6 | 4 1 9 | 7 3 5 |
+-------+-------+-------+
```

AutoHotkey

```#SingleInstance, Force
SetBatchLines, -1
SetTitleMatchMode, 3

Loop 9 {
r := A_Index, y := r*17-8 + (A_Index >= 7 ? 4 : A_Index >= 4 ? 2 : 0)
Loop 9 {
c := A_Index, x := c*17+5 + (A_Index >= 7 ? 4 : A_Index >= 4 ? 2 : 0)
Gui, Add, Edit, x%x% y%y% w17 h17 v%r%_%c% Center Number Limit1 gNext
}
}
Gui, Add, Button, vButton gSolve w175 x10 Center, Solve
Gui, Add, Text, vMsg r3, Enter Sudoku puzzle and click Solve
Gui, Show,, Sudoku Solver
Return

Solve:
Gui, Submit, NoHide
Loop 9
{
r := A_Index
Loop 9
If (%r%_%A_Index% = "")
puzzle .= "@"
Else
puzzle .= %r%_%A_Index%
}
s := A_TickCount
iterations := ErrorLevel
e := A_TickCount
seconds := (e-s)/1000
Loop 9
{
r := A_Index
Loop 9
{
b := (r*9)+A_Index-9
GuiControl,, %r%_%A_Index%, % a%b%
}
}
GuiControl,, Msg, Solved!`nTime: %seconds%s`nIterations: %iterations%
else
GuiControl,, Msg, Failed! :(`nTime: %seconds%s`nIterations: %iterations%
GuiControl,, Button, Again!
GuiControl, +gAgain, Button
return

GuiClose:
ExitApp

Again:

#IfWinActive, Sudoku Solver
~*Enter::GoSub % GetKeyState( "Shift", "P" ) ? "~Up" : "~Down"
~Up::
GuiControlGet, f, focus
StringTrimLeft, f, f, 4
f := ((f >= 1 && f <= 9) ? f+72 : f-9)
GuiControl, Focus, Edit%f%
return
~Down::
GuiControlGet, f, focus
StringTrimLeft, f, f, 4
f := ((f >= 73 && f <= 81) ? f-72 : f + 9)
GuiControl, Focus, Edit%f%
return
~Left::
GuiControlGet, f, focus
StringTrimLeft, f, f, 4
f := Mod(f + 79, 81) + 1
GuiControl, Focus, Edit%f%
return
Next:
~Right::
GuiControlGet, f, focus
StringTrimLeft, f, f, 4
f := Mod(f, 81) + 1
GuiControl, Focus, Edit%f%
return
#IfWinActive

; Functions Start here

Sudoku( p ) { ;ErrorLevel contains the number of iterations
p := RegExReplace(p, "[^1-9@]"), ErrorLevel := 0 ;format puzzle as single line string
return Sudoku_Display(Sudoku_Solve(p))
}

Sudoku_Solve( p, d = 0 ) { ;d is 0-based
;   http://www.autohotkey.com/forum/topic46679.html
;   p: 81 character puzzle string
;      (concat all 9 rows of 9 chars each)
;      givens represented as chars 1-9
;      fill-ins as any non-null, non 1-9 char
;   d: used internally. omit on initial call
;
;   returns: 81 char string with non-givens replaced with valid solution
;
If (d >= 81), ErrorLevel++
return p  ;this is 82nd iteration, so it has successfully finished iteration 81
If InStr( "123456789", SubStr(p, d+1, 1) ) ;this depth is a given, skip through
return Sudoku_Solve(p, d+1)
m := Sudoku_Constraints(p,d) ;a string of this level's constraints.
; (these will not change for all 9 loops)
Loop 9
{
If InStr(m, A_Index)
Continue
NumPut(Asc(A_Index), p, d, "Char")
If r := Sudoku_Solve(p, d+1)
return r
}
return 0
}

Sudoku_Constraints( ByRef p, d ) {
; returns a string of the constraints for a particular position
c := Mod(d,9)
, r := (d - c) // 9
, b := r//3*27 + c//3*3 + 1
;convert to 1-based
, c++
return ""
; row:
. SubStr(p, r * 9 + 1, 9)
; column:
. SubStr(p,c   ,1) SubStr(p,c+9 ,1) SubStr(p,c+18,1)
. SubStr(p,c+27,1) SubStr(p,c+36,1) SubStr(p,c+45,1)
. SubStr(p,c+54,1) SubStr(p,c+63,1) SubStr(p,c+72,1)
;box
. SubStr(p, b, 3) SubStr(p, b+9, 3) SubStr(p, b+18, 3)
}

Sudoku_Display( p ) {
If StrLen(p) = 81
loop 81
r .= SubStr(p, A_Index, 1) . "|"
return r
}
```

AWK

```# syntax: GAWK -f SUDOKU_RC.AWK
BEGIN {
#             row1      row2      row3      row4      row5      row6      row7      row8      row9
#   puzzle = "111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111" # NG duplicate hints
#   puzzle = "1........ ..274.... ...5....4 .3....... 75....... .....96.. .4...6... .......71 .....1.30" # NG can't use zero
#   puzzle = "1........ ..274.... ...5....4 .3....... 75....... .....96.. .4...6... .......71 .....1.39" # no solution
#   puzzle = "1........ ..274.... ...5....4 .3....... 75....... .....96.. .4...6... .......71 .....1.3." # OK
puzzle = "123456789 456789123 789123456 ......... ......... ......... ......... ......... ........." # OK
gsub(/ /,"",puzzle)
if (length(puzzle) != 81) { error("length of puzzle is not 81") }
if (puzzle !~ /^[1-9\.]+\$/) { error("only 1-9 and . are valid") }
if (gsub(/[1-9]/,"&",puzzle) < 17) { error("too few hints") }
if (errors > 0) {
exit(1)
}
plot(puzzle,"unsolved")
if (dup_hints_check(puzzle) == 1) {
if (solve(puzzle) == 1) {
dup_hints_check(sos)
plot(sos,"solved")
printf("\nbef: %s\naft: %s\n",puzzle,sos)
exit(0)
}
else {
error("no solution")
}
}
exit(1)
}
function dup_hints_check(ss,  esf,msg,Rarr,Carr,Barr,i,r_row,r_col,r_pos,r_hint,c_row,c_col,c_pos,c_hint,box) {
esf = errors                       # errors so far
for (i=0; i<81; i++) {
# row
r_row = int(i/9) + 1             # determine row: 1..9
r_col = i%9 + 1                  # determine column: 1..9
r_pos = i + 1                    # determine hint position: 1..81
r_hint = substr(ss,r_pos,1)      # extract 1 character; the hint
Rarr[r_row,r_hint]++             # save row
# column
c_row = i%9 + 1                  # determine row: 1..9
c_col = int(i/9) + 1             # determine column: 1..9
c_pos = (c_row-1) * 9 + c_col    # determine hint position: 1..81
c_hint = substr(ss,c_pos,1)      # extract 1 character; the hint
Carr[c_col,c_hint]++             # save column
# box (there has to be a better way)
if      ((r_row r_col) ~ /[123][123]/) { box = 1 }
else if ((r_row r_col) ~ /[123][456]/) { box = 2 }
else if ((r_row r_col) ~ /[123][789]/) { box = 3 }
else if ((r_row r_col) ~ /[456][123]/) { box = 4 }
else if ((r_row r_col) ~ /[456][456]/) { box = 5 }
else if ((r_row r_col) ~ /[456][789]/) { box = 6 }
else if ((r_row r_col) ~ /[789][123]/) { box = 7 }
else if ((r_row r_col) ~ /[789][456]/) { box = 8 }
else if ((r_row r_col) ~ /[789][789]/) { box = 9 }
else { box = 0 }
Barr[box,r_hint]++               # save box
}
dup_hints_print(Rarr,"row")
dup_hints_print(Carr,"column")
dup_hints_print(Barr,"box")
return((errors == esf) ? 1 : 0)
}
function dup_hints_print(arr,rcb,  hint,i) {
# rcb - Row Column Box
for (i=1; i<=9; i++) {             # "i" is either the row, column, or box
for (hint=1; hint<=9; hint++) {  # 1..9 only; don't care about "." place holder
if (arr[i,hint]+0 > 1) {       # was a digit specified more than once
error(sprintf("duplicate hint in %s %d",rcb,i))
}
}
}
}
function plot(ss,text1,text2,  a,b,c,d,ou) {
# 1st call prints the unsolved puzzle.
# 2nd call prints the solved puzzle
printf("| - - - + - - - + - - - | %s\n",text1)
for (a=0; a<3; a++) {
for (b=0; b<3; b++) {
ou = "|"
for (c=0; c<3; c++) {
for (d=0; d<3; d++) {
ou = sprintf("%s %1s",ou,substr(ss,1+d+3*c+9*b+27*a,1))
}
ou = ou " |"
}
print(ou)
}
printf("| - - - + - - - + - - - | %s\n",(a==2)?text2:"")
}
}
function solve(ss,  a,b,c,d,e,r,co,ro,bi,bl,nss) {
i = 0
# first, use some simple logic to fill grid as much as possible
do {
i++
didit = 0
delete nr
delete nc
delete nb
delete ca
for (a=0; a<81; a++) {
b = substr(ss,a+1,1)
if (b == ".") {                # construct row, column and block at cell
c = a % 9
r = int(a/9)
ro = substr(ss,r*9+1,9)
co = ""
for (d=0; d<9; d++) { co = co substr(ss,d*9+c+1,1) }
bi = int(c/3)*3+(int(r/3)*3)*9+1
bl = ""
for (d=0; d<3; d++) { bl = bl substr(ss,bi+d*9,3) }
e = 0
# count non-occurrences of digits 1-9 in combined row, column and block, per row, column and block, and flag cell/digit as candidate
for (d=1; d<10; d++) {
if (index(ro co bl, d) == 0) {
e++
nr[r,d]++
nc[c,d]++
nb[bi,d]++
ca[c,r,d] = bi
}
}
if (e == 0) {                # in case no candidate is available, give up
return(0)
}
}
}
# go through all cell/digit candidates
# hidden singles
for (crd in ca) {
# a candidate may have been deleted after the loop started
if (ca[crd] != "") {
split(crd,spl,SUBSEP)
c = spl[1]
r = spl[2]
d = spl[3]
bi = ca[crd]
a = c + r * 9
# unique solution if at least one non-occurrence counter is exactly 1
if ((nr[r,d] == 1) || (nc[c,d] == 1) || (nb[bi,d] == 1)) {
ss = substr(ss,1,a) d substr(ss,a+2,length(ss))
didit = 1
# remove candidates from current row, column, block
for (e=0; e<9; e++) {
delete ca[c,e,d]
delete ca[e,r,d]
}
for (e=0; e<3; e++) {
for (b=0; b<3; b++) {
delete ca[int(c/3)*3+b,int(r/3)*3+e,d]
}
}
}
}
}
} while (didit == 1)
# second, pick a viable solution for the next empty cell and see if it leads to a solution
a = index(ss,".")-1
if (a == -1) {                     # no more empty cells, done
sos = ss
return(1)
}
else {
c = a % 9
r = int(a/9)
# concatenate current row, column and block
# row
co = substr(ss,r*9+1,9)
# column
for (d=0; d<9; d++) { co = co substr(ss,d*9+c+1,1) }
# block
c = int(c/3)*3+(int(r/3)*3)*9+1
for (d=0; d<3; d++) { co = co substr(ss,c+d*9,3) }
for (b=1; b<10; b++) {           # get a viable digit
if (index(co,b) == 0) {        # check if candidate digit already exists
# is viable, put in cell
nss = substr(ss,1,a) b substr(ss,a+2,length(ss))
d = solve(nss)               # try to solve
if (d == 1) {                # if successful, return
return(1)
}
}
}
}
return(0)                          # no digits viable, no solution
}
function error(message) { printf("error: %s\n",message) ; errors++ }
```
Output:
```| - - - + - - - + - - - | unsolved
| 1 2 3 | 4 5 6 | 7 8 9 |
| 4 5 6 | 7 8 9 | 1 2 3 |
| 7 8 9 | 1 2 3 | 4 5 6 |
| - - - + - - - + - - - |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| - - - + - - - + - - - |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| - - - + - - - + - - - |
| - - - + - - - + - - - | solved
| 1 2 3 | 4 5 6 | 7 8 9 |
| 4 5 6 | 7 8 9 | 1 2 3 |
| 7 8 9 | 1 2 3 | 4 5 6 |
| - - - + - - - + - - - |
| 2 1 4 | 3 6 5 | 8 9 7 |
| 3 6 5 | 8 9 7 | 2 1 4 |
| 8 9 7 | 2 1 4 | 3 6 5 |
| - - - + - - - + - - - |
| 5 3 1 | 6 4 2 | 9 7 8 |
| 6 4 2 | 9 7 8 | 5 3 1 |
| 9 7 8 | 5 3 1 | 6 4 2 |
| - - - + - - - + - - - |

bef: 123456789456789123789123456......................................................
aft: 123456789456789123789123456214365897365897214897214365531642978642978531978531642
```

BBC BASIC

```      VDU 23,22,453;453;8,20,16,128
*FONT Arial,28

DIM Board%(8,8)
Board%() = %111111111

FOR L% = 0 TO 9:P% = L%*100
LINE 2,P%+2,902,P%+2
IF (L% MOD 3)=0 LINE 2,P%,902,P% : LINE 2,P%+4,902,P%+4
LINE P%+2,2,P%+2,902
IF (L% MOD 3)=0 LINE P%,2,P%,902 : LINE P%+4,2,P%+4,902
NEXT

DATA "  4 5  6 "
DATA " 6 1  8 9"
DATA "3    7   "
DATA " 8    5  "
DATA "   4 3   "
DATA "  6    7 "
DATA "   2    6"
DATA "1 5  4 3 "
DATA " 2  7 1  "

FOR R% = 8 TO 0 STEP -1
FOR C% = 0 TO 8
A% = ASCMID\$(A\$,C%+1) AND 15
IF A% Board%(R%,C%) = 1 << (A%-1)
NEXT
NEXT R%

GCOL 4
PROCshow
WAIT 200
dummy% = FNsolve(Board%(), TRUE)
GCOL 2
PROCshow
REPEAT WAIT 1 : UNTIL FALSE
END

DEF PROCshow
LOCAL C%,P%,R%
FOR C% = 0 TO 8
FOR R% = 0 TO 8
P% = Board%(R%,C%)
IF (P% AND (P%-1)) = 0 THEN
IF P% P% = LOGP%/LOG2+1.5
MOVE C%*100+30,R%*100+90
VDU 5,P%+48,4
ENDIF
NEXT
NEXT
ENDPROC

DEF FNsolve(P%(),F%)
LOCAL C%,D%,M%,N%,R%,X%,Y%,Q%()
DIM Q%(8,8)
REPEAT
Q%() = P%()
FOR R% = 0 TO 8
FOR C% = 0 TO 8
D% = P%(R%,C%)
IF (D% AND (D%-1))=0 THEN
M% = NOT D%
FOR X% = 0 TO 8
IF X%<>C% P%(R%,X%) AND= M%
IF X%<>R% P%(X%,C%) AND= M%
NEXT
FOR X% = C%DIV3*3 TO C%DIV3*3+2
FOR Y% = R%DIV3*3 TO R%DIV3*3+2
IF X%<>C% IF Y%<>R% P%(Y%,X%) AND= M%
NEXT
NEXT
ENDIF
NEXT
NEXT
Q%() -= P%()
UNTIL SUMQ%()=0
M% = 10
FOR R% = 0 TO 8
FOR C% = 0 TO 8
D% = P%(R%,C%)
IF D%=0 M% = 0
IF D% AND (D%-1) THEN
N% = 0
REPEAT N% += D% AND 1
D% DIV= 2
UNTIL D% = 0
IF N%<M% M% = N% : X% = C% : Y% = R%
ENDIF
NEXT
NEXT
IF M%=0 THEN = 0
IF M%=10 THEN = 1
D% = 0
FOR M% = 0 TO 8
IF P%(Y%,X%) AND (2^M%) THEN
Q%() = P%()
Q%(Y%,X%) = 2^M%
C% = FNsolve(Q%(),F%)
D% += C%
IF C% IF F% P%() = Q%() : = D%
ENDIF
NEXT
= D%
```

BCPL

```// This can be run using Cintcode BCPL freely available from www.cl.cam.ac.uk/users/mr10.
// Implemented by Martin Richards.

// If you have cintcode BCPL installed on a Linux system you can compile and run this program
// execute the following sequence of commands.

// cd \$BCPLROOT
// cintsys
// c bc sudoku
// sudoku

// This is a really naive program to solve SuDoku problems. Even so it is usually quite fast.

// SuDoku consists of a 9x9 grid of cells. Each cell should contain
// a digit in the range 1..9. Every row, column and major 3x3
// square should contain all the digits 1..9. Some cells have
// given values. The problem is to find digits to place in
// the unspecified cells satisfying the constraints.

// A typical problem is:

//  - - -   6 3 8   - - -
//  7 - 6   - - -   3 - 5
//  - 1 -   - - -   - 4 -

//  - - 8   7 1 2   4 - -
//  - 9 -   - - -   - 5 -
//  - - 2   5 6 9   1 - -

//  - 3 -   - - -   - 1 -
//  1 - 5   - - -   6 - 8
//  - - -   1 8 4   - - -

SECTION "sudoku"

GET "libhdr"

GLOBAL { count:ug

// The 9x9 board

a1; a2; a3; a4; a5; a6; a7; a8; a9
b1; b2; b3; b4; b5; b6; b7; b8; b9
c1; c2; c3; c4; c5; c6; c7; c8; c9
d1; d2; d3; d4; d5; d6; d7; d8; d9
e1; e2; e3; e4; e5; e6; e7; e8; e9
f1; f2; f3; f4; f5; f6; f7; f8; f9
g1; g2; g3; g4; g5; g6; g7; g8; g9
h1; h2; h3; h4; h5; h6; h7; h8; h9
i1; i2; i3; i4; i5; i6; i7; i8; i9
}

MANIFEST {
N1=1<<0; N2=1<<1; N3=1<<2;
N4=1<<3; N5=1<<4; N6=1<<5;
N7=1<<6; N8=1<<7; N9=1<<8
}

LET start() = VALOF
{ count := 0
initboard()
prboard()
ta1()
writef("*n*nTotal number of solutions: %n*n", count)
RESULTIS 0
}

AND initboard() BE {
a1, a2, a3, a4, a5, a6, a7, a8, a9 :=  0, 0, 0, N6,N3,N8,  0, 0, 0
b1, b2, b3, b4, b5, b6, b7, b8, b9 := N7, 0,N6,  0, 0, 0, N3, 0,N5
c1, c2, c3, c4, c5, c6, c7, c8, c9 :=  0,N1, 0,  0, 0, 0,  0,N4, 0
d1, d2, d3, d4, d5, d6, d7, d8, d9 :=  0, 0,N8, N7,N1,N2, N4, 0, 0
e1, e2, e3, e4, e5, e6, e7, e8, e9 :=  0,N9, 0,  0, 0, 0,  0,N5, 0
f1, f2, f3, f4, f5, f6, f7, f8, f9 :=  0, 0,N2, N5,N6,N9, N1, 0, 0
g1, g2, g3, g4, g5, g6, g7, g8, g9 :=  0,N3, 0,  0, 0, 0,  0,N1, 0
h1, h2, h3, h4, h5, h6, h7, h8, h9 := N1, 0,N5,  0, 0, 0, N6, 0,N8
i1, i2, i3, i4, i5, i6, i7, i8, i9 :=  0, 0, 0, N1,N8,N4,  0, 0, 0

// Un-comment the following to test that the backtracking works
// giving 184 solutions.
//h1, h2, h3, h4, h5, h6, h7, h8, h9 := N1, 0,N5,  0, 0, 0, N6, 0, 0
//i1, i2, i3, i4, i5, i6, i7, i8, i9 :=  0, 0, 0,  0, 0, 0,  0, 0, 0
}

AND c(n) = VALOF SWITCHON n INTO
{ DEFAULT:    RESULTIS '?'
CASE  0:    RESULTIS '-'
CASE N1:    RESULTIS '1'
CASE N2:    RESULTIS '2'
CASE N3:    RESULTIS '3'
CASE N4:    RESULTIS '4'
CASE N5:    RESULTIS '5'
CASE N6:    RESULTIS '6'
CASE N7:    RESULTIS '7'
CASE N8:    RESULTIS '8'
CASE N9:    RESULTIS '9'
}

AND prboard() BE
{ LET form = "%c %c %c   %c %c %c   %c %c %c*n"
writef("*ncount = %n*n", count)
newline()
writef(form, c(a1),c(a2),c(a3),c(a4),c(a5),c(a6),c(a7),c(a8),c(a9))
writef(form, c(b1),c(b2),c(b3),c(b4),c(b5),c(b6),c(b7),c(b8),c(b9))
writef(form, c(c1),c(c2),c(c3),c(c4),c(c5),c(c6),c(c7),c(c8),c(c9))
newline()
writef(form, c(d1),c(d2),c(d3),c(d4),c(d5),c(d6),c(d7),c(d8),c(d9))
writef(form, c(e1),c(e2),c(e3),c(e4),c(e5),c(e6),c(e7),c(e8),c(e9))
writef(form, c(f1),c(f2),c(f3),c(f4),c(f5),c(f6),c(f7),c(f8),c(f9))
newline()
writef(form, c(g1),c(g2),c(g3),c(g4),c(g5),c(g6),c(g7),c(g8),c(g9))
writef(form, c(h1),c(h2),c(h3),c(h4),c(h5),c(h6),c(h7),c(h8),c(h9))
writef(form, c(i1),c(i2),c(i3),c(i4),c(i5),c(i6),c(i7),c(i8),c(i9))

newline()

//abort(1000)
}

AND try(p, f, row, col, sq) BE
{ LET x = !p
TEST x
THEN f()
ELSE { LET bits = row|col|sq
//prboard()
//              writef("x=%n %b9*n", x, bits)
//abort(1000)
IF (N1&bits)=0 DO { !p:=N1; f() }
IF (N2&bits)=0 DO { !p:=N2; f() }
IF (N3&bits)=0 DO { !p:=N3; f() }
IF (N4&bits)=0 DO { !p:=N4; f() }
IF (N5&bits)=0 DO { !p:=N5; f() }
IF (N6&bits)=0 DO { !p:=N6; f() }
IF (N7&bits)=0 DO { !p:=N7; f() }
IF (N8&bits)=0 DO { !p:=N8; f() }
IF (N9&bits)=0 DO { !p:=N9; f() }
!p := 0
}
}

AND ta1() BE try(@a1, ta2, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND ta2() BE try(@a2, ta3, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND ta3() BE try(@a3, ta4, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND ta4() BE try(@a4, ta5, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND ta5() BE try(@a5, ta6, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND ta6() BE try(@a6, ta7, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND ta7() BE try(@a7, ta8, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
a7+a8+a9+b7+b8+b9+c7+c8+c9)
AND ta8() BE try(@a8, ta9, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
a7+a8+a9+b7+b8+b9+c7+c8+c9)
AND ta9() BE try(@a9, tb1, a1+a2+a3+a4+a5+a6+a7+a8+a9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
a7+a8+a9+b7+b8+b9+c7+c8+c9)

AND tb1() BE try(@b1, tb2, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND tb2() BE try(@b2, tb3, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND tb3() BE try(@b3, tb4, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND tb4() BE try(@b4, tb5, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND tb5() BE try(@b5, tb6, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND tb6() BE try(@b6, tb7, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND tb7() BE try(@b7, tb8, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
a7+a8+a9+b7+b8+b9+c7+c8+c9)
AND tb8() BE try(@b8, tb9, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
a7+a8+a9+b7+b8+b9+c7+c8+c9)
AND tb9() BE try(@b9, tc1, b1+b2+b3+b4+b5+b6+b7+b8+b9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
a7+a8+a9+b7+b8+b9+c7+c8+c9)

AND tc1() BE try(@c1, tc2, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND tc2() BE try(@c2, tc3, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND tc3() BE try(@c3, tc4, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
a1+a2+a3+b1+b2+b3+c1+c2+c3)
AND tc4() BE try(@c4, tc5, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND tc5() BE try(@c5, tc6, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND tc6() BE try(@c6, tc7, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
a4+a5+a6+b4+b5+b6+c4+c5+c6)
AND tc7() BE try(@c7, tc8, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
a7+a8+a9+b7+b8+b9+c7+c8+c9)
AND tc8() BE try(@c8, tc9, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
a7+a8+a9+b7+b8+b9+c7+c8+c9)
AND tc9() BE try(@c9, td1, c1+c2+c3+c4+c5+c6+c7+c8+c9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
a7+a8+a9+b7+b8+b9+c7+c8+c9)

AND td1() BE try(@d1, td2, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND td2() BE try(@d2, td3, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND td3() BE try(@d3, td4, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND td4() BE try(@d4, td5, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND td5() BE try(@d5, td6, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND td6() BE try(@d6, td7, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND td7() BE try(@d7, td8, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
d7+d8+d9+e7+e8+e9+f7+f8+f9)
AND td8() BE try(@d8, td9, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
d7+d8+d9+e7+e8+e9+f7+f8+f9)
AND td9() BE try(@d9, te1, d1+d2+d3+d4+d5+d6+d7+d8+d9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
d7+d8+d9+e7+e8+e9+f7+f8+f9)

AND te1() BE try(@e1, te2, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND te2() BE try(@e2, te3, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND te3() BE try(@e3, te4, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND te4() BE try(@e4, te5, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND te5() BE try(@e5, te6, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND te6() BE try(@e6, te7, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND te7() BE try(@e7, te8, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
d7+d8+d9+e7+e8+e9+f7+f8+f9)
AND te8() BE try(@e8, te9, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
d7+d8+d9+e7+e8+e9+f7+f8+f9)
AND te9() BE try(@e9, tf1, e1+e2+e3+e4+e5+e6+e7+e8+e9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
d7+d8+d9+e7+e8+e9+f7+f8+f9)

AND tf1() BE try(@f1, tf2, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND tf2() BE try(@f2, tf3, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND tf3() BE try(@f3, tf4, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
d1+d2+d3+e1+e2+e3+f1+f2+f3)
AND tf4() BE try(@f4, tf5, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND tf5() BE try(@f5, tf6, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND tf6() BE try(@f6, tf7, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
d4+d5+d6+e4+e5+e6+f4+f5+f6)
AND tf7() BE try(@f7, tf8, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
d7+d8+d9+e7+e8+e9+f7+f8+f9)
AND tf8() BE try(@f8, tf9, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
d7+d8+d9+e7+e8+e9+f7+f8+f9)
AND tf9() BE try(@f9, tg1, f1+f2+f3+f4+f5+f6+f7+f8+f9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
d7+d8+d9+e7+e8+e9+f7+f8+f9)

AND tg1() BE try(@g1, tg2, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND tg2() BE try(@g2, tg3, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND tg3() BE try(@g3, tg4, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND tg4() BE try(@g4, tg5, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND tg5() BE try(@g5, tg6, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND tg6() BE try(@g6, tg7, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND tg7() BE try(@g7, tg8, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
g7+g8+g9+h7+h8+h9+i7+i8+i9)
AND tg8() BE try(@g8, tg9, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
g7+g8+g9+h7+h8+h9+i7+i8+i9)
AND tg9() BE try(@g9, th1, g1+g2+g3+g4+g5+g6+g7+g8+g9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
g7+g8+g9+h7+h8+h9+i7+i8+i9)

AND th1() BE try(@h1, th2, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND th2() BE try(@h2, th3, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND th3() BE try(@h3, th4, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND th4() BE try(@h4, th5, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND th5() BE try(@h5, th6, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND th6() BE try(@h6, th7, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND th7() BE try(@h7, th8, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
g7+g8+g9+h7+h8+h9+i7+i8+i9)
AND th8() BE try(@h8, th9, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
g7+g8+g9+h7+h8+h9+i7+i8+i9)
AND th9() BE try(@h9, ti1, h1+h2+h3+h4+h5+h6+h7+h8+h9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
g7+g8+g9+h7+h8+h9+i7+i8+i9)

AND ti1() BE try(@i1, ti2, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a1+b1+c1+d1+e1+f1+g1+h1+i1,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND ti2() BE try(@i2, ti3, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a2+b2+c2+d2+e2+f2+g2+h2+i2,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND ti3() BE try(@i3, ti4, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a3+b3+c3+d3+e3+f3+g3+h3+i3,
g1+g2+g3+h1+h2+h3+i1+i2+i3)
AND ti4() BE try(@i4, ti5, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a4+b4+c4+d4+e4+f4+g4+h4+i4,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND ti5() BE try(@i5, ti6, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a5+b5+c5+d5+e5+f5+g5+h5+i5,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND ti6() BE try(@i6, ti7, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a6+b6+c6+d6+e6+f6+g6+h6+i6,
g4+g5+g6+h4+h5+h6+i4+i5+i6)
AND ti7() BE try(@i7, ti8, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a7+b7+c7+d7+e7+f7+g7+h7+i7,
g7+g8+g9+h7+h8+h9+i7+i8+i9)
AND ti8() BE try(@i8, ti9, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a8+b8+c8+d8+e8+f8+g8+h8+i8,
g7+g8+g9+h7+h8+h9+i7+i8+i9)
AND ti9() BE try(@i9, suc, i1+i2+i3+i4+i5+i6+i7+i8+i9,
a9+b9+c9+d9+e9+f9+g9+h9+i9,
g7+g8+g9+h7+h8+h9+i7+i8+i9)

AND suc() BE
{ count := count + 1
prboard()
}```

Befunge

Input should be provided as a sequence of 81 digits (optionally separated by whitespace), with zero representing an unknown value.

```99*>1-:0>:#\$"0"\# #~`#\$_"0"-\::9%:9+00p3/\9/:99++10p3vv%2g\g01<
2%v|:p+9/9\%9:\p\g02\1p\g01\1:p\g00\1:+8:\p02+*93+*3/<>\20g\g#:
v<+>:0\`>v >\::9%:9+00p3/\9/:99++10p3/3*+39*+20p\:8+::00g\g2%\^
v^+^pppp\$_:|v<::<_>1-::9%\9/9+g.::9%!\3%+>>#v_>" "v..v,<<<+55<<
03!\$v9:_>1v\$>9%\v^|:<_v#<%<9<:<<_v#+%*93\!::<,,"|"<\/>:#^_>>>v^
p|<\$0.0^!g+:#9/9<^@ ^,>#+5<5_>#!<>#\$0"------+-------+-----":#<^
<>v\$v1:::0<>"P"`!^>0g#0v#p+9/9\%9:p04:\pg03g021pg03g011pg03g001
::>^_:#<0#!:p#-\#1:#g0<>30g010g30g020g30g040g:9%\:9/9+\01-\1+0:
```
Input:
```8 5 0 0 0 2 4 0 0
7 2 0 0 0 0 0 0 9
0 0 4 0 0 0 0 0 0
0 0 0 1 0 7 0 0 2
3 0 5 0 0 0 9 0 0
0 4 0 0 0 0 0 0 0
0 0 0 0 8 0 0 7 0
0 1 7 0 0 0 0 0 0
0 0 0 0 3 6 0 4 0```
Output:
```8 5 9 | 6 1 2 | 4 3 7
7 2 3 | 8 5 4 | 1 6 9
1 6 4 | 3 7 9 | 5 2 8
------+-------+------
9 8 6 | 1 4 7 | 3 5 2
3 7 5 | 2 6 8 | 9 1 4
2 4 1 | 5 9 3 | 7 8 6
------+-------+------
4 3 2 | 9 8 1 | 6 7 5
6 1 7 | 4 2 5 | 8 9 3
5 9 8 | 7 3 6 | 2 4 1```

Bracmat

The program:

```{sudokuSolver.bra

Solves any 9x9 sudoku, using backtracking.
Not a simple brute force algorithm!}

sudokuSolver=
( sudoku
=   ( new
=   create
.   ( create
=   a
.     !arg:%(<3:?a) ?arg
&   ( !a
.     !arg:
& 1 2 3 4 5 6 7 8 9
| create\$!arg
)
create\$(!a+1 !arg)
|
)
& create\$(0 0 0 0):?(its.Tree)
& ( init
=   cell remainingCells remainingRows x y
.       !arg
: ( ?y
. ?x
. (.%?cell ?remainingCells) ?remainingRows
)
&   (   !cell:#
& ( !cell
.   mod\$(!x,3)
div\$(!x,3)
mod\$(!y,3)
div\$(!y,3)
)
|
)
(   !remainingCells:
& init\$(!y+1.0.!remainingRows)
|   init
\$ ( !y
. !x+1
. (.!remainingCells) !remainingRows
)
)
|
)
& out\$!arg
&   (its.Set)\$(!(its.Tree).init\$(0.0.!arg))
: ?(its.Tree)
)
( Display
=   val
.     put\$(str\$("|~~~|~~~|~~~|" \n))
&   !(its.Tree)
:   ?
( ?
.     ?
( ?&put\$"|"
.     ?
( ?
.     ?
( ( ?
.     ?val
& !val:% %
& put\$"-"
|   !val:
& put\$" "
| put\$!val
)
& ~
)
?
| ?&put\$"|"&~
)
?
| ?&put\$\n&~
)
?
|   ?
& put\$(str\$("|~~~|~~~|~~~|" \n))
& ~
)
?
|
)
( Set
=     update certainValue a b c d
, tree branch todo DOING loop dcba minlen len minp
.   ( update
=     path rempath value tr
, k z x y trc p v branch s n
.   !arg:(?path.?value.?tr.?trc)
& (   !path:%?path ?rempath
& `(     !tr
: ?k (!path:?p.?branch) ?z
& `(   update\$(!rempath.!value.!branch.!p !trc)
: ?s
&     update
\$ (!path !rempath.!value.!z.!trc)
: ?n
& !k (!p.!s) !n
)
| !tr
)
| !DOING:(?.!trc)&!value
|   !tr:?x !value ?y
& `( !x !y
: (   ~:@
& (   !todo:? (?v.!trc) ?
& ( !v:!x !y
|     out
\$ (mismatch v !v "<>" x y !x !y)
& get'
)
| (!x !y.!trc) !todo:?todo
)
| % %
| &!DOING:(?.!trc)
)
)
| !tr
)
)
& !arg:(?tree.?todo)
& ( loop
=   !todo:
|     !todo
: ((?certainValue.%?d %?c %?b %?a):?DOING) ?todo
&   update\$(!a ? !c ?.!certainValue.!tree.)
: ?tree
&   update\$(!a !b <>!c ?.!certainValue.!tree.)
: ?tree
&   update\$(<>!a ? !c !d.!certainValue.!tree.)
: ?tree
& !loop
)
& !loop
& ( ~( !tree
:   ?
(?.? (?.? (?.? (?.% %) ?) ?) ?)
?
)
|   9:?minlen
& :?minp
& ( len
=
.   !arg:% %?arg&1+len\$!arg
| 1
)
& (   !tree
:   ?
( ?a
.   ?
( ?b
.   ?
( ?c
.   ?
( ?d
.   % %:?p
& len\$!p:<!minlen:?minlen
& !d !c !b !a:?dcba
& !p:?:?minp
& ~
)
?
)
?
)
?
)
?
|   !minp
:   ?
( %@?n
& (its.Set)\$(!tree.!n.!dcba):?tree
)
?
)
)
& !tree
)
(Tree=)
)
( new
=   puzzle
.   new\$((its.sudoku),!arg):?puzzle
& (puzzle..Display)\$
);```

Solve a sudoku that is hard for a brute force solver:

```new'( sudokuSolver
, (.- - - - - - - - -)
(.- - - - - 3 - 8 5)
(.- - 1 - 2 - - - -)
(.- - - 5 - 7 - - -)
(.- - 4 - - - 1 - -)
(.- 9 - - - - - - -)
(.5 - - - - - - 7 3)
(.- - 2 - 1 - - - -)
(.- - - - 4 - - - 9)
);```

Solution:

```|~~~|~~~|~~~|
|987|654|321|
|246|173|985|
|351|928|746|
|~~~|~~~|~~~|
|128|537|694|
|634|892|157|
|795|461|832|
|~~~|~~~|~~~|
|519|286|473|
|472|319|568|
|863|745|219|
|~~~|~~~|~~~|```

C

See e.g. this GPLed solver written in C.

The following code is really only good for size 3 puzzles. A longer, even less readable version here could handle size 4s.

```#include <stdio.h>

void show(int *x)
{
int i, j;
for (i = 0; i < 9; i++) {
if (!(i % 3)) putchar('\n');
for (j = 0; j < 9; j++)
printf(j % 3 ? "%2d" : "%3d", *x++);
putchar('\n');
}
}

int trycell(int *x, int pos)
{
int row = pos / 9;
int col = pos % 9;
int i, j, used = 0;

if (pos == 81) return 1;
if (x[pos]) return trycell(x, pos + 1);

for (i = 0; i < 9; i++)
used |= 1 << (x[i * 9 + col] - 1);

for (j = 0; j < 9; j++)
used |= 1 << (x[row * 9 + j] - 1);

row = row / 3 * 3;
col = col / 3 * 3;
for (i = row; i < row + 3; i++)
for (j = col; j < col + 3; j++)
used |= 1 << (x[i * 9 + j] - 1);

for (x[pos] = 1; x[pos] <= 9; x[pos]++, used >>= 1)
if (!(used & 1) && trycell(x, pos + 1)) return 1;

x[pos] = 0;
return 0;
}

void solve(const char *s)
{
int i, x[81];
for (i = 0; i < 81; i++)
x[i] = s[i] >= '1' && s[i] <= '9' ? s[i] - '0' : 0;

if (trycell(x, 0))
show(x);
else
puts("no solution");
}

int main(void)
{
solve(	"5x..7...."
"6..195..."
".98....6."
"8...6...3"
"4..8.3..1"
"7...2...6"
".6....28."
"...419..5"
"....8..79"	);

return 0;
}
```

C#

Backtracking

Translation of: Java
```using System;

class SudokuSolver
{
private int[] grid;

public SudokuSolver(String s)
{
grid = new int[81];
for (int i = 0; i < s.Length; i++)
{
grid[i] = int.Parse(s[i].ToString());
}
}

public void solve()
{
try
{
placeNumber(0);
Console.WriteLine("Unsolvable!");
}
catch (Exception ex)
{
Console.WriteLine(ex.Message);
Console.WriteLine(this);
}
}

public void placeNumber(int pos)
{
if (pos == 81)
{
throw new Exception("Finished!");
}
if (grid[pos] > 0)
{
placeNumber(pos + 1);
return;
}
for (int n = 1; n <= 9; n++)
{
if (checkValidity(n, pos % 9, pos / 9))
{
grid[pos] = n;
placeNumber(pos + 1);
grid[pos] = 0;
}
}
}

public bool checkValidity(int val, int x, int y)
{
for (int i = 0; i < 9; i++)
{
if (grid[y * 9 + i] == val || grid[i * 9 + x] == val)
return false;
}
int startX = (x / 3) * 3;
int startY = (y / 3) * 3;
for (int i = startY; i < startY + 3; i++)
{
for (int j = startX; j < startX + 3; j++)
{
if (grid[i * 9 + j] == val)
return false;
}
}
return true;
}

public override string ToString()
{
string sb = "";
for (int i = 0; i < 9; i++)
{
for (int j = 0; j < 9; j++)
{
sb += (grid[i * 9 + j] + " ");
if (j == 2 || j == 5)
sb += ("| ");
}
sb += ('\n');
if (i == 2 || i == 5)
sb += ("------+-------+------\n");
}
return sb;
}

public static void Main(String[] args)
{
new SudokuSolver("850002400" +
"720000009" +
"004000000" +
"000107002" +
"305000900" +
"040000000" +
"000080070" +
"017000000" +
"000036040").solve();
}
}
```

Best First Search

```using System.Linq;
using static System.Linq.Enumerable;
using System.Collections.Generic;
using System;
using System.Runtime.CompilerServices;

namespace SodukoFastMemoBFS {
internal readonly record struct Square (int Row, int Col);
internal record Constraints (IEnumerable<int> ConstrainedRange, Square Square);
internal class Cache : Dictionary<Square, Constraints> { };
internal record CacheGrid (int[][] Grid, Cache Cache);

internal static class SudokuFastMemoBFS {
internal static U Fwd<T, U>(this T data, Func<T, U> f) => f(data);

[MethodImpl(MethodImplOptions.AggressiveInlining)]
private static int RowCol(int rc) => rc <= 2 ? 0 : rc <= 5 ? 3 : 6;

private static bool Solve(this CacheGrid cg, Constraints constraints, int finished) {
var (row, col) = constraints.Square;
foreach (var i in constraints.ConstrainedRange) {
cg.Grid[row][col] = i;
if (cg.Cache.Count == finished || cg.Solve(cg.Next(constraints.Square), finished))
return true;
}
cg.Grid[row][col] = 0;
return false;
}

private static readonly int[] domain = Range(0, 9).ToArray();
private static readonly int[] range = Range(1, 9).ToArray();

private static bool Valid(this int[][] grid, int row, int col, int val) {
for (var i = 0; i < 9; i++)
if (grid[row][i] == val || grid[i][col] == val)
return false;
for (var r = RowCol(row); r < RowCol(row) + 3; r++)
for (var c = RowCol(col); c < RowCol(col) + 3; c++)
if (grid[r][c] == val)
return false;
return true;
}

private static IEnumerable<int> Constraints(this int[][] grid, int row, int col) =>
range.Where(val => grid.Valid(row, col, val));

private static Constraints Next(this CacheGrid cg, Square square) =>
cg.Cache.ContainsKey(square)
? cg.Cache[square]
: cg.Cache[square]=cg.Grid.SortedCells();

private static Constraints SortedCells(this int[][] grid) =>
(from row in domain
from col in domain
where grid[row][col] == 0
let cell = new Constraints(grid.Constraints(row, col), new Square(row, col))
orderby cell.ConstrainedRange.Count() ascending
select cell).First();

private static CacheGrid Parse(string input) =>
input
.Select((c, i) => (index: i, val: int.Parse(c.ToString())))
.GroupBy(id => id.index / 9)
.Select(grp => grp.Select(id => id.val).ToArray())
.ToArray()
.Fwd(grid => new CacheGrid(grid, new Cache()));

public static string AsString(this int[][] grid) =>
string.Join('\n', grid.Select(row => string.Concat(row)));

public static int[][] Run(string input) {
var cg = Parse(input);
var marked = cg.Grid.SelectMany(row => row.Where(c => c > 0)).Count();
return cg.Solve(cg.Grid.SortedCells(), 80 - marked) ? cg.Grid : new int[][] { Array.Empty<int>() };
}
}
}
```

Usage

```using System.Linq;
using static System.Linq.Enumerable;
using static System.Console;
using System.IO;

namespace SodukoFastMemoBFS {
static class Program {

static void Main(string[] args) {
var num = int.Parse(args[0]);
var single = puzzles.First();

var watch = new System.Diagnostics.Stopwatch();
watch.Start();
WriteLine(SudokuFastMemoBFS.Run(single).AsString());
watch.Stop();
WriteLine(\$"{single}: {watch.ElapsedMilliseconds} ms");

WriteLine(\$"Doing {num} puzzles");
var total = 0.0;
watch.Start();
foreach (var puzzle in puzzles) {
watch.Reset();
watch.Start();
SudokuFastMemoBFS.Run(puzzle);
watch.Stop();
total += watch.ElapsedMilliseconds;
Write(".");
}
watch.Stop();
WriteLine(\$"\nPuzzles:{num}, Total:{total} ms, Average:{total / num:0.00} ms");
}
}
}
```

Output

```693784512
487512936
125963874
932651487
568247391
741398625
319475268
856129743
274836159
000000010400000000020000000000050407008000300001090000300400200050100000000806000: 336 ms
Doing 100 puzzles
....................................................................................................
Puzzles:100, Total:5316 ms, Average:53.16 ms```

Solver

```using Microsoft.SolverFoundation.Solvers;

namespace Sudoku
{
class Program
{
private static int[,] B = new int[,] {{9,7,0, 3,0,0, 0,6,0},
{0,6,0, 7,5,0, 0,0,0},
{0,0,0, 0,0,8, 0,5,0},

{0,0,0, 0,0,0, 6,7,0},
{0,0,0, 0,3,0, 0,0,0},
{0,5,3, 9,0,0, 2,0,0},

{7,0,0, 0,2,5, 0,0,0},
{0,0,2, 0,1,0, 0,0,8},
{0,4,0, 0,0,7, 3,0,0}};

private static CspTerm[] GetSlice(CspTerm[][] sudoku, int Ra, int Rb, int Ca, int Cb)
{
CspTerm[] slice = new CspTerm[9];
int i = 0;
for (int row = Ra; row < Rb + 1; row++)
for (int col = Ca; col < Cb + 1; col++)
{
{
slice[i++] = sudoku[row][col];
}
}
return slice;
}

static void Main(string[] args)
{
ConstraintSystem S = ConstraintSystem.CreateSolver();
CspDomain Z = S.CreateIntegerInterval(1, 9);
CspTerm[][] sudoku = S.CreateVariableArray(Z, "cell", 9, 9);
for (int row = 0; row < 9; row++)
{
for (int col = 0; col < 9; col++)
{
if (B[row, col] > 0)
{
}
}
}
for (int col = 0; col < 9; col++)
{
}
for (int a = 0; a < 3; a++)
{
for (int b = 0; b < 3; b++)
{
S.AddConstraints(S.Unequal(GetSlice(sudoku, a * 3, a * 3 + 2, b * 3, b * 3 + 2)));
}
}
ConstraintSolverSolution soln = S.Solve();
object[] h = new object[9];
for (int row = 0; row < 9; row++)
{
if ((row % 3) == 0) System.Console.WriteLine();
for (int col = 0; col < 9; col++)
{
soln.TryGetValue(sudoku[row][col], out h [col]);
}
System.Console.WriteLine("{0}{1}{2} {3}{4}{5} {6}{7}{8}", h[0],h[1],h[2],h[3],h[4],h[5],h[6],h[7],h[8]);
}
}
}
}
```

Produces:

```975 342 861
861 759 432
324 168 957

219 584 673
487 236 519
653 971 284

738 425 196
592 613 748
146 897 325
```

```using System;
using System.Collections.Generic;
using System.Text;
using static System.Linq.Enumerable;

public class Sudoku
{
public static void Main2() {
string puzzle = "....7.94.....9...53....5.7...74..1..463...........7.8.8........7......28.5.26....";
string solution = new Sudoku().Solutions(puzzle).FirstOrDefault() ?? puzzle;
Print(puzzle, solution);
}

private DLX dlx;

public void Build() {
const int rows = 9 * 9 * 9, columns = 4 * 9 * 9;
dlx = new DLX(rows, columns);

for (int cell = 0, row = 0; row < 9; row++) {
for (int column = 0; column < 9; column++) {
int box = row / 3 * 3 + column / 3;
for (int digit = 0; digit < 9; digit++) {
dlx.AddRow(cell, 81 + row * 9 + digit, 2 * 81 + column * 9 + digit, 3 * 81 + box * 9 + digit);
}
cell++;
}
}
}

public IEnumerable<string> Solutions(string puzzle) {
if (puzzle == null) throw new ArgumentNullException(nameof(puzzle));
if (puzzle.Length != 81) throw new ArgumentException("The input is not of the correct length.");
if (dlx == null) Build();

for (int i = 0; i < puzzle.Length; i++) {
if (puzzle[i] == '0' || puzzle[i] == '.') continue;
if (puzzle[i] < '1' && puzzle[i] > '9') throw new ArgumentException(\$"Input contains an invalid character: ({puzzle[i]})");
int digit = puzzle[i] - '0' - 1;
dlx.Give(i * 9 + digit);
}
return Iterator();

IEnumerable<string> Iterator() {
var sb = new StringBuilder(new string('.', 81));
foreach (int[] rows in dlx.Solutions()) {
foreach (int r in rows) {
sb[r / 81 * 9 + r / 9 % 9] = (char)(r % 9 + '1');
}
yield return sb.ToString();
}
}
}

static void Print(string left, string right) {
foreach (string line in GetPrintLines(left).Zip(GetPrintLines(right), (l, r) => l + "\t" + r)) {
Console.WriteLine(line);
}

IEnumerable<string> GetPrintLines(string s) {
int r = 0;
foreach (string row in s.Cut(9)) {
yield return r == 0
? "╔═══╤═══╤═══╦═══╤═══╤═══╦═══╤═══╤═══╗"
: r % 3 == 0
? "╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣"
: "╟───┼───┼───╫───┼───┼───╫───┼───┼───╢";
yield return "║ " + row.Cut(3).Select(segment => segment.DelimitWith(" │ ")).DelimitWith(" ║ ") + " ║";
r++;
}
yield return "╚═══╧═══╧═══╩═══╧═══╧═══╩═══╧═══╧═══╝";
}
}

}

public class DLX //Some functionality elided
{
private readonly Stack<Node> solutionNodes = new Stack<Node>();
private int initial = 0;

public DLX(int rowCapacity, int columnCapacity) {
rows = new List<Node>(rowCapacity);
}

h.AttachLeftRight();
}

public void AddRow(params int[] newRow) {
Node first = null;
if (newRow != null) {
for (int i = 0; i < newRow.Length; i++) {
if (newRow[i] < 0) continue;
if (first == null) first = AddNode(rows.Count, newRow[i]);
}
}
}

private Node AddNode(int row, int column) {
Node n = new Node(null, null, columns[column].Up, columns[column], columns[column], row);
n.AttachUpDown();
return n;
}

private void AddNode(Node firstNode, int column) {
Node n = new Node(firstNode.Left, firstNode, columns[column].Up, columns[column], columns[column], firstNode.Row);
n.AttachLeftRight();
n.AttachUpDown();
}

public void Give(int row) {
solutionNodes.Push(rows[row]);
CoverMatrix(rows[row]);
initial++;
}

public IEnumerable<int[]> Solutions() {
try {
Node node = ChooseSmallestColumn().Down;
do {
if (node == root) {
yield return solutionNodes.Select(n => n.Row).ToArray();
}
if (solutionNodes.Count > initial) {
node = solutionNodes.Pop();
UncoverMatrix(node);
node = node.Down;
}
} else {
solutionNodes.Push(node);
CoverMatrix(node);
node = ChooseSmallestColumn().Down;
}
} while(solutionNodes.Count > initial || node != node.Head);
} finally {
Restore();
}
}

private void Restore() {
while (solutionNodes.Count > 0) UncoverMatrix(solutionNodes.Pop());
initial = 0;
}

Header traveller = root, choice = root;
do {
if (traveller.Size < choice.Size) choice = traveller;
} while (traveller != root && choice.Size > 0);
return choice;
}

private void CoverRow(Node row) {
Node traveller = row.Right;
while (traveller != row) {
traveller.DetachUpDown();
traveller = traveller.Right;
}
}

private void UncoverRow(Node row) {
Node traveller = row.Left;
while (traveller != row) {
traveller.AttachUpDown();
traveller = traveller.Left;
}
}

column.DetachLeftRight();
Node traveller = column.Down;
while (traveller != column) {
CoverRow(traveller);
traveller = traveller.Down;
}
}

Node traveller = column.Up;
while (traveller != column) {
UncoverRow(traveller);
traveller = traveller.Up;
}
column.AttachLeftRight();
}

private void CoverMatrix(Node node) {
Node traveller = node;
do {
traveller = traveller.Right;
} while (traveller != node);
}

private void UncoverMatrix(Node node) {
Node traveller = node;
do {
traveller = traveller.Left;
} while (traveller != node);
}

private class Node
{
public Node(Node left, Node right, Node up, Node down, Header head, int row) {
Left = left ?? this;
Right = right ?? this;
Up = up ?? this;
Down = down ?? this;
Row = row;
}

public Node Left   { get; set; }
public Node Right  { get; set; }
public Node Up     { get; set; }
public Node Down   { get; set; }
public int Row     { get; }

public void AttachLeftRight() {
this.Left.Right = this;
this.Right.Left = this;
}

public void AttachUpDown() {
this.Up.Down = this;
this.Down.Up = this;
}

public void DetachLeftRight() {
this.Left.Right = this.Right;
this.Right.Left = this.Left;
}

public void DetachUpDown() {
this.Up.Down = this.Down;
this.Down.Up = this.Up;
}

}

{
public Header(Node left, Node right) : base(left, right, null, null, null, -1) { }

public int Size { get; set; }
}

}

static class Extensions
{
public static IEnumerable<string> Cut(this string input, int length)
{
for (int cursor = 0; cursor < input.Length; cursor += length) {
if (cursor + length > input.Length) yield return input.Substring(cursor);
else yield return input.Substring(cursor, length);
}
}

public static string DelimitWith<T>(this IEnumerable<T> source, string separator) => string.Join(separator, source);
}
```
Output:
```╔═══╤═══╤═══╦═══╤═══╤═══╦═══╤═══╤═══╗	╔═══╤═══╤═══╦═══╤═══╤═══╦═══╤═══╤═══╗
║ . │ . │ . ║ . │ 7 │ . ║ 9 │ 4 │ . ║	║ 2 │ 1 │ 5 ║ 8 │ 7 │ 6 ║ 9 │ 4 │ 3 ║
╟───┼───┼───╫───┼───┼───╫───┼───┼───╢	╟───┼───┼───╫───┼───┼───╫───┼───┼───╢
║ . │ . │ . ║ . │ 9 │ . ║ . │ . │ 5 ║	║ 6 │ 7 │ 8 ║ 3 │ 9 │ 4 ║ 2 │ 1 │ 5 ║
╟───┼───┼───╫───┼───┼───╫───┼───┼───╢	╟───┼───┼───╫───┼───┼───╫───┼───┼───╢
║ 3 │ . │ . ║ . │ . │ 5 ║ . │ 7 │ . ║	║ 3 │ 4 │ 9 ║ 1 │ 2 │ 5 ║ 8 │ 7 │ 6 ║
╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣	╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣
║ . │ . │ 7 ║ 4 │ . │ . ║ 1 │ . │ . ║	║ 5 │ 8 │ 7 ║ 4 │ 3 │ 2 ║ 1 │ 6 │ 9 ║
╟───┼───┼───╫───┼───┼───╫───┼───┼───╢	╟───┼───┼───╫───┼───┼───╫───┼───┼───╢
║ 4 │ 6 │ 3 ║ . │ . │ . ║ . │ . │ . ║	║ 4 │ 6 │ 3 ║ 9 │ 8 │ 1 ║ 7 │ 5 │ 2 ║
╟───┼───┼───╫───┼───┼───╫───┼───┼───╢	╟───┼───┼───╫───┼───┼───╫───┼───┼───╢
║ . │ . │ . ║ . │ . │ 7 ║ . │ 8 │ . ║	║ 1 │ 9 │ 2 ║ 6 │ 5 │ 7 ║ 3 │ 8 │ 4 ║
╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣	╠═══╪═══╪═══╬═══╪═══╪═══╬═══╪═══╪═══╣
║ 8 │ . │ . ║ . │ . │ . ║ . │ . │ . ║	║ 8 │ 2 │ 6 ║ 7 │ 4 │ 3 ║ 5 │ 9 │ 1 ║
╟───┼───┼───╫───┼───┼───╫───┼───┼───╢	╟───┼───┼───╫───┼───┼───╫───┼───┼───╢
║ 7 │ . │ . ║ . │ . │ . ║ . │ 2 │ 8 ║	║ 7 │ 3 │ 4 ║ 5 │ 1 │ 9 ║ 6 │ 2 │ 8 ║
╟───┼───┼───╫───┼───┼───╫───┼───┼───╢	╟───┼───┼───╫───┼───┼───╫───┼───┼───╢
║ . │ 5 │ . ║ 2 │ 6 │ . ║ . │ . │ . ║	║ 9 │ 5 │ 1 ║ 2 │ 6 │ 8 ║ 4 │ 3 │ 7 ║
╚═══╧═══╧═══╩═══╧═══╧═══╩═══╧═══╧═══╝	╚═══╧═══╧═══╩═══╧═══╧═══╩═══╧═══╧═══╝```

C++

Translation of: Java
```#include <iostream>
using namespace std;

class SudokuSolver {
private:
int grid[81];

public:

SudokuSolver(string s) {
for (unsigned int i = 0; i < s.length(); i++) {
grid[i] = (int) (s[i] - '0');
}
}

void solve() {
try {
placeNumber(0);
cout << "Unsolvable!" << endl;
} catch (char* ex) {
cout << ex << endl;
cout << this->toString() << endl;
}
}

void placeNumber(int pos) {
if (pos == 81) {
throw (char*) "Finished!";
}
if (grid[pos] > 0) {
placeNumber(pos + 1);
return;
}
for (int n = 1; n <= 9; n++) {
if (checkValidity(n, pos % 9, pos / 9)) {
grid[pos] = n;
placeNumber(pos + 1);
grid[pos] = 0;
}
}
}

bool checkValidity(int val, int x, int y) {
for (int i = 0; i < 9; i++) {
if (grid[y * 9 + i] == val || grid[i * 9 + x] == val)
return false;
}
int startX = (x / 3) * 3;
int startY = (y / 3) * 3;
for (int i = startY; i < startY + 3; i++) {
for (int j = startX; j < startX + 3; j++) {
if (grid[i * 9 + j] == val)
return false;
}
}
return true;
}

string toString() {
string sb;
for (int i = 0; i < 9; i++) {
for (int j = 0; j < 9; j++) {
char c[2];
c[0] = grid[i * 9 + j] + '0';
c[1] = '\0';
sb.append(c);
sb.append(" ");
if (j == 2 || j == 5)
sb.append("| ");
}
sb.append("\n");
if (i == 2 || i == 5)
sb.append("------+-------+------\n");
}
return sb;
}

};

int main() {
SudokuSolver ss("850002400"
"720000009"
"004000000"
"000107002"
"305000900"
"040000000"
"000080070"
"017000000"
"000036040");
ss.solve();
return EXIT_SUCCESS;
}
```

Clojure

```(ns rosettacode.sudoku
(:use [clojure.pprint :only (cl-format)]))

(defn- compatible? [m x y n]
(let [n= #(= n (get-in m [%1 %2]))]
(or (n= y x)
(let [c (count m)]
(and (zero? (get-in m [y x]))
(not-any? #(or (n= y %) (n= % x)) (range c))
(let [zx (* c (quot x c)), zy (* c (quot y c))]
(every? false?
(map n= (range zy (+ zy c)) (range zx (+ zx c))))))))))

(defn solve [m]
(let [c (count m)]
(loop [m m, x 0, y 0]
(if (= y c) m
(let [ng (->> (range 1 c)
(filter #(compatible? m x y %))
first
(assoc-in m [y x]))]
(if (= x (dec c))
(recur ng 0 (inc y))
(recur ng (inc x) y)))))))
```
```sudoku>(cl-format true "~{~{~a~^ ~}~%~}"
(solve [[3 9 4 0 0 2 6 7 0]
[0 0 0 3 0 0 4 0 0]
[5 0 0 6 9 0 0 2 0]
[0 4 5 0 0 0 9 0 0]
[6 0 0 0 0 0 0 0 7]
[0 0 7 0 0 0 5 8 0]
[0 1 0 0 6 7 0 0 8]
[0 0 9 0 0 8 0 0 0]
[0 2 6 4 0 0 7 3 5]])
3  9  4  8  5  2  6  7  1
2  6  8  3  7  1  4  5  9
5  7  1  6  9  4  8  2  3
1  4  5  7  8  3  9  6  2
6  8  2  9  4  5  3  1  7
9  3  7  1  2  6  5  8  4
4  1  3  5  6  7  2  9  8
7  5  9  2  3  8  1  4  6
8  2  6  4  1  9  7  3  5

nil
```

Common Lisp

A simple solver without optimizations (except for pre-computing the possible entries of a cell).

```(defun row-neighbors (row column grid &aux (neighbors '()))
(dotimes (i 9 neighbors)
(let ((x (aref grid row i)))
(unless (or (eq '_ x) (= i column))
(push x neighbors)))))

(defun column-neighbors (row column grid &aux (neighbors '()))
(dotimes (i 9 neighbors)
(let ((x (aref grid i column)))
(unless (or (eq x '_) (= i row))
(push x neighbors)))))

(defun square-neighbors (row column grid &aux (neighbors '()))
(let* ((rmin (* 3 (floor row 3)))    (rmax (+ rmin 3))
(cmin (* 3 (floor column 3))) (cmax (+ cmin 3)))
(do ((r rmin (1+ r))) ((= r rmax) neighbors)
(do ((c cmin (1+ c))) ((= c cmax))
(let ((x (aref grid r c)))
(unless (or (eq x '_) (= r row) (= c column))
(push x neighbors)))))))

(defun choices (row column grid)
(nset-difference
(list 1 2 3 4 5 6 7 8 9)
(nconc (row-neighbors row column grid)
(column-neighbors row column grid)
(square-neighbors row column grid))))

(defun solve (grid &optional (row 0) (column 0))
(cond
((= row 9)
grid)
((= column 9)
(solve grid (1+ row) 0))
((not (eq '_ (aref grid row column)))
(solve grid row (1+ column)))
(t (dolist (choice (choices row column grid) (setf (aref grid row column) '_))
(setf (aref grid row column) choice)
(when (eq grid (solve grid row (1+ column)))
(return grid))))))
```

Example:

```> (defparameter *puzzle*
#2A((3 9 4    _ _ 2    6 7 _)
(_ _ _    3 _ _    4 _ _)
(5 _ _    6 9 _    _ 2 _)

(_ 4 5    _ _ _    9 _ _)
(6 _ _    _ _ _    _ _ 7)
(_ _ 7    _ _ _    5 8 _)

(_ 1 _    _ 6 7    _ _ 8)
(_ _ 9    _ _ 8    _ _ _)
(_ 2 6    4 _ _    7 3 5)))
*PUZZLE*

> (pprint (solve *puzzle*))

#2A((3 9 4 8 5 2 6 7 1)
(2 6 8 3 7 1 4 5 9)
(5 7 1 6 9 4 8 2 3)
(1 4 5 7 8 3 9 6 2)
(6 8 2 9 4 5 3 1 7)
(9 3 7 1 2 6 5 8 4)
(4 1 3 5 6 7 2 9 8)
(7 5 9 2 3 8 1 4 6)
(8 2 6 4 1 9 7 3 5))```

Crystal

Based on the Java implementation presented in the video "Create a Sudoku Solver In Java...".

```GRID_SIZE = 9

def isNumberInRow(board, number, row)
board[row].includes?(number)
end
def isNumberInColumn(board, number, column)
board.any?{|row| row[column] == number }
end
def isNumberInBox(board, number, row, column)
localBoxRow = row - row % 3
localBoxColumn = column - column % 3
(localBoxRow...(localBoxRow+3)).each do |i|
(localBoxColumn...(localBoxColumn+3)).each do |j|
return true if board[i][j] == number
end
end
false
end

def isValidPlacement(board, number, row, column)
return !isNumberInRow(board, number, row) &&
!isNumberInColumn(board, number, column) &&
!isNumberInBox(board, number, row, column)
end

def solveBoard(board)
board.each_with_index do |row, i|
row.each_with_index do |cell, j|
if(cell == 0)
(1..GRID_SIZE).each do |n|
if(isValidPlacement(board,n,i,j))
board[i][j]=n
if(solveBoard(board))
return true
else
board[i][j]=0
end
end
end
return false
end
end
end
return true
end

def printBoard(board)
board.each_with_index do |row, i|
row.each_with_index do |cell, j|
print cell
print '|' if j == 2 || j == 5
print '\n' if j == 8
end
print "-"*11 + '\n' if i == 2 || i == 5
end
print '\n'
end

board = [
[7, 0, 2, 0, 5, 0, 6, 0, 0],
[0, 0, 0, 0, 0, 3, 0, 0, 0],
[1, 0, 0, 0, 0, 9, 5, 0, 0],
[8, 0, 0, 0, 0, 0, 0, 9, 0],
[0, 4, 3, 0, 0, 0, 7, 5, 0],
[0, 9, 0, 0, 0, 0, 0, 0, 8],
[0, 0, 9, 7, 0, 0, 0, 0, 5],
[0, 0, 0, 2, 0, 0, 0, 0, 0],
[0, 0, 7, 0, 4, 0, 2, 0, 3]]

printBoard(board)
if(solveBoard(board))
printBoard(board)
end
```
Output:
```702|050|600
000|003|000
100|009|500
-----------
800|000|090
043|000|750
090|000|008
-----------
009|700|005
000|200|000
007|040|203

732|458|619
956|173|824
184|629|537
-----------
871|564|392
643|892|751
295|317|468
-----------
329|786|145
418|235|976
567|941|283
```

Curry

Copied from Curry: Example Programs.

```-----------------------------------------------------------------------------
--- Solving Su Doku puzzles in Curry with FD constraints
---
--- @author Michael Hanus
--- @version December 2005
-----------------------------------------------------------------------------

import CLPFD
import List

-- Solving a Su Doku puzzle represented as a matrix of numbers (possibly free
-- variables):
sudoku :: [[Int]] -> Success
sudoku m =
domain (concat m) 1 9 &                         -- define domain of all digits
foldr1 (&) (map allDifferent m)  &             -- all rows contain different digits
foldr1 (&) (map allDifferent (transpose m))  & -- all columns have different digits
foldr1 (&) (map allDifferent (squaresOfNine m)) & -- all 3x3 squares are different
labeling [FirstFailConstrained] (concat m)

-- translate a matrix into a list of small 3x3 squares
squaresOfNine :: [[a]] -> [[a]]
squaresOfNine [] = []
squaresOfNine (l1:l2:l3:ls) = group3Rows [l1,l2,l3] ++ squaresOfNine ls

group3Rows l123 = if null (head l123) then [] else
concatMap (take 3) l123 : group3Rows (map (drop 3) l123)

-- read a Su Doku specification written as a list of strings containing digits
-- and spaces
readSudoku s = map (map transDigit) s
where
transDigit c = if c==' ' then x else ord c - ord '0'
where x free

-- show a solved Su Doku matrix
showSudoku :: [[Int]] -> String
showSudoku = unlines . map (concatMap (\i->[chr (i + ord '0'),' ']))

-- the main function, e.g., evaluate (main s1):
main s | sudoku m = putStrLn (showSudoku m)

s1 = ["9  2  5  ",
" 4  6  3 ",
"  3     6",
"   9  2  ",
"    5  8 ",
"  7  4  3",
"7     1  ",
" 5  2  4 ",
"  1  6  9"]

s2 = ["819  5   ",
"  2   75 ",
" 371 4 6 ",
"4  59 1  ",
"7  3 8  2",
"  3 62  7",
" 5 7 921 ",
" 64   9  ",
"   2  438"]```

Alternative version

Works with: PAKCS

Minimal w/o read or show utilities.

```import CLPFD
import Constraint (allC)
import List (transpose)

sudoku :: [[Int]] -> Success
sudoku rows =
domain (concat rows) 1 9
& different rows
& different (transpose rows)
& different blocks
& labeling [] (concat rows)
where
different = allC allDifferent

blocks = [concat ys | xs <- each3 rows
, ys <- transpose \$ map each3 xs
]
each3 xs = case xs of
(x:y:z:rest) -> [x,y,z] : each3 rest
rest         -> [rest]

test = [ [_,_,3,_,_,_,_,_,_]
, [4,_,_,_,8,_,_,3,6]
, [_,_,8,_,_,_,1,_,_]
, [_,4,_,_,6,_,_,7,3]
, [_,_,_,9,_,_,_,_,_]
, [_,_,_,_,_,2,_,_,5]
, [_,_,4,_,7,_,_,6,8]
, [6,_,_,_,_,_,_,_,_]
, [7,_,_,6,_,_,5,_,_]
]
main | sudoku xs = xs where xs = test```
Output:
```Execution time: 0 msec. / elapsed: 10 msec.
[[1,2,3,4,5,6,7,8,9],[4,5,7,1,8,9,2,3,6],[9,6,8,3,2,7,1,5,4],[2,4,9,5,6,1,8,7,3],[5,7,6,9,3,8,4,1,2],[8,3,1,7,4,2,6,9,5],[3,1,4,2,7,5,9,6,8],[6,9,5,8,1,4,3,2,7],[7,8,2,6,9,3,5,4,1]]```

D

Translation of: C++

A little over-engineered solution, that shows some strong static typing useful in larger programs.

```import std.stdio, std.range, std.string, std.algorithm, std.array,
std.ascii, std.typecons;

struct Digit {
immutable char d;

this(in char d_) pure nothrow @safe @nogc
in { assert(d_ >= '0' && d_ <= '9'); }
body { this.d = d_; }

this(in int d_) pure nothrow @safe @nogc
in { assert(d_ >= '0' && d_ <= '9'); }
body { this.d = cast(char)d_; } // Required cast.

alias d this;
}

enum size_t sudokuUnitSide = 3;
enum size_t sudokuSide = sudokuUnitSide ^^ 2; // Sudoku grid side.
alias SudokuTable = Digit[sudokuSide ^^ 2];

Nullable!SudokuTable sudokuSolver(in ref SudokuTable problem)
pure nothrow {
alias Tgrid = uint;
Tgrid[SudokuTable.length] grid = void;
problem[].map!(c => c - '0').copy(grid[]);

// DMD doesn't inline this function. Performance loss.
Tgrid access(in size_t x, in size_t y) nothrow @safe @nogc {
return grid[y * sudokuSide + x];
}

// DMD doesn't inline this function. If you want to retain
// the same performance as the C++ entry and you use the DMD
// compiler then this function must be manually inlined.
bool checkValidity(in Tgrid val, in size_t x, in size_t y)
pure nothrow @safe @nogc {
/*static*/ foreach (immutable i; staticIota!(0, sudokuSide))
if (access(i, y) == val || access(x, i) == val)
return false;

immutable startX = (x / sudokuUnitSide) * sudokuUnitSide;
immutable startY = (y / sudokuUnitSide) * sudokuUnitSide;

/*static*/ foreach (immutable i; staticIota!(0, sudokuUnitSide))
/*static*/ foreach (immutable j; staticIota!(0, sudokuUnitSide))
if (access(startX + j, startY + i) == val)
return false;

return true;
}

bool canPlaceNumbers(in size_t pos=0) nothrow @safe @nogc {
if (pos == SudokuTable.length)
return true;
if (grid[pos] > 0)
return canPlaceNumbers(pos + 1);

foreach (immutable n; 1 .. sudokuSide + 1)
if (checkValidity(n, pos % sudokuSide, pos / sudokuSide)) {
grid[pos] = n;
if (canPlaceNumbers(pos + 1))
return true;
grid[pos] = 0;
}

return false;
}

if (canPlaceNumbers) {
//return typeof(return)(grid[]
//                      .map!(c => Digit(c + '0'))
//                      .array);
immutable SudokuTable result = grid[]
.map!(c => Digit(c + '0'))
.array;
return typeof(return)(result);
} else
return typeof(return)();
}

string representSudoku(in ref SudokuTable sudo)
pure nothrow @safe out(result) {
assert(result.countchars("1-9") == sudo[].count!q{a != '0'});
assert(result.countchars(".") == sudo[].count!q{a == '0'});
} body {
static assert(sudo.length == 81,
"representSudoku works only with a 9x9 Sudoku.");
string result;

foreach (immutable i; 0 .. sudokuSide) {
foreach (immutable j; 0 .. sudokuSide) {
result ~= sudo[i * sudokuSide + j];
result ~= ' ';
if (j == 2 || j == 5)
result ~= "| ";
}
result ~= "\n";
if (i == 2 || i == 5)
result ~= "------+-------+------\n";
}

return result.replace("0", ".");
}

void main() {
enum ValidateCells(string s) = s.map!Digit.array;

immutable SudokuTable problem = ValidateCells!("
850002400
720000009
004000000
000107002
305000900
040000000
000080070
017000000
000036040".removechars(whitespace));
problem.representSudoku.writeln;

immutable solution = problem.sudokuSolver;
if (solution.isNull)
writeln("Unsolvable!");
else
solution.get.representSudoku.writeln;
}
```
Output:
```8 5 . | . . 2 | 4 . .
7 2 . | . . . | . . 9
. . 4 | . . . | . . .
------+-------+------
. . . | 1 . 7 | . . 2
3 . 5 | . . . | 9 . .
. 4 . | . . . | . . .
------+-------+------
. . . | . 8 . | . 7 .
. 1 7 | . . . | . . .
. . . | . 3 6 | . 4 .

8 5 9 | 6 1 2 | 4 3 7
7 2 3 | 8 5 4 | 1 6 9
1 6 4 | 3 7 9 | 5 2 8
------+-------+------
9 8 6 | 1 4 7 | 3 5 2
3 7 5 | 2 6 8 | 9 1 4
2 4 1 | 5 9 3 | 7 8 6
------+-------+------
4 3 2 | 9 8 1 | 6 7 5
6 1 7 | 4 2 5 | 8 9 3
5 9 8 | 7 3 6 | 2 4 1 ```

Short Version

```import std.stdio, std.algorithm, std.range;

const(int)[] solve(immutable int[] s) pure nothrow @safe {
immutable i = s.countUntil(0);
if (i == -1)
return s;

enum B = (int i, int j) => i / 27 ^ j / 27 | (i%9 / 3 ^ j%9 / 3);
immutable c = iota(81)
.filter!(j => !((i - j) % 9 * (i/9 ^ j/9) * B(i, j)))
.map!(j => s[j]).array;

foreach (immutable v; 1 .. 10)
if (!c.canFind(v)) {
const r = solve(s[0 .. i] ~ v ~ s[i + 1 .. \$]);
if (!r.empty)
return r;
}
return null;
}

void main() {
immutable problem = [
8, 5, 0, 0, 0, 2, 4, 0, 0,
7, 2, 0, 0, 0, 0, 0, 0, 9,
0, 0, 4, 0, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 7, 0, 0, 2,
3, 0, 5, 0, 0, 0, 9, 0, 0,
0, 4, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 8, 0, 0, 7, 0,
0, 1, 7, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 3, 6, 0, 4, 0];
writefln("%(%s\n%)", problem.solve.chunks(9));
}
```
Output:
```[8, 5, 9, 6, 1, 2, 4, 3, 7]
[7, 2, 3, 8, 5, 4, 1, 6, 9]
[1, 6, 4, 3, 7, 9, 5, 2, 8]
[9, 8, 6, 1, 4, 7, 3, 5, 2]
[3, 7, 5, 2, 6, 8, 9, 1, 4]
[2, 4, 1, 5, 9, 3, 7, 8, 6]
[4, 3, 2, 9, 8, 1, 6, 7, 5]
[6, 1, 7, 4, 2, 5, 8, 9, 3]
[5, 9, 8, 7, 3, 6, 2, 4, 1]```

No-Heap Version

This version is similar to the precedent one, but it shows idioms to avoid memory allocations on the heap. This is enforced by the use of the @nogc attribute.

```import std.stdio, std.algorithm, std.range, std.typecons;

Nullable!(const ubyte[81]) solve(in ubyte[81] s) pure nothrow @safe @nogc {
immutable i = s[].countUntil(0);
if (i == -1)
return typeof(return)(s);

static immutable B = (in int i, in int j) pure nothrow @safe @nogc =>
i / 27 ^ j / 27 | (i % 9 / 3 ^ j % 9 / 3);

ubyte[81] c = void;
size_t len = 0;
foreach (immutable int j; 0 .. c.length)
if (!((i - j) % 9 * (i/9 ^ j/9) * B(i, j)))
c[len++] = s[j];

foreach (immutable ubyte v; 1 .. 10)
if (!c[0 .. len].canFind(v)) {
ubyte[81] s2 = void;
s2[0 .. i] = s[0 .. i];
s2[i] = v;
s2[i + 1 .. \$] = s[i + 1 .. \$];
const r = solve(s2);
if (!r.isNull)
return typeof(return)(r);
}
return typeof(return)();
}

void main() {
immutable ubyte[81] problem = [
8, 5, 0, 0, 0, 2, 4, 0, 0,
7, 2, 0, 0, 0, 0, 0, 0, 9,
0, 0, 4, 0, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 7, 0, 0, 2,
3, 0, 5, 0, 0, 0, 9, 0, 0,
0, 4, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 8, 0, 0, 7, 0,
0, 1, 7, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 3, 6, 0, 4, 0];
writefln("%(%s\n%)", problem.solve.get[].chunks(9));
}
```

Same output.

Delphi

Example taken from C++

```type
TIntArray = array of Integer;

{ TSudokuSolver }

TSudokuSolver = class
private
FGrid: TIntArray;

function CheckValidity(val: Integer; x: Integer; y: Integer): Boolean;
function ToString: string; reintroduce;
function PlaceNumber(pos: Integer): Boolean;
public
constructor Create(s: string);

procedure Solve;
end;

implementation

uses
Dialogs;

{ TSudokuSolver }

function TSudokuSolver.CheckValidity(val: Integer; x: Integer; y: Integer
): Boolean;
var
i: Integer;
j: Integer;
StartX: Integer;
StartY: Integer;
begin
for i := 0 to 8 do
begin
if (FGrid[y * 9 + i] = val) or
(FGrid[i * 9 + x] = val) then
begin
Result := False;
Exit;
end;
end;
StartX := (x div 3) * 3;
StartY := (y div 3) * 3;
for i := StartY to Pred(StartY + 3) do
begin
for j := StartX to Pred(StartX + 3) do
begin
if FGrid[i * 9 + j] = val then
begin
Result := False;
Exit;
end;
end;
end;
Result := True;
end;

function TSudokuSolver.ToString: string;
var
sb: string;
i: Integer;
j: Integer;
c: char;
begin
sb := '';
for i := 0 to 8 do
begin
for j := 0 to 8 do
begin
c := (IntToStr(FGrid[i * 9 + j]) + '0')[1];
sb := sb + c + ' ';
if (j = 2) or (j = 5) then sb := sb + '| ';
end;
sb := sb + #13#10;
if (i = 2) or (i = 5) then
sb := sb + '-----+-----+-----' + #13#10;
end;
Result := sb;
end;

function TSudokuSolver.PlaceNumber(pos: Integer): Boolean;
var
n: Integer;
begin
Result := False;
if Pos = 81 then
begin
Result := True;
Exit;
end;
if FGrid[pos] > 0 then
begin
Result := PlaceNumber(Succ(pos));
Exit;
end;
for n := 1 to 9 do
begin
if CheckValidity(n, pos mod 9, pos div 9) then
begin
FGrid[pos] := n;
Result := PlaceNumber(Succ(pos));
if not Result then
FGrid[pos] := 0;
end;
end;
end;

constructor TSudokuSolver.Create(s: string);
var
lcv: Cardinal;
begin
SetLength(FGrid, 81);
for lcv := 0 to Pred(Length(s)) do
FGrid[lcv] := StrToInt(s[Succ(lcv)]);
end;

procedure TSudokuSolver.Solve;
begin
if not PlaceNumber(0) then
ShowMessage('Unsolvable')
else
ShowMessage('Solved!');
end;
end;
```

Usage:

```var
SudokuSolver: TSudokuSolver;
begin
SudokuSolver := TSudokuSolver.Create('850002400' +
'720000009' +
'004000000' +
'000107002' +
'305000900' +
'040000000' +
'000080070' +
'017000000' +
'000036040');
try
SudokuSolver.Solve;
finally
FreeAndNil(SudokuSolver);
end;
end;
```

EasyLang

```len row[] 90
len col[] 90
len box[] 90
len grid[] 82
#
func init . .
for pos = 1 to 81
if pos mod 9 = 1
s\$ = input
if s\$ = ""
s\$ = input
.
len inp[] 0
for i = 1 to len s\$
if substr s\$ i 1 <> " "
inp[] &= number substr s\$ i 1
.
.
.
dig = number inp[(pos - 1) mod 9 + 1]
if dig > 0
grid[pos] = dig
r = (pos - 1) div 9
c = (pos - 1) mod 9
b = r div 3 * 3 + c div 3
row[r * 10 + dig] = 1
col[c * 10 + dig] = 1
box[b * 10 + dig] = 1
.
.
.
call init
#
func display . .
for i = 1 to 81
write grid[i] & " "
if i mod 3 = 0
write " "
.
if i mod 9 = 0
print ""
.
if i mod 27 = 0
print ""
.
.
.
#
func solve pos . .
while grid[pos] <> 0
pos += 1
.
if pos > 81
# solved
call display
break 1
.
r = (pos - 1) div 9
c = (pos - 1) mod 9
b = r div 3 * 3 + c div 3
r *= 10
c *= 10
b *= 10
for d = 1 to 9
if row[r + d] = 0 and col[c + d] = 0 and box[b + d] = 0
grid[pos] = d
row[r + d] = 1
col[c + d] = 1
box[b + d] = 1
call solve pos + 1
row[r + d] = 0
col[c + d] = 0
box[b + d] = 0
.
.
grid[pos] = 0
.
call solve 1
#
input_data
5 3 0  0 2 4  7 0 0
0 0 2  0 0 0  8 0 0
1 0 0  7 0 3  9 0 2

0 0 8  0 7 2  0 4 9
0 2 0  9 8 0  0 7 0
7 9 0  0 0 0  0 8 0

0 0 0  0 3 0  5 0 6
9 6 0  0 1 0  3 0 0
0 5 0  6 9 0  0 1 0
```

Elixir

Translation of: Erlang
```defmodule Sudoku do
def display( grid ), do: ( for y <- 1..9, do: display_row(y, grid) )

def start( knowns ), do: Enum.into( knowns, Map.new )

def solve( grid ) do
sure = solve_all_sure( grid )
solve_unsure( potentials(sure), sure )
end

IO.puts "start"
start = start( knowns )
display( start )
IO.puts "solved"
solved = solve( start )
display( solved )
IO.puts ""
end

defp bt( grid ), do: bt_reject( is_not_allowed(grid), grid )

defp bt_accept( true, board ), do: throw( {:ok, board} )
defp bt_accept( false, grid ), do: bt_loop( potentials_one_position(grid), grid )

defp bt_loop( {position, values}, grid ), do: ( for x <- values, do: bt( Map.put(grid, position, x) ) )

defp bt_reject( true, _grid ), do: :backtrack
defp bt_reject( false, grid ), do: bt_accept( is_all_correct(grid), grid )

defp display_row( row, grid ) do
for x <- [1, 4, 7], do: display_row_group( x, row, grid )
display_row_nl( row )
end

defp display_row_group( start, row, grid ) do
Enum.each(start..start+2, &IO.write " #{Map.get( grid, {&1, row}, ".")}")
IO.write " "
end

defp display_row_nl( n ) when n in [3,6,9], do: IO.puts "\n"
defp display_row_nl( _n ), do: IO.puts ""

defp is_all_correct( grid ), do: map_size( grid ) == 81

defp is_not_allowed( grid ) do
is_not_allowed_rows( grid ) or is_not_allowed_columns( grid ) or is_not_allowed_groups( grid )
end

defp is_not_allowed_columns( grid ), do: values_all_columns(grid) |> Enum.any?(&is_not_allowed_values/1)

defp is_not_allowed_groups( grid ),  do: values_all_groups(grid)  |> Enum.any?(&is_not_allowed_values/1)

defp is_not_allowed_rows( grid ),    do: values_all_rows(grid)    |> Enum.any?(&is_not_allowed_values/1)

defp is_not_allowed_values( values ), do: length( values ) != length( Enum.uniq(values) )

defp group_positions( {x, y} ) do
for colum <- group_positions_close(x), row <- group_positions_close(y), do: {colum, row}
end

defp group_positions_close( n ) when n < 4, do: [1,2,3]
defp group_positions_close( n ) when n < 7, do: [4,5,6]
defp group_positions_close( _n )          , do: [7,8,9]

defp positions_not_in_grid( grid ) do
keys = Map.keys( grid )
for x <- 1..9, y <- 1..9, not {x, y} in keys, do: {x, y}
end

defp potentials_one_position( grid ) do
Enum.min_by( potentials( grid ), fn {_position, values} -> length(values) end )
end

defp potentials( grid ), do: List.flatten( for x <- positions_not_in_grid(grid), do: potentials(x, grid) )

defp potentials( position, grid ) do
useds = potentials_used_values( position, grid )
{position, Enum.to_list(1..9) -- useds }
end

defp potentials_used_values( {x, y}, grid ) do
row_values    = (for row <- 1..9, row != x, do: {row, y})          |> potentials_values( grid )
column_values = (for column <- 1..9, column != y, do: {x, column}) |> potentials_values( grid )
group_values  = group_positions({x, y}) -- [ {x, y} ]              |> potentials_values( grid )
row_values ++ column_values ++ group_values
end

defp potentials_values( keys, grid ) do
for x <- keys, val = grid[x], do: val
end

defp values_all_columns( grid ) do
for x <- 1..9, do:
( for y <- 1..9, do: {x, y} ) |> potentials_values( grid )
end

defp values_all_groups( grid ) do
[[g1,g2,g3], [g4,g5,g6], [g7,g8,g9]] = for x <- [1,4,7], do: values_all_groups(x, grid)
[g1,g2,g3,g4,g5,g6,g7,g8,g9]
end

defp values_all_groups( x, grid ) do
for x_offset <- x..x+2, do: values_all_groups(x, x_offset, grid)
end

defp values_all_groups( _x, x_offset, grid ) do
( for y_offset <- group_positions_close(x_offset), do: {x_offset, y_offset} )
|> potentials_values( grid )
end

defp values_all_rows( grid ) do
for y <- 1..9, do:
( for x <- 1..9, do: {x, y} ) |> potentials_values( grid )
end

defp solve_all_sure( grid ), do: solve_all_sure( solve_all_sure_values(grid), grid )

defp solve_all_sure( [], grid ), do: grid
defp solve_all_sure( sures, grid ) do
solve_all_sure( Enum.reduce(sures, grid, &solve_all_sure_store/2) )
end

defp solve_all_sure_values( grid ), do: (for{position, [value]} <- potentials(grid), do: {position, value} )

defp solve_all_sure_store( {position, value}, acc ), do: Map.put( acc, position, value )

defp solve_unsure( [], grid ), do: grid
defp solve_unsure( _potentials, grid ) do
try do
bt( grid )
catch
{:ok, board} -> board
end
end
end

simple = [{{1, 1}, 3}, {{2, 1}, 9}, {{3, 1},4}, {{6, 1}, 2}, {{7, 1}, 6}, {{8, 1}, 7},
{{4, 2}, 3}, {{7, 2}, 4},
{{1, 3}, 5}, {{4, 3}, 6}, {{5, 3}, 9}, {{8, 3}, 2},
{{2, 4}, 4}, {{3, 4}, 5}, {{7, 4}, 9},
{{1, 5}, 6}, {{9, 5}, 7},
{{3, 6}, 7}, {{7, 6}, 5}, {{8, 6}, 8},
{{2, 7}, 1}, {{5, 7}, 6}, {{6, 7}, 7}, {{9, 7}, 8},
{{3, 8}, 9}, {{6, 8}, 8},
{{2, 9}, 2}, {{3, 9}, 6}, {{4, 9}, 4}, {{7, 9}, 7}, {{8, 9}, 3}, {{9, 9}, 5}]

difficult = [{{6, 2}, 3}, {{8, 2}, 8}, {{9, 2}, 5},
{{3, 3}, 1}, {{5, 3}, 2},
{{4, 4}, 5}, {{6, 4}, 7},
{{3, 5}, 4}, {{7, 5}, 1},
{{2, 6}, 9},
{{1, 7}, 5}, {{8, 7}, 7}, {{9, 7}, 3},
{{3, 8}, 2}, {{5, 8}, 1},
{{5, 9}, 4}, {{9, 9}, 9}]
```
Output:
```start
3 9 4  . . 2  6 7 .
. . .  3 . .  4 . .
5 . .  6 9 .  . 2 .

. 4 5  . . .  9 . .
6 . .  . . .  . . 7
. . 7  . . .  5 8 .

. 1 .  . 6 7  . . 8
. . 9  . . 8  . . .
. 2 6  4 . .  7 3 5

solved
3 9 4  8 5 2  6 7 1
2 6 8  3 7 1  4 5 9
5 7 1  6 9 4  8 2 3

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

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

start
. . .  . . .  . . .
. . .  . . 3  . 8 5
. . 1  . 2 .  . . .

. . .  5 . 7  . . .
. . 4  . . .  1 . .
. 9 .  . . .  . . .

5 . .  . . .  . 7 3
. . 2  . 1 .  . . .
. . .  . 4 .  . . 9

solved
9 8 7  6 5 4  3 2 1
2 4 6  1 7 3  9 8 5
3 5 1  9 2 8  7 4 6

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

5 1 9  2 8 6  4 7 3
4 7 2  3 1 9  5 6 8
8 6 3  7 4 5  2 1 9
```

Erlang

I first try to solve the Sudoku grid without guessing. For the guessing part I eschew spawning a process for each guess, instead opting for backtracking. It is fun trying new things.

```-module( sudoku ).

-export( [display/1, start/1, solve/1, task/0] ).

display( Grid ) -> [display_row(Y, Grid) || Y <- lists:seq(1, 9)].
%% A known value is {{Column, Row}, Value}
%% Top left corner is {1, 1}, Bottom right corner is {9,9}
start( Knowns ) -> dict:from_list( Knowns ).

solve( Grid ) ->
Sure = solve_all_sure( Grid ),
solve_unsure( potentials(Sure), Sure ).

Simple = [{{1, 1}, 3}, {{2, 1}, 9}, {{3, 1},4}, {{6, 1}, 2}, {{7, 1}, 6}, {{8, 1}, 7},
{{4, 2}, 3}, {{7, 2}, 4},
{{1, 3}, 5}, {{4, 3}, 6}, {{5, 3}, 9}, {{8, 3}, 2},
{{2, 4}, 4}, {{3, 4}, 5}, {{7, 4}, 9},
{{1, 5}, 6}, {{9, 5}, 7},
{{3, 6}, 7}, {{7, 6}, 5}, {{8, 6}, 8},
{{2, 7}, 1}, {{5, 7}, 6}, {{6, 7}, 7}, {{9, 7}, 8},
{{3, 8}, 9}, {{6, 8}, 8},
{{2, 9}, 2}, {{3, 9}, 6}, {{4, 9}, 4}, {{7, 9}, 7}, {{8, 9}, 3}, {{9, 9}, 5}],
Difficult = [{{6, 2}, 3}, {{8, 2}, 8}, {{9, 2}, 5},
{{3, 3}, 1}, {{5, 3}, 2},
{{4, 4}, 5}, {{6, 4}, 7},
{{3, 5}, 4}, {{7, 5}, 1},
{{2, 6}, 9},
{{1, 7}, 5}, {{8, 7}, 7}, {{9, 7}, 3},
{{3, 8}, 2}, {{5, 8}, 1},
{{5, 9}, 4}, {{9, 9}, 9}],

bt( Grid ) -> bt_reject( is_not_allowed(Grid), Grid ).

bt_accept( true, Board ) -> erlang:throw( {ok, Board} );
bt_accept( false, Grid ) -> bt_loop( potentials_one_position(Grid), Grid ).

bt_loop( {Position, Values}, Grid ) -> [bt( dict:store(Position, X, Grid) ) || X <- Values].

bt_reject( true, _Grid ) -> backtrack;
bt_reject( false, Grid ) -> bt_accept( is_all_correct(Grid), Grid ).

display_row( Row, Grid ) ->
[display_row_group( X, Row, Grid ) || X <- [1, 4, 7]],
display_row_nl( Row ).

display_row_group( Start, Row, Grid ) ->
[io:fwrite(" ~c", [display_value(X, Row, Grid)]) || X <- [Start, Start+1, Start+2]],
io:fwrite( " " ).

display_row_nl( N ) when N =:= 3; N =:= 6; N =:= 9 -> io:nl(), io:nl();
display_row_nl( _N ) -> io:nl().

display_value( X, Y, Grid ) -> display_value( dict:find({X, Y}, Grid) ).

display_value( error ) -> \$.;
display_value( {ok, Value} ) -> Value + \$0.

is_all_correct( Grid ) -> dict:size( Grid ) =:= 81.

is_not_allowed( Grid ) ->
is_not_allowed_rows( Grid )
orelse is_not_allowed_columns( Grid )
orelse is_not_allowed_groups( Grid ).

is_not_allowed_columns( Grid ) -> lists:any( fun is_not_allowed_values/1, values_all_columns(Grid) ).

is_not_allowed_groups( Grid ) -> lists:any( fun is_not_allowed_values/1, values_all_groups(Grid) ).

is_not_allowed_rows( Grid ) -> lists:any( fun is_not_allowed_values/1, values_all_rows(Grid) ).

is_not_allowed_values( Values ) -> erlang:length( Values ) =/= erlang:length( lists:usort(Values) ).

group_positions( {X, Y} ) -> [{Colum, Row} || Colum <- group_positions_close(X), Row <- group_positions_close(Y)].

group_positions_close( N ) when N < 4 -> [1,2,3];
group_positions_close( N ) when N < 7 -> [4,5,6];
group_positions_close( _N ) -> [7,8,9].

positions_not_in_grid( Grid ) ->
Keys = dict:fetch_keys( Grid ),
[{X, Y} || X <- lists:seq(1, 9), Y <- lists:seq(1, 9), not lists:member({X, Y}, Keys)].

potentials_one_position( Grid ) ->
[{_Shortest, Position, Values} | _T] = lists:sort( [{erlang:length(Values), Position, Values} || {Position, Values} <- potentials( Grid )] ),
{Position, Values}.

potentials( Grid ) -> lists:flatten( [potentials(X, Grid)  || X <- positions_not_in_grid(Grid)] ).

potentials( Position, Grid ) ->
Useds = potentials_used_values( Position, Grid ),
{Position, [Value || Value <- lists:seq(1, 9) -- Useds]}.

potentials_used_values( {X, Y}, Grid ) ->
Row_positions = [{Row, Y} || Row <- lists:seq(1, 9), Row =/= X],
Row_values = potentials_values( Row_positions, Grid ),
Column_positions = [{X, Column} || Column <- lists:seq(1, 9), Column =/= Y],
Column_values = potentials_values( Column_positions, Grid ),
Group_positions = lists:delete( {X, Y}, group_positions({X, Y}) ),
Group_values = potentials_values( Group_positions, Grid ),
Row_values ++ Column_values ++ Group_values.

potentials_values( Keys, Grid ) ->
Row_values_unfiltered = [dict:find(X, Grid) || X <- Keys],
[Value || {ok, Value} <- Row_values_unfiltered].

values_all_columns( Grid ) -> [values_all_columns(X, Grid) || X <- lists:seq(1, 9)].

values_all_columns( X, Grid ) ->
Positions = [{X, Y} || Y <- lists:seq(1, 9)],
potentials_values( Positions, Grid ).

values_all_groups( Grid ) ->
[G123, G456, G789] = [values_all_groups(X, Grid) || X <- [1, 4, 7]],
[G1,G2,G3] = G123,
[G4,G5,G6] = G456,
[G7,G8,G9] = G789,
[G1,G2,G3,G4,G5,G6,G7,G8,G9].

values_all_groups( X, Grid ) ->[values_all_groups(X, X_offset, Grid) || X_offset <- [X, X+1, X+2]].

values_all_groups( _X, X_offset, Grid ) ->
Positions = [{X_offset, Y_offset} || Y_offset <- group_positions_close(X_offset)],
potentials_values( Positions, Grid ).

values_all_rows( Grid ) ->[values_all_rows(Y, Grid) || Y <- lists:seq(1, 9)].

values_all_rows( Y, Grid ) ->
Positions = [{X, Y} || X <- lists:seq(1, 9)],
potentials_values( Positions, Grid ).

solve_all_sure( Grid ) -> solve_all_sure( solve_all_sure_values(Grid), Grid ).

solve_all_sure( [], Grid ) -> Grid;
solve_all_sure( Sures, Grid ) -> solve_all_sure( lists:foldl(fun solve_all_sure_store/2, Grid, Sures) ).

solve_all_sure_values( Grid ) -> [{Position, Value} || {Position, [Value]} <- potentials(Grid)].

solve_all_sure_store( {Position, Value}, Acc ) -> dict:store( Position, Value, Acc ).

solve_unsure( [], Grid ) -> Grid;
solve_unsure( _Potentials, Grid ) ->
try
bt( Grid )

catch
_:{ok, Board} -> Board

end.

io:fwrite( "Start~n" ),
Start = start( Knowns ),
display( Start ),
io:fwrite( "Solved~n" ),
Solved = solve( Start ),
display( Solved ),
io:nl().
```
Output:
```5> sudoku:task().
Start
3 9 4  . . 2  6 7 .
. . .  3 . .  4 . .
5 . .  6 9 .  . 2 .

. 4 5  . . .  9 . .
6 . .  . . .  . . 7
. . 7  . . .  5 8 .

. 1 .  . 6 7  . . 8
. . 9  . . 8  . . .
. 2 6  4 . .  7 3 5

Solved
3 9 4  8 5 2  6 7 1
2 6 8  3 7 1  4 5 9
5 7 1  6 9 4  8 2 3

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

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

Start
. . .  . . .  . . .
. . .  . . 3  . 8 5
. . 1  . 2 .  . . .

. . .  5 . 7  . . .
. . 4  . . .  1 . .
. 9 .  . . .  . . .

5 . .  . . .  . 7 3
. . 2  . 1 .  . . .
. . .  . 4 .  . . 9

Solved
9 8 7  6 5 4  3 2 1
2 4 6  1 7 3  9 8 5
3 5 1  9 2 8  7 4 6

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

5 1 9  2 8 6  4 7 3
4 7 2  3 1 9  5 6 8
8 6 3  7 4 5  2 1 9
```

ERRE

Sudoku solver. Program solves Sudoku grid with an iterative method: it's taken from ERRE distribution disk and so comments are in Italian. Grid data are contained in the file SUDOKU.TXT

Example of SUDOKU.TXT

503600009

010002600

900000080

000700005

006804100

200003000

030000008

004300050

800006702

0 is the empty cell.

```!--------------------------------------------------------------------
! risolve Sudoku: in input il file SUDOKU.TXT
! Metodo seguito : cancellazioni successive e quando non possibile
!                  ricerca combinatoria sulle celle con due valori
!                  possibili - max. 30 livelli di ricorsione
!                  Non risolve se,dopo l'analisi per la cancellazione,
!                  restano solo celle a 4 valori
!--------------------------------------------------------------------

PROGRAM SUDOKU

LABEL 76,77,88,91,97,99

DIM TAV\$[9,9]             ! 81 caselle in nove quadranti
! cella non definita --> 0/. nel file SUDOKU.TXT
! diventa 123456789 dopo LEGGI_SCHEMA

!---------------------------------------------------------------------------
! tabelle per gestire la ricerca combinatoria
! (primo indice--> livelli ricorsione)
!---------------------------------------------------------------------------
DIM TAV2\$[30,9,9],INFO[30,4]

!\$INCLUDE="PC.LIB"

PROCEDURE MESSAGGI(MEX%)
CASE MEX% OF
1-> LOCATE(21,1) PRINT("Cancellazione successiva - liv. 1") END ->
2-> LOCATE(21,1) PRINT("Cancellazione successiva - liv. 2") END ->
3-> LOCATE(22,1) PRINT("Ricerca combinatoria - liv.";LIVELLO;"   ") END ->
END CASE
END PROCEDURE

PROCEDURE VISUALIZZA_SCHEMA
LOCATE(1,1)
PRINT("+---+---+---+---+---+---+---+---+----+")
FOR I=1 TO 9 DO
FOR J=1 TO 9 DO
PRINT("|";)
IF LEN(TAV\$[I,J])=1 THEN
PRINT(" ";TAV\$[I,J];" ";)
ELSE
PRINT("   ";)
END IF
END FOR
PRINT("³")
IF I<>9 THEN PRINT("+---+---+---+---+---+---+---+---+----+") END IF
END FOR
PRINT("+---+---+---+---+---+---+---+---+----+")
END PROCEDURE

!------------------------------------------------------------------------
! in input  la cella (riga,colonna)
! in output se ha un valore definito
!------------------------------------------------------------------------
PROCEDURE VALORE_DEFINITO
FLAG%=FALSE
IF LEN(TAV\$[RIGA,COLONNA])=1 THEN FLAG%=TRUE END IF
END PROCEDURE

PROCEDURE SALVA_CONFIG
LIVELLO=LIVELLO+1
FOR R=1 TO 9 DO
FOR S=1 TO 9 DO
TAV2\$[LIVELLO,R,S]=TAV\$[R,S]
END FOR
END FOR
INFO[LIVELLO,0]=1 INFO[LIVELLO,1]=RIGA INFO[LIVELLO,2]=COLONNA
INFO[LIVELLO,3]=SECOND INFO[LIVELLO,4]=THIRD
END PROCEDURE

PROCEDURE RIPRISTINA_CONFIG
91:
LIVELLO=LIVELLO-1
IF INFO[LIVELLO,0]=3 THEN GOTO 91 END IF
FOR R=1 TO 9 DO
FOR S=1 TO 9 DO
TAV\$[R,S]=TAV2\$[LIVELLO,R,S]
END FOR
END FOR
RIGA=INFO[LIVELLO,1] COLONNA=INFO[LIVELLO,2]
SECOND=INFO[LIVELLO,3] THIRD=INFO[LIVELLO,4]
IF INFO[LIVELLO,0]=1 THEN
TAV\$[RIGA,COLONNA]=MID\$(STR\$(SECOND),2)
END IF
IF INFO[LIVELLO,0]=2 THEN
IF THIRD<>0 THEN
TAV\$[RIGA,COLONNA]=MID\$(STR\$(THIRD),2)
ELSE
GOTO 91
END IF
END IF
INFO[LIVELLO,0]=INFO[LIVELLO,0]+1
VISUALIZZA_SCHEMA
END PROCEDURE

PROCEDURE VERIFICA_SE_FINITO
COMPLETO%=TRUE
FOR RIGA=1 TO 9 DO
PRD#=1
FOR COLONNA=1 TO 9 DO
PRD#=PRD#*VAL(TAV\$[RIGA,COLONNA])
END FOR
IF PRD#<>362880 THEN COMPLETO%=FALSE EXIT END IF
END FOR
IF NOT COMPLETO% THEN EXIT PROCEDURE END IF
FOR COLONNA=1 TO 9 DO
PRD#=1
FOR RIGA=1 TO 9 DO
PRD#=PRD#*VAL(TAV\$[RIGA,COLONNA])
END FOR
IF PRD#<>362880 THEN COMPLETO%=FALSE EXIT END IF
END FOR
END PROCEDURE

!-------------------------------------------------------------------
! toglie i valore certi dalle celle sulla
!-------------------------------------------------------------------
PROCEDURE TOGLI_VALORE

!iniziamo a togliere il valore dalla stessa riga ....
FOR J=1 TO 9 DO
CH\$=TAV\$[RIGA,J] CH=VAL(Z\$)
IF LEN(CH\$)<>1 THEN
CHANGE(CH\$,CH,"-"->CH\$)
TAV\$[RIGA,J]=CH\$
END IF
END FOR
!... iniziamo a togliere il valore dalla stessa colonna ...
FOR I=1 TO 9 DO
CH\$=TAV\$[I,COLONNA] CH=VAL(Z\$)
IF LEN(CH\$)<>1 THEN
CHANGE(CH\$,CH,"-"->CH\$)
TAV\$[I,COLONNA]=CH\$
END IF
END FOR
!... iniziamo a togliere il valore dallo stesso quadrante
R=INT(RIGA/3.1)*3+1
S=INT(COLONNA/3.1)*3+1
FOR I=R TO R+2 DO
FOR J=S TO S+2 DO
CH\$=TAV\$[I,J] CH=VAL(Z\$)
IF LEN(CH\$)<>1 THEN
CHANGE(CH\$,CH,"-"->CH\$)
TAV\$[I,J]=CH\$
END IF
END FOR
END FOR
MESSAGGI(1)
END PROCEDURE

PROCEDURE ESAMINA_SCHEMA
FOR RIGA=1 TO 9 DO
FOR COLONNA=1 TO 9 DO
VALORE_DEFINITO
IF FLAG% THEN
Z\$=TAV\$[RIGA,COLONNA]
TOGLI_VALORE
END IF
END FOR
END FOR
END PROCEDURE

PROCEDURE IDENTIFICA_UNICO
FOR KL=1 TO 9 DO
KL\$=MID\$(STR\$(KL),2)
NN=0
FOR H=1 TO LEN(ZZ\$) DO
IF MID\$(ZZ\$,H,1)=KL\$ THEN NN=NN+1 END IF
END FOR
IF NN=1 THEN Q=INSTR(ZZ\$,KL\$) KL=9 END IF
END FOR
END PROCEDURE

!----------------------------------------------------------------------------
! intercetta i valori unici per le celle ancora non definite
!----------------------------------------------------------------------------
PROCEDURE TOGLI_VALORE2

MESSAGGI(2)
! iniziamo dalle righe ....
OK%=FALSE
FOR RIGA=1 TO 9 DO
ZZ\$=""
FOR COLONNA=1 TO 9 DO
IF LEN(TAV\$[RIGA,COLONNA])<>1 THEN
ZZ\$=ZZ\$+TAV\$[RIGA,COLONNA]
ELSE
ZZ\$=ZZ\$+STRING\$(9," ")
END IF
END FOR
Q=0 IDENTIFICA_UNICO
IF Q<>0 THEN
COLONNA=INT(Q/9.1)+1
TAV\$[RIGA,COLONNA]=KL\$
OK%=TRUE EXIT
END IF
END FOR
IF OK% THEN GOTO 76 END IF

! .... poi dalle colonne ....
FOR COLONNA=1 TO 9 DO
ZZ\$=""
FOR RIGA=1 TO 9 DO
IF LEN(TAV\$[RIGA,COLONNA])<>1 THEN
ZZ\$=ZZ\$+TAV\$[RIGA,COLONNA]
ELSE
ZZ\$=ZZ\$+STRING\$(9," ")
END IF
END FOR
Q=0 IDENTIFICA_UNICO
IF Q<>0 THEN
RIGA=INT(Q/9.1)+1
TAV\$[RIGA,COLONNA]=KL\$ OK%=TRUE EXIT
END IF
END FOR
IF OK% THEN GOTO 76 END IF

ZZ\$=""
1-> R=1 S=1 END ->
2-> R=1 S=4 END ->
3-> R=1 S=7 END ->
4-> R=4 S=1 END ->
5-> R=4 S=4 END ->
6-> R=4 S=7 END ->
7-> R=7 S=1 END ->
8-> R=7 S=4 END ->
9-> R=7 S=7 END ->
END CASE
FOR RIGA=R TO R+2 DO
FOR COLONNA=S TO S+2 DO
IF LEN(TAV\$[RIGA,COLONNA])<>1 THEN
ZZ\$=ZZ\$+TAV\$[RIGA,COLONNA]
ELSE
ZZ\$=ZZ\$+STRING\$(9," ")
END IF
END FOR
END FOR
Q=0 IDENTIFICA_UNICO
IF Q<>0 THEN
CASE Q OF
1..9->   ALFA=R   BETA=S   END ->
10..18-> ALFA=R   BETA=S+1 END ->
19..27-> ALFA=R   BETA=S+2 END ->
28..36-> ALFA=R+1 BETA=S   END ->
37..45-> ALFA=R+1 BETA=S+1 END ->
46..54-> ALFA=R+1 BETA=S+2 END ->
55..63-> ALFA=R+2 BETA=S   END ->
64..72-> ALFA=R+2 BETA=S+1 END ->
OTHERWISE
ALFA=R+2 BETA=S+2
END CASE
77:
TAV\$[ALFA,BETA]=KL\$ EXIT
END IF
END FOR
76:
MESSAGGI(2)
END PROCEDURE

PROCEDURE CONVERTI_VALORE
FINE%=TRUE NESSUNO%=TRUE
FOR RIGA=1 TO 9 DO
FOR COLONNA=1 TO 9 DO
CH\$=TAV\$[RIGA,COLONNA]
IF LEN(CH\$)<>1 THEN
FINE%=FALSE ! flag per fine partita -- trovati tutti
Q=0         ! conta i '-' nella stringa se ce ne sono 8,
! trovato valore
FOR Z=1 TO LEN(CH\$) DO
IF MID\$(CH\$,Z,1)="-" THEN Q=Q+1 ELSE LAST=Z END IF
END FOR
IF Q=8 THEN
CH\$=MID\$(STR\$(LAST),2)
TAV\$[RIGA,COLONNA]=CH\$
NESSUNO%=FALSE
END IF
END IF
END FOR
END FOR
END PROCEDURE

PROCEDURE LEGGI_SCHEMA
OPEN("I",1,"sudoku.txt")
FOR I=1 TO 9 DO
INPUT(LINE,#1,RIGA\$)
FOR J=1 TO 9 DO
CH\$=MID\$(RIGA\$,J,1)
IF CH\$="0" OR CH\$="." THEN
TAV\$[I,J]="123456789"
ELSE
TAV\$[I,J]=CH\$
END IF
END FOR
END FOR
CLOSE(1)
END PROCEDURE

!---------------------------------------------------------------------------
! Praticamente - visita di un albero binario (caso con cella a 2 valori
!                                             possibili)
!---------------------------------------------------------------------------
PROCEDURE RICERCA_COMBINATORIA
TRE%=TRUE
FOR RIGA=1 TO 9 DO
FOR COLONNA=1 TO 9 DO
CH\$=TAV\$[RIGA,COLONNA]
IF LEN(CH\$)<>1 THEN
Q=0 FIRST=0 SECOND=0 THIRD=0
FOR Z=1 TO LEN(CH\$) DO
IF MID\$(CH\$,Z,1)="-" THEN
Q=Q+1
ELSE
IF FIRST=0 THEN
FIRST=Z
ELSE
SECOND=Z
END IF
END IF
END FOR
IF Q=7 THEN
SALVA_CONFIG
TAV\$[RIGA,COLONNA]=MID\$(STR\$(FIRST),2)
TRE%=FALSE
GOTO 97
END IF
END IF
END FOR
END FOR
IF TRE% THEN GOTO 88 END IF
97:
MESSAGGI(3)
EXIT PROCEDURE
88:
QUATTRO%=TRUE
FOR RIGA=1 TO 9 DO
FOR COLONNA=1 TO 9 DO
CH\$=TAV\$[RIGA,COLONNA]
IF LEN(CH\$)<>1 THEN
Q=0 FIRST=0 SECOND=0 THIRD=0
FOR Z=1 TO LEN(CH\$) DO
IF MID\$(CH\$,Z,1)="-" THEN
Q=Q+1
ELSE
IF FIRST=0 THEN
FIRST=Z
ELSE
IF SECOND=0 THEN
SECOND=Z
ELSE
THIRD=Z
END IF
END IF
END IF
END FOR
IF Q=6 THEN
SALVA_CONFIG
TAV\$[RIGA,COLONNA]=MID\$(STR\$(FIRST),2)
QUATTRO%=FALSE
GOTO 97
END IF
END IF
END FOR
END FOR
IF QUATTRO% THEN
LIVELLO=LIVELLO+1
RIPRISTINA_CONFIG
GOTO 97
END IF
! se restano solo celle con 4 valori,forza la chiusura del ramo dell'albero
!\$RCODE="STOP"
END PROCEDURE

BEGIN
CLS
LIVELLO=1 NZ%=0
LEGGI_SCHEMA
WHILE TRUE DO
VISUALIZZA_SCHEMA
99:
NZ%=NZ%+1
ESAMINA_SCHEMA
CONVERTI_VALORE
EXIT IF FINE%
IF NESSUNO% THEN
TOGLI_VALORE2
IF OK%=0 THEN
RICERCA_COMBINATORIA  ! cerca altri celle da assegnare
END IF
END IF
END WHILE
VISUALIZZA_SCHEMA
VERIFICA_SE_FINITO
IF NOT COMPLETO% THEN
LIVELLO=LIVELLO+1
RIPRISTINA_CONFIG
GOTO 99
END IF
END PROGRAM```

F#

Backtracking

```module SudokuBacktrack

//Helpers
let tuple2 a b = a,b
let flip  f a b = f b a
let (>>=) f g = Option.bind g f

/// "A1" to "I9" squares as key in values dictionary
let key a b = \$"{a}{b}"

/// Cross product of elements in ax and elements in bx
let cross ax bx = [| for a in ax do for b in bx do key a b |]

// constants
let valid   = "1234567890.,"
let rows    = "ABCDEFGHI"
let cols    = "123456789"
let squares = cross rows cols

// List of all row, cols and boxes:  aka units
let unitList =
[for c in cols do cross rows (string c) ]@ // row units
[for r in rows do cross (string r) cols ]@ // col units
[for rs in ["ABC";"DEF";"GHI"] do for cs in ["123";"456";"789"] do cross rs cs ] // box units

/// Dictionary of units for each square
let units =
[for s in squares do s, [| for u in unitList do if u |> Array.contains s then u |] ] |> Map.ofSeq

/// Dictionary of all peer squares in the relevant units wrt square in question
let peers =
[for s in squares do units[s] |> Array.concat |> Array.distinct |> Array.except [s] |> tuple2 s] |> Map.ofSeq

/// Should parse grid in many input formats or return None
let parseGrid grid =
let ints = [for c in grid do if valid |> Seq.contains c then if ",." |> Seq.contains c then 0 else (c |> string |> int)]
if Seq.length ints = 81 then ints |> Seq.zip squares |> Map.ofSeq |> Some else None

/// Outputs single line puzzle with 0 as empty squares
let asString  =  function
| Some values -> values |> Map.toSeq |> Seq.map (snd>>string) |> String.concat ""
| _ ->  "No solution or Parse Failure"

/// Outputs puzzle in 2D format with 0 as empty squares
let prettyPrint = function
| Some (values:Map<_,_>) ->
[for r in rows do [for c in cols do (values[key r c] |> string) ] |> String.concat " " ] |> String.concat "\n"
| _ ->  "No solution or Parse Failure"

/// Is digit allowed in the square in question? !!! hot path !!!!
/// Array/Array2D no faster and they need explicit copy since not immutable
let constraints (values:Map<_,_>) s d = peers[s] |> Seq.map (fun p -> values[p]) |> Seq.exists ((=) d) |> not

/// Move to next square or None if out of bounds
let next s = squares |> Array.tryFindIndex ((=)s) |> function Some i when i + 1 < 81 -> Some squares[i + 1] | _ -> None

/// Backtrack recursively and immutably from index
let rec backtracker (values:Map<_,_>) = function
| None -> Some values // solved!
| Some s when values[s] > 0 -> backtracker values (next s)  // square not empty
| Some s ->
let rec tracker  = function
| [] -> None
| d::dx ->
values
|> Map.change s (Option.map (fun _ -> d))
|> flip backtracker (next s)
|> function
| None ->  tracker dx
| success -> success
[for d in 1..9 do if constraints values s d then d] |> tracker

/// solve sudoku using simple backtracking
let solve grid = grid |> parseGrid >>= flip backtracker (Some "A1")
```

Usage:

```open System
open SudokuBacktrack

[<EntryPoint>]
let main argv =
let puzzle =  "000028000800010000000000700000600403200004000100700000030400500000000010060000000"
puzzle |> printfn "Puzzle:\n%s"
puzzle |> parseGrid |> prettyPrint |> printfn "Formatted:\n%s"
puzzle |> solve |> prettyPrint |> printfn "Solution:\n%s"

printfn "Press any key to exit"
0
```
Output:
```
Puzzle:
000028000800010000000000700000600403200004000100700000030400500000000010060000000
Formatted:
0 0 0 0 2 8 0 0 0
8 0 0 0 1 0 0 0 0
0 0 0 0 0 0 7 0 0
0 0 0 6 0 0 4 0 3
2 0 0 0 0 4 0 0 0
1 0 0 7 0 0 0 0 0
0 3 0 4 0 0 5 0 0
0 0 0 0 0 0 0 1 0
0 6 0 0 0 0 0 0 0
Solution:
6 1 7 3 2 8 9 4 5
8 9 4 5 1 7 2 3 6
3 2 5 9 4 6 7 8 1
9 7 8 6 5 1 4 2 3
2 5 6 8 3 4 1 7 9
1 4 3 7 9 2 6 5 8
7 3 1 4 8 9 5 6 2
4 8 9 2 6 5 3 1 7
5 6 2 1 7 3 8 9 4
Press any key to exit

```

Constraint Satisfaction (Norvig)

```// https://norvig.com/sudoku.html
// using array O(1) lookup & mutable  instead of map O(logn) immutable - now 6 times faster
module SudokuCPSArray
open System

/// from 11 to 99 as squares key maps to 0 to 80 in arrays
let key a b = (9*a + b) - 10

/// Keys generator
let cross ax bx = [| for a in ax do for b in bx do key a b |]

let digits  = [|1..9|]
let rows    = digits
let cols    = digits
let empty   = "0,."
let valid   = "123456789"+empty
let boxi    = [for b in 1..3..9 -> [|b..b+2|]]
let squares = cross rows cols

/// List of all row, cols and boxes:  aka units
let unitlist =
[for c in cols -> cross rows [|c|] ]@
[for r in rows -> cross [|r|] cols ]@
[for rs in boxi do for cs in boxi do cross rs cs ]

/// Dictionary of units for each square
let units =
[|for s in squares do [| for u in unitlist do if u |> Array.contains s then u |] |]

/// Dictionary of all peer squares in the relevant units wrt square in question
let peers =
[| for s in squares do units[s] |> Array.concat |> Array.distinct |> Array.except [s] |]

/// folds folder returning Some on completion or returns None if not
let rec all folder state source =
match state, source with
| None, _ -> None
| Some st, [] -> Some st
| Some st , hd::rest -> folder st hd |> (fun st1 -> all folder st1 rest)

/// Assign digit d to values[s] and propagate (via eliminate)
/// Return Some values, except None if a contradiction is detected.
let rec assign (values:int[][]) (s) d =
values[s]
|> Array.filter ((<>)d)
|> List.ofArray |> all (fun vx d1 -> eliminate vx s d1) (Some values)

/// Eliminate digit d from values[s] and propagate when values[s] size is 1.
/// Return Some values, except return None if a contradiction is detected.
and eliminate (values:int[][]) s d =
let peerElim (values1:int[][]) = // If a square s is reduced to one value d, then *eliminate* d from the peers.
match Seq.length values1[s] with
| 0 -> None // contradiction - removed last value
| 1 -> peers[s] |> List.ofArray |> all (fun vx1 s1 ->  eliminate vx1 s1 (values1[s] |> Seq.head) ) (Some values1)
| _ -> Some values1

let unitsElim values1 = // If a unit u is reduced to only one place for a value d, then *assign* it there.
units[s]
|> List.ofArray
|> all (fun (vx1:int[][]) u ->
let sx = [for s in u do if vx1[s] |> Seq.contains d then s]
match Seq.length sx with
| 0 -> None
| 1 -> assign vx1 (Seq.head sx) d
| _ -> Some vx1) (Some values1)

match values[s] |> Seq.contains d with
| false ->  Some values // Already eliminated, nothing to do
| true ->
values[s] <- values[s]|> Array.filter ((<>)d)
values
|> peerElim
|> Option.bind unitsElim

/// Convert grid into a Map of {square: char} with "0","."or"," for empties.
let parseGrid grid =
let cells = [for c in grid do if valid |> Seq.contains c then if empty |> Seq.contains c then 0 else ((string>>int)c)]
if Seq.length cells = 81 then cells |> Seq.zip squares |> Map.ofSeq |> Some  else None

/// Convert grid to a Map of constraint propagated possible values, or return None if a contradiction is detected.
let applyCPS (parsedGrid:Map<_,_>) =
let values = [| for s in squares do digits |]
parsedGrid
|> Seq.filter (fun (KeyValue(_,d)) -> digits |> Seq.contains d)
|> List.ofSeq
|> all (fun vx (KeyValue(s,d)) -> assign vx s d) (Some values)

/// Calculate string centre for each square - which can contain more than 1 digit when debugging
let centre s width =
let n = width - (Seq.length s)
if n <= 0 then s
else
let half = n/2 + (if (n%2>0 && width%2>0) then 1 else 0)
sprintf "%s%s%s" (String(' ',half)) s (String(' ', n - half))

/// Display these values as a 2-D grid. Used for debugging
let prettyPrint (values:int[][]) =
let asString = Seq.map string >> String.concat ""
let width = 1 + ([for s in squares do Seq.length values[s]] |> List.max)
let line = sprintf "%s\n" ((String('-',width*3) |> Seq.replicate 3) |> String.concat "+")
[for r in rows do
for c in cols do
sprintf "%s%s" (centre (asString values[key r c]) width) (if List.contains c [3;6] then "|" else "")
sprintf "\n%s"(if List.contains r [3;6] then line else "") ]
|> String.concat ""

/// Outputs single line puzzle with 0 as empty squares
let asString values = values |> Map.toSeq |> Seq.map (snd>>string) |> String.concat ""

let copy values = values |> Array.map Array.copy

/// Using depth-first search and propagation, try all possible values.
let rec search (values:int[][])=
[for s in squares do if Seq.length values[s] > 1 then Seq.length values[s] ,s]
|> function
| [] -> Some values // Solved!
| list -> // tryPick ~ Norvig's `some`
list |> List.minBy fst
|> fun (_,s) -> values[s] |> Seq.tryPick (fun d -> assign (copy values) s d |> (Option.bind search))

let run n g f = parseGrid >> function None -> n | Some m -> f m |> g
let solver = run "Parse Error" (Option.fold (fun _ t -> t |> prettyPrint) "No Solution")
let solveNoSearch: string -> string = solver applyCPS
let solveWithSearch: string -> string = solver (applyCPS >> (Option.bind search))
let solveWithSearchToMapOnly:string -> int[][] option = run None id (applyCPS >> (Option.bind search))
```
Usage
```open System
open SudokuCPSArray
open System.Diagnostics
open System.IO

[<EntryPoint>]
let main argv =
printfn "Easy board solution automatic with constraint propagation"
let easy = "..3.2.6..9..3.5..1..18.64....81.29..7.......8..67.82....26.95..8..2.3..9..5.1.3.."
easy |> solveNoSearch |> printfn "%s"

printfn "Simple elimination not possible"
let simple = "4.....8.5.3..........7......2.....6.....8.4......1.......6.3.7.5..2.....1.4......"
simple |> run "Parse Error" asString id |> printfn "%s"
simple |> solveNoSearch |> printfn "%s"

printfn "Try again with search:"
simple |> solveWithSearch |> printfn "%s"

let watch = Stopwatch()

let hard = "85...24..72......9..4.........1.7..23.5...9...4...........8..7..17..........36.4."
printfn "Hard"
watch.Start()
hard |> solveWithSearch |> printfn "%s"
watch.Stop()
printfn \$"Elapsed milliseconds = {watch.ElapsedMilliseconds } ms"
watch.Reset()

let puzzles  =
if Seq.length argv = 1 then
let num = argv[0] |> int
printfn \$"First {num} puzzles in sudoku17 (http://staffhome.ecm.uwa.edu.au/~00013890/sudoku17)"
else
printfn \$"All puzzles in sudoku17 (http://staffhome.ecm.uwa.edu.au/~00013890/sudoku17)"
watch.Start()
let result = puzzles |> Array.map solveWithSearchToMapOnly
watch.Stop()
if result |> Seq.forall Option.isSome then
let total = watch.ElapsedMilliseconds
let avg = (float total) /(float result.Length)
printfn \$"\nPuzzles:{result.Length}, Total:%.2f{((float)total)/1000.0} s, Average:%.2f{avg} ms"
else
printfn "Some sudoku17 puzzles failed"
0
```
Output:
Timings run on i7500U @2.75Ghz CPU, 16GB RAM
```Easy board solution automatic with constraint propagation
4 8 3 |9 2 1 |6 5 7
9 6 7 |3 4 5 |8 2 1
2 5 1 |8 7 6 |4 9 3
------+------+------
5 4 8 |1 3 2 |9 7 6
7 2 9 |5 6 4 |1 3 8
1 3 6 |7 9 8 |2 4 5
------+------+------
3 7 2 |6 8 9 |5 1 4
8 1 4 |2 5 3 |7 6 9
6 9 5 |4 1 7 |3 8 2
Simple elimination not possible
400000805030000000000700000020000060000080400000010000000603070500200000104000000

4      1679   12679  |  139     2369    269   |   8      1239     5
26789     3    1256789 | 14589   24569   245689 | 12679    1249   124679
2689   15689   125689 |   7     234569  245689 | 12369   12349   123469
------------------------+------------------------+------------------------
3789     2     15789  |  3459   34579    4579  | 13579     6     13789
3679   15679   15679  |  359      8     25679  |   4     12359   12379
36789     4     56789  |  359      1     25679  | 23579   23589   23789
------------------------+------------------------+------------------------
289      89     289   |   6      459      3    |  1259     7     12489
5      6789     3    |   2      479      1    |   69     489     4689
1      6789     4    |  589     579     5789  | 23569   23589   23689

Try again with search:
4 1 7 |3 6 9 |8 2 5
6 3 2 |1 5 8 |9 4 7
9 5 8 |7 2 4 |3 1 6
------+------+------
8 2 5 |4 3 7 |1 6 9
7 9 1 |5 8 6 |4 3 2
3 4 6 |9 1 2 |7 5 8
------+------+------
2 8 9 |6 4 3 |5 7 1
5 7 3 |2 9 1 |6 8 4
1 6 4 |8 7 5 |2 9 3
Hard
8 5 9 |6 1 2 |4 3 7
7 2 3 |8 5 4 |1 6 9
1 6 4 |3 7 9 |5 2 8
------+------+------
9 8 6 |1 4 7 |3 5 2
3 7 5 |2 6 8 |9 1 4
2 4 1 |5 9 3 |7 8 6
------+------+------
4 3 2 |9 8 1 |6 7 5
6 1 7 |4 2 5 |8 9 3
5 9 8 |7 3 6 |2 4 1
Elapsed milliseconds = 8 ms
All puzzles in sudoku17 (http://staffhome.ecm.uwa.edu.au/~00013890/sudoku17)

Puzzles:49151, Total:80.99 s, Average:1.65 ms```

SLPsolve

```// Solve Sudoku Like Puzzles. Nigel Galloway: September 6th., 2018
let fN y n g=let _q n' g'=[for n in n*n'..n*n'+n-1 do for g in g*g'..g*g'+g-1 do yield (n,g)]
let N=[|for n in 0..(y/n)-1 do for g in 0..(y/g)-1 do yield _q n g|]
(fun n' g'->N.[((n'/n)*n+g'/g)])
let fI n=let _q g=[for n in 0..n-1 do yield (g,n)]
let N=[|for n in 0..n-1 do yield _q n|]
(fun g (_:int)->N.[g])
let fG n=let _q g=[for n in 0..n-1 do yield (n,g)]
let N=[|for n in 0..n-1 do yield _q n|]
(fun (_:int) n->N.[n])
let fE v z n g fn=let N,G,B,fn=fI z,fG z,fN z n g,readCSV ',' fn|>List.ofSeq
let fG n g mask=List.except (N n g@G n g@B n g) mask
let b=List.except(List.map(fun n->(n.row,n.col))fn)[for n in 0..z-1 do for g in 0..z-1 do yield (n,g)]
let q=Map.ofList[for v' in v do yield ((v'),List.choose(fun n->if n.value=v' then Some(n.row,n.col) else None)fn|>List.fold(fun z (n,g)->(n,g)::fG n g z)b)]
(fG,(fun n->Map.find n q),z,v)
let SLPsolve (N,G,z,l)=
let rec nQueens col mask res=[
if col=z then yield res else
let rec sudoku l res=seq{
match l with
|h::t->let n=nQueens 0 (List.except (List.concat res) (G h)) []
yield! n|>Seq.collect(fun n->sudoku t (n::res))
|_->yield res}
sudoku l []
let printSLP n g=
List.map2(fun n g->List.map(fun(n',g')->((n',g'),n))g) (List.rev n) g|>List.concat|>List.sortBy (fun ((_,n),_)->n)|>List.groupBy(fun ((n,_),_)->n)|>List.sortBy(fun(n,_)->n)
|>List.iter(fun (_,n)->n|>Seq.fold(fun z ((_,g),v)->[z..g-1]|>Seq.iter(fun _->printf " |");printf "%s|" v; g+1 ) 0 |>ignore;printfn "")
```

Usage: Given sud1.csv:

```7,1,,4,,6,,2
,,,,7,,,,3
4,,,,,1,,8
,,,,1,3,,,9
,,1,,,,7
2,,,8,6
,2,,1,,,,,4
9,,,,8
,7,,6,,4,,5,2
```

then

```let n=SLPsolve (fE ([1..9]|>List.map(string)) 9 3 3 "sud1.csv")
printSLP ([1..9]|>List.map(string)) (Seq.item 0 n)
```
Output:
```7|1|8|4|3|6|9|2|5|
5|6|2|9|7|8|4|1|3|
4|3|9|5|2|1|6|8|7|
6|8|5|7|1|3|2|4|9|
3|9|1|2|4|5|7|6|8|
2|4|7|8|6|9|5|3|1|
8|2|6|1|5|7|3|9|4|
9|5|4|3|8|2|1|7|6|
1|7|3|6|9|4|8|5|2|
```

Forth

Works with: 4tH version 3.60.0
```include lib/interprt.4th
include lib/istype.4th
include lib/argopen.4th

\  ---------------------
\  Variables
\  ---------------------

81 string sudokugrid
9 array sudoku_row
9 array sudoku_col
9 array sudoku_box

\  -------------
\  4tH interface
\  -------------

: >grid                                ( n2 a1 n1 -- n3)
rot dup >r 9 chars * sudokugrid + dup >r swap
0 do                                 ( a1 a2)
over i chars + c@ dup is-digit     ( a1 a2 c f)
if [char] 0 - over c! char+ else drop then
loop                                 ( a1 a2)
nip r> - 9 /  r> +                   ( n3)
;

0
s" 090004007" >grid
s" 000007900" >grid
s" 800000000" >grid
s" 405800000" >grid
s" 300000002" >grid
s" 000009706" >grid
s" 000000004" >grid
s" 003500000" >grid
s" 200600080" >grid
drop

\  ---------------------
\  Logic
\  ---------------------
\  Basically :
\     Grid is parsed. All numbers are put into sets, which are
\     implemented as bitmaps (sudoku_row, sudoku_col, sudoku_box)
\     which represent sets of numbers in each row, column, box.
\     only one specific instance of a number can exist in a
\     particular set.

\     SOLVER is recursively called
\     SOLVER looks for the next best guess using FINDNEXTSPACE
\     tries this trail down... if fails, backtracks... and tries
\     again.

\ Grid Related

: xy 9 * + ;                           \  x y -- offset ;
: getrow 9 / ;
: getcol 9 mod ;
: getbox dup getrow 3 / 3 * swap getcol 3 / + ;

\ Puts and gets numbers from/to grid only
: setnumber sudokugrid + c! ;          \ n position --
: getnumber sudokugrid + c@ ;

: cleargrid sudokugrid 81 bounds do 0 i c! loop ;

\ --------------
\ Set related: sets are sudoku_row, sudoku_col, sudoku_box

\ ie x y --   ;  adds x into bitmap y
: addbits_row cells sudoku_row + dup @ rot 1 swap lshift or swap ! ;
: addbits_col cells sudoku_col + dup @ rot 1 swap lshift or swap ! ;
: addbits_box cells sudoku_box + dup @ rot 1 swap lshift or swap ! ;

\ ie x y --  ; remove number x from bitmap y
: removebits_row cells sudoku_row + dup @ rot 1 swap lshift invert and swap ! ;
: removebits_col cells sudoku_col + dup @ rot 1 swap lshift invert and swap ! ;
: removebits_box cells sudoku_box + dup @ rot 1 swap lshift invert and swap ! ;

\ clears all bitsmaps to 0
: clearbitmaps 9 0 do i cells
0 over sudoku_row + !
0 over sudoku_col + !
0 swap sudoku_box + !
loop ;

\ Adds number to grid and sets
: addnumber                            \ number position --
2dup setnumber
;

\ Remove number from grid, and sets
: removenumber                         \ position --
dup getnumber swap
2dup getrow removebits_row
2dup getcol removebits_col
2dup getbox removebits_box
nip 0 swap setnumber
;

\ gets bitmap at position, ie
\ position -- bitmap

: getrow_bits getrow cells sudoku_row + @ ;
: getcol_bits getcol cells sudoku_col + @ ;
: getbox_bits getbox cells sudoku_box + @ ;

\ position -- composite bitmap  (or'ed)
: getbits
dup getrow_bits
over getcol_bits
rot getbox_bits or or
;

\ algorithm from c.l.f circa 1995 ? Will Baden
: countbits    ( number -- bits )
[HEX] DUP  55555555 AND  SWAP  1 RSHIFT  55555555 AND  +
DUP  33333333 AND  SWAP  2 RSHIFT  33333333 AND  +
DUP  0F0F0F0F AND  SWAP  4 RSHIFT  0F0F0F0F AND  +
[DECIMAL] 255 MOD
;

\ Try tests a number in a said position of grid
\ Returns true if it's possible, else false.
: try                                  \ number position -- true/false
getbits 1 rot lshift and 0=
;

\ --------------
: parsegrid                            \ Parses Grid to fill sets.. Run before solver.
sudokugrid                          \ to ensure all numbers are parsed into sets/bitmaps
81 0 do
dup i + c@
dup if
dup i try if
else
unloop drop drop FALSE exit
then
else
drop
then
loop
drop
TRUE
;

\ Morespaces? manually checks for spaces ...
\ Obviously this can be optimised to a count var, done initially
\ a 'spaces' variable.

: morespaces?
0  sudokugrid 81 bounds do i c@  0= if 1+ then loop ;

: findnextmove                         \  -- n ; n = index next item, if -1 finished.

-1  10                              \  index  prev_possibilities  --
\  err... yeah... local variables, kind of...

81 0 do
i sudokugrid + c@ 0= IF
i getbits countbits 9 swap -

\ get bitmap and see how many possibilities
\ stack diagram:
\ index prev_possibilities  new_possiblities --

2dup > if
\ if new_possibilities < prev_possibilities...
nip nip i swap
\ new_index new_possibilies --

else                      \ else prev_possibilities < new possibilities, so:

drop                  \ new_index new_possibilies --

then
THEN
loop
drop
;

\ findnextmove returns index of best next guess OR returns -1
\ if no more guesses. You then have to check to see if there are
\ spaces left on the board unoccupied. If this is the case, you
\ need to back up the recursion and try again.

: solver
findnextmove
dup 0< if
morespaces? if
drop false exit
else
drop true exit
then
then

10 1 do
i over try if
recurse  if
drop unloop TRUE EXIT
else
dup removenumber
then
then
loop

drop FALSE
;

\ SOLVER

: startsolving
clearbitmaps                        \ reparse bitmaps and reparse grid
parsegrid                           \ just in case..
solver
AND
;

\  ---------------------
\  Display Grid
\  ---------------------

\ Prints grid nicely

: .sudokugrid
CR CR
sudokugrid
81 0 do
dup i + c@ .
i 1+
dup 3 mod 0= if
dup 9 mod 0= if
CR
dup 27 mod 0= if
dup 81 < if ." ------+-------+------" CR then
then
else
." | "
then
then
drop
loop
drop
CR
;

\  ---------------------
\  Higher Level Words
\  ---------------------

: checkifoccupied                      ( offset -- t/f)
sudokugrid + c@
;

: add                                  ( n x y --)
xy 2dup
dup checkifoccupied if
dup removenumber
then
try if
.sudokugrid
else
CR ." Not a valid move. " CR
2drop
then
;

: rm
xy removenumber
.sudokugrid
;

: clearit
cleargrid
clearbitmaps
.sudokugrid
;

: solveit
CR
startsolving
if
." Solution found!" CR .sudokugrid
else
." No solution found!" CR CR
then
;

: showit .sudokugrid ;

: help
CR
." Type clearit     ; to clear grid " CR
."      1-9 x y add ; to add 1-9 to grid at x y (0 based) " CR
."      x y rm      ; to remove number at x y " CR
."      showit      ; redisplay grid " CR
."      solveit     ; to solve " CR
."      help        ; for help " CR
CR
;

\  ---------------------
\  Execution starts here
\  ---------------------

: godoit
clearbitmaps
parsegrid if
CR ." Grid valid!"
else
CR ." Warning: grid invalid!"
then
.sudokugrid
help
;

\  -------------
\  4tH interface
\  -------------

input 1 arg-open 0
begin dup 9 < while refill while 0 parse >grid repeat
drop close
;

: bye quit ;

create wordlist                        \ dictionary
," clearit" ' clearit ,
," rm"      ' rm ,
," showit"  ' showit ,
," solveit" ' solveit ,
," quit"    ' bye ,
," exit"    ' bye ,
," bye"     ' bye ,
," q"       ' bye ,
," help"    ' help ,
NULL ,

wordlist to dictionary
:noname ." Unknown command '" type ." '" cr ; is NotFound
\ sudoku interpreter
: sudoku
argn 1 > if read-sudoku then
godoit
begin
." OK" cr
refill drop ['] interpret
catch if ." Error" cr then
again
;

sudoku
```

Fortran

Works with: Fortran version 90 and later

This implementation uses a brute force method. The subroutine `solve` recursively checks valid entries using the rules defined in the function `is_safe`. When `solve` is called beyond the end of the sudoku, we know that all the currently entered values are valid. Then the result is displayed.

```program sudoku

implicit none
integer, dimension (9, 9) :: grid
integer, dimension (9, 9) :: grid_solved
grid = reshape ((/               &
& 0, 0, 3, 0, 2, 0, 6, 0, 0,   &
& 9, 0, 0, 3, 0, 5, 0, 0, 1,   &
& 0, 0, 1, 8, 0, 6, 4, 0, 0,   &
& 0, 0, 8, 1, 0, 2, 9, 0, 0,   &
& 7, 0, 0, 0, 0, 0, 0, 0, 8,   &
& 0, 0, 6, 7, 0, 8, 2, 0, 0,   &
& 0, 0, 2, 6, 0, 9, 5, 0, 0,   &
& 8, 0, 0, 2, 0, 3, 0, 0, 9,   &
& 0, 0, 5, 0, 1, 0, 3, 0, 0/), &
& shape = (/9, 9/),            &
& order = (/2, 1/))
call pretty_print (grid)
call solve (1, 1)
write (*, *)
call pretty_print (grid_solved)

contains

recursive subroutine solve (i, j)
implicit none
integer, intent (in) :: i
integer, intent (in) :: j
integer :: n
integer :: n_tmp
if (i > 9) then
grid_solved = grid
else
do n = 1, 9
if (is_safe (i, j, n)) then
n_tmp = grid (i, j)
grid (i, j) = n
if (j == 9) then
call solve (i + 1, 1)
else
call solve (i, j + 1)
end if
grid (i, j) = n_tmp
end if
end do
end if
end subroutine solve

function is_safe (i, j, n) result (res)
implicit none
integer, intent (in) :: i
integer, intent (in) :: j
integer, intent (in) :: n
logical :: res
integer :: i_min
integer :: j_min
if (grid (i, j) == n) then
res = .true.
return
end if
if (grid (i, j) /= 0) then
res = .false.
return
end if
if (any (grid (i, :) == n)) then
res = .false.
return
end if
if (any (grid (:, j) == n)) then
res = .false.
return
end if
i_min = 1 + 3 * ((i - 1) / 3)
j_min = 1 + 3 * ((j - 1) / 3)
if (any (grid (i_min : i_min + 2, j_min : j_min + 2) == n)) then
res = .false.
return
end if
res = .true.
end function is_safe

subroutine pretty_print (grid)
implicit none
integer, dimension (9, 9), intent (in) :: grid
integer :: i
integer :: j
character (*), parameter :: bar = '+-----+-----+-----+'
character (*), parameter :: fmt = '(3 ("|", i0, 1x, i0, 1x, i0), "|")'
write (*, '(a)') bar
do j = 0, 6, 3
do i = j + 1, j + 3
write (*, fmt) grid (i, :)
end do
write (*, '(a)') bar
end do
end subroutine pretty_print

end program sudoku
```
Output:
```
+-----+-----+-----+
|0 0 3|0 2 0|6 0 0|
|9 0 0|3 0 5|0 0 1|
|0 0 1|8 0 6|4 0 0|
+-----+-----+-----+
|0 0 8|1 0 2|9 0 0|
|7 0 0|0 0 0|0 0 8|
|0 0 6|7 0 8|2 0 0|
+-----+-----+-----+
|0 0 2|6 0 9|5 0 0|
|8 0 0|2 0 3|0 0 9|
|0 0 5|0 1 0|3 0 0|
+-----+-----+-----+

+-----+-----+-----+
|4 8 3|9 2 1|6 5 7|
|9 6 7|3 4 5|8 2 1|
|2 5 1|8 7 6|4 9 3|
+-----+-----+-----+
|5 4 8|1 3 2|9 7 6|
|7 2 9|5 6 4|1 3 8|
|1 3 6|7 9 8|2 4 5|
+-----+-----+-----+
|3 7 2|6 8 9|5 1 4|
|8 1 4|2 5 3|7 6 9|
|6 9 5|4 1 7|3 8 2|

+-----+-----+-----+```

FreeBASIC

Translation of: VBA
```Dim Shared As Integer cuadricula(9, 9), cuadriculaResuelta(9, 9)

Function isSafe(i As Integer, j As Integer, n As Integer) As Boolean
Dim As Integer iMin, jMin, f, c

'cuadricula(i, j) es una celda vacía. Compruebe si n está OK
'primero revisa la fila i
For f = 1 To 9
If cuadricula(i, f) = n Then Return False
Next f

'ahora comprueba la columna j
For c = 1 To 9
If cuadricula(c, j) = n Then Return False
Next c

iMin = 1 + 3 * Int((i - 1) / 3)
jMin = 1 + 3 * Int((j - 1) / 3)
For c = iMin To iMin + 2
For f = jMin To jMin + 2
If cuadricula(c, f) = n Then Return False
Next f
Next c

'todas las pruebas estuvieron OK
Return True
End Function

Sub Resolver(i As Integer, j As Integer)
Dim As Integer f, c, n, temp
If i > 9 Then
For c = 1 To 9
For f = 1 To 9
Next f
Next c
Exit Sub
End If
For n = 1 To 9
If isSafe(i, j, n) Then
If j = 9 Then
Resolver i + 1, 1
Else
Resolver i, j + 1
End If
End If
Next n
End Sub

Dim As String s(9)
s(1) = "001005070"
s(2) = "920600000"
s(3) = "008000600"
s(4) = "090020401"
s(5) = "000000000"
s(6) = "304080090"
s(7) = "007000300"
s(8) = "000007069"
s(9) = "010800700"

Dim As Integer i, j
For i = 1 To 9
For j = 1 To 9
cuadricula(i, j) = Int(Val(Mid(s(i), j, 1)))
Next j
Next i

Resolver 1, 1
Print "Solucion:"
Color 12: Print "---------+---------+---------"
For i = 1 To 9
For j = 1 To 9
Color 7: Print cuadriculaResuelta(i, j); " ";
Color 12
If (j Mod 3 = 0) And (j <> 9) Then Color 12: Print "|";
Next j
If (i Mod 3 = 0) Then Print !"\n---------+---------+---------" Else Print
Next i
Sleep
```
Output:
```Solucion:
---------+---------+---------
6  3  1 | 2  4  5 | 9  7  8
9  2  5 | 6  7  8 | 1  4  3
4  7  8 | 3  1  9 | 6  5  2
---------+---------+---------
7  9  6 | 5  2  3 | 4  8  1
1  8  2 | 9  6  4 | 5  3  7
3  5  4 | 7  8  1 | 2  9  6
---------+---------+---------
8  6  7 | 4  9  2 | 3  1  5
2  4  3 | 1  5  7 | 8  6  9
5  1  9 | 8  3  6 | 7  2  4
---------+---------+---------
```

FutureBasic

First is a short version:

```include "ConsoleWindow"
include "NSLog.incl"
include "Util_Containers.incl"

begin globals
dim as container gC
end globals

BeginCDeclaration
short solve_sudoku(short i);
short check_sudoku(short r, short c);
CFMutableStringRef print_sudoku();
EndC

BeginCFunction
short sudoku[9][9] = {
{3,0,0,0,0,1,4,0,9},
{7,0,0,0,0,4,2,0,0},
{0,5,0,2,0,0,0,1,0},
{5,7,0,0,4,3,0,6,0},
{0,9,0,0,0,0,0,3,0},
{0,6,0,7,9,0,0,8,5},
{0,8,0,0,0,5,0,4,0},
{0,0,6,4,0,0,0,0,7},
{9,0,5,6,0,0,0,0,3},
};

short check_sudoku( short r, short c )
{
short i;
short rr, cc;

for (i = 0; i < 9; i++)
{
if (i != c && sudoku[r][i] == sudoku[r][c]) return 0;
if (i != r && sudoku[i][c] == sudoku[r][c]) return 0;
rr = r/3 * 3 + i/3;
cc = c/3 * 3 + i%3;
if ((rr != r || cc != c) && sudoku[rr][cc] == sudoku[r][c]) return 0;
}
return -1;
}

short solve_sudoku( short i )
{
short r, c;

if (i < 0) return 0;
else if (i >= 81) return -1;

r = i / 9;
c = i % 9;

if (sudoku[r][c])
return check_sudoku(r, c) && solve_sudoku(i + 1);
else
for (sudoku[r][c] = 9; sudoku[r][c] > 0; sudoku[r][c]--)
{
if ( solve_sudoku(i) ) return -1;
}
return 0;
}

CFMutableStringRef print_sudoku()
{
short i, j;
CFMutableStringRef mutStr;

mutStr = CFStringCreateMutable( kCFAllocatorDefault, 0 );

for (i = 0; i < 9; i++)
{
for (j = 0; j < 9; j++)
{
CFStringAppendFormat( mutStr, NULL, (CFStringRef)@" %d", sudoku[i][j] );
}
CFStringAppendFormat( mutStr, NULL, (CFStringRef)@"\r" );
}
return( mutStr );
}
EndC

toolbox fn solve_sudoku( short i ) = short
toolbox fn check_sudoku( short r, short c ) = short
toolbox fn print_sudoku() = CFMutableStringRef

dim as short solution
dim as CFMutableStringRef cfRef

gC = " "
cfRef = fn print_sudoku()
fn ContainerCreateWithCFString( cfRef, gC )
print : print "Sudoku challenge:" : print : print gC

solution = fn solve_sudoku(0)

print : print "Sudoku solved:" : print
if ( solution )
gC = " "
cfRef = fn print_sudoku()
fn ContainerCreateWithCFString( cfRef, gC )
print gC
else
print "No solution found"
end if```

Output:

```Sudoku challenge:

3 0 0 0 0 1 4 0 9
7 0 0 0 0 4 2 0 0
0 5 0 2 0 0 0 1 0
5 7 0 0 4 3 0 6 0
0 9 0 0 0 0 0 3 0
0 6 0 7 9 0 0 8 5
0 8 0 0 0 5 0 4 0
0 0 6 4 0 0 0 0 7
9 0 5 6 0 0 0 0 3

Sudoku solved:

3 2 8 5 6 1 4 7 9
7 1 9 3 8 4 2 5 6
6 5 4 2 7 9 3 1 8
5 7 1 8 4 3 9 6 2
8 9 2 1 5 6 7 3 4
4 6 3 7 9 2 1 8 5
2 8 7 9 3 5 6 4 1
1 3 6 4 2 8 5 9 7
9 4 5 6 1 7 8 2 3
```

More code in this one, but faster execution:

```include "ConsoleWindow"
include "Tlbx Timer.incl"

begin globals
_digits = 9
_setH = 3
_setV = 3
_nSetH = 3
_nSetV = 3

begin record Board
dim as boolean f(_digits,_digits,_digits)
dim as char    match(_digits,_digits)
dim as pointer previousBoard // singly-linked list used to discover repetitions
dim &&
end record

dim quiz as board
dim as long t
dim as double       sProgStartTime

end globals

// 'ordinary' timer used for playing
local fn Milliseconds as long // time in ms since prog start
'~'1
dim as UnsignedWide us

long if ( sProgStartTime == 0.0 )
Microseconds( @us )
sProgStartTime = 4294967296.0*us.hi + us.lo
end if
Microseconds( @us )
end fn = (4294967296.0*us.hi + us.lo - sProgStartTime)'*1e-3

local fn InitMilliseconds
'~'1
sProgStartTime = 0.0
fn Milliseconds
end fn

local mode
local fn CopyBoard( source as ^Board, dest as ^Board )
'~'1
BlockMoveData( source, dest, sizeof( Board ) )
dest.previousBoard = source // linked list
end fn

local fn prepare( b as ^Board )
'~'1
dim as short i, j, n

for i = 1 to _digits
for j = 1 to _digits
for n = 1 to _digits
b.match[i, j] = 0
b.f[i, j, n] = _true
next n
next j
next i
end fn

local fn printBoard( b as ^Board )
'~'1
dim as short i, j

for i = 1 to _digits
for j = 1 to _digits
Print b.match[i, j];
next j
print
next i
end fn

local fn verifica( b as ^Board )
'~'1
dim as short i, j, n, first, x, y, ii
dim as boolean check

check = _true

for i = 1 to _digits
for j = 1 to _digits
long if ( b.match[i, j] == 0 )
check = _false
for n = 1 to _digits
long if ( b.f[i, j, n] != _false )
check = _true
end if
next n
if ( check == _false ) then exit fn
end if
next j
next i

check = _true
for j = 1 to _digits
for n = 1 to _digits
first = 0
for i = 1 to _digits
long if ( b.match[i, j] == n )
long if ( first == 0 )
first = i
xelse
check = _false
exit fn
end if
end if
next i
next n
next j

for i = 1 to _digits
for n = 1 to _digits
first = 0
for j = 1 to _digits
long if ( b.match[i, j] == n )
long if ( first == 0 )
first = j
xelse
check = _false
exit fn
end if
end if
next j
next n
next i

for x = 0 to ( _nSetH - 1 )
for y = 0 to ( _nSetV - 1 )
first = 0
for ii = 0 to ( _digits - 1 )
i = x * _setH + ii mod _setH + 1
j = y * _setV + ii / _setH + 1
long if ( b.match[i, j] == n )
long if ( first == 0 )
first = j
xelse
check = _false
exit fn
end if
end if
next ii
next y
next x

end fn = check

local fn setCell( b as ^Board, x as short, y as short, n as short) as boolean
dim as short   i, j, rx, ry
dim as boolean check

b.match[x, y] = n
for i = 1 to _digits
b.f[x, i, n] = _false
b.f[i, y, n] = _false
next i

rx = (x - 1) / _setH
ry = (y - 1) / _setV

for i = 1 to _setH
for j = 1 to _setV
b.f[ rx * _setH + i, ry * _setV + j, n ] = _false
next j
next i

check = fn verifica( #b )
if ( check == _false ) then exit fn

end fn = check

local fn solve( b as ^Board )
dim as short i, j, n, first, x, y, ii, ppi, ppj
dim as boolean check

check = _true

for i = 1 to _digits
for j = 1 to _digits
long if ( b.match[i, j] == 0 )
first = 0
for n = 1 to _digits
long if ( b.f[i, j, n] != _false )
long if ( first == 0 )
first = n
xelse
first = -1
exit for
end if
end if
next n

long if ( first > 0 )
check = fn setCell( #b, i, j, first )
if ( check == _false ) then exit fn
check = fn solve(#b)
if ( check == _false ) then exit fn
end if

end if
next j
next i

for i = 1 to _digits
for n = 1 to _digits
first = 0

for j = 1 to _digits
if ( b.match[i, j] == n ) then exit for

long if ( b.f[i, j, n] != _false ) and ( b.match[i, j] == 0 )
long if ( first == 0 )
first = j
xelse
first = -1
exit for
end if

end if

next j

long if ( first > 0 )
check = fn setCell( #b, i, first, n )
if ( check == _false ) then exit fn
check = fn solve(#b)
if ( check == _false ) then exit fn
end if

next n
next i

for j = 1 to _digits
for n = 1 to _digits
first = 0

for i = 1 to _digits
if ( b.match[i, j] == n ) then exit for

long if ( b.f[i, j, n] != _false ) and ( b.match[i, j] == 0 )
long if ( first == 0 )
first = i
xelse
first = -1
exit for
end if

end if

next i

long if ( first > 0 )
check = fn setCell( #b, first, j, n )
if ( check == _false ) then exit fn
check = fn solve(#b)
if ( check == _false ) then exit fn
end if

next n
next j

for x = 0 to ( _nSetH - 1 )
for y = 0 to ( _nSetV - 1 )

for n = 1 to _digits
first = 0

for ii = 0 to ( _digits - 1 )

i = x * _setH + ii mod _setH + 1
j = y * _setV + ii / _setH + 1

if ( b.match[i, j] == n ) then exit for

long if ( b.f[i, j, n] != _false ) and ( b.match[i, j] == 0 )
long if ( first == 0 )
first = n
ppi = i
ppj = j
xelse
first = -1
exit for
end if
end if

next ii

long if ( first > 0 )
check = fn setCell( #b, ppi, ppj, n )
if ( check == _false ) then exit fn
check = fn solve(#b)
if ( check == _false ) then exit fn
end if

next n

next y
next x

end fn = check

local fn resolve( b as ^Board )
dim as boolean check, daFinire
dim as long i, j, n
dim as board localBoard

check = fn solve(b)

long if ( check == _false )
exit fn
end if

daFinire = _false

for i = 1 to _digits
for j = 1 to _digits
long if ( b.match[i, j] == 0 )

daFinire = _true

for n = 1 to _digits
long if ( b.f[i, j, n] != _false )

fn CopyBoard( b, @localBoard )

check = fn setCell(@localBoard, i, j, n)

long if ( check != _false )
check = fn resolve( @localBoard )
long if ( check == -1 )
fn CopyBoard( @localBoard, b )

exit fn
end if
end if

end if

next n

end if
next j
next i

long if daFinire
xelse
check = -1
end if

end fn = check

fn InitMilliseconds

fn prepare( @quiz )

DATA 0,0,0,0,2,9,0,8,7
DATA 0,9,7,3,0,0,0,0,0
DATA 0,0,2,0,0,0,4,0,9
DATA 0,0,3,9,0,1,0,0,6
DATA 0,4,0,0,0,0,0,9,0
DATA 9,0,0,7,0,3,1,0,0
DATA 0,0,9,0,0,0,6,0,0
DATA 0,0,0,0,0,5,8,2,0
DATA 2,8,0,1,3,0,0,0,0

dim as short i, j, d
for i = 1 to _digits
for j = 1 to _digits
fn setCell(@quiz, j, i, d)
next j
next i

Print : print "quiz:"
fn printBoard( @quiz )
print : print "-------------------" : print
dim as boolean check

t = fn Milliseconds
check = fn resolve(@quiz)
t = fn Milliseconds - t

if ( check )
print "solution:"; str\$( t/1000.0 ) + " ms"
else
print "No solution found"
end if
fn printBoard( @quiz )
```

Output:

```quiz:
0 0 0 0 0 9 0 0 2
0 9 0 0 4 0 0 0 8
0 7 2 3 0 0 9 0 0
0 3 0 9 0 7 0 0 1
2 0 0 0 0 0 0 0 3
9 0 0 1 0 3 0 5 0
0 0 4 0 0 1 6 8 0
8 0 0 0 9 0 0 2 0
7 0 9 6 0 0 0 0 0

-------------------

solution: 6.956 ms
3 8 6 5 7 9 4 1 2
1 9 5 2 4 6 3 7 8
4 7 2 3 1 8 9 6 5
6 3 8 9 5 7 2 4 1
2 5 1 8 6 4 7 9 3
9 4 7 1 2 3 8 5 6
5 2 4 7 3 1 6 8 9
8 6 3 4 9 5 1 2 7
7 1 9 6 8 2 5 3 4
```

Go

Solution using Knuth's DLX. This code follows his paper fairly closely. Input to function solve is an 81 character string. This seems to be a conventional computer representation for Sudoku puzzles.

```package main

import "fmt"

// sudoku puzzle representation is an 81 character string
var puzzle = "" +
"394  267 " +
"   3  4  " +
"5  69  2 " +
" 45   9  " +
"6       7" +
"  7   58 " +
" 1  67  8" +
"  9  8   " +
" 264  735"

func main() {
printGrid("puzzle:", puzzle)
if s := solve(puzzle); s == "" {
fmt.Println("no solution")
} else {
printGrid("solved:", s)
}
}

// print grid (with title) from 81 character string
func printGrid(title, s string) {
fmt.Println(title)
for r, i := 0, 0; r < 9; r, i = r+1, i+9 {
fmt.Printf("%c %c %c | %c %c %c | %c %c %c\n", s[i], s[i+1], s[i+2],
s[i+3], s[i+4], s[i+5], s[i+6], s[i+7], s[i+8])
if r == 2 || r == 5 {
fmt.Println("------+-------+------")
}
}
}

// solve puzzle in 81 character string format.
// if solved, result is 81 character string.
// if not solved, result is the empty string.
func solve(u string) string {
// construct an dlx object with 324 constraint columns.
// other than the number 324, this is not specific to sudoku.
d := newDlxObject(324)
// now add constraints that define sudoku rules.
for r, i := 0, 0; r < 9; r++ {
for c := 0; c < 9; c, i = c+1, i+1 {
b := r/3*3 + c/3
n := int(u[i] - '1')
if n >= 0 && n < 9 {
d.addRow([]int{i, 81 + r*9 + n, 162 + c*9 + n,
243 + b*9 + n})
} else {
for n = 0; n < 9; n++ {
d.addRow([]int{i, 81 + r*9 + n, 162 + c*9 + n,
243 + b*9 + n})
}
}
}
}
// run dlx.  not sudoku specific.
d.search()
// extract the sudoku-specific 81 character result from the dlx solution.
return d.text()
}

// Knuth's data object
type x struct {
c          *y
u, d, l, r *x
// except x0 is not Knuth's.  it's pointer to first constraint in row,
// so that the sudoku string can be constructed from the dlx solution.
x0 *x
}

// Knuth's column object
type y struct {
x
s int // size
n int // name
}

// an object to hold the matrix and solution
type dlx struct {
ch []y  // all column headers
h  *y   // ch[0], the root node
o  []*x // solution
}

// constructor creates the column headers but no rows.
func newDlxObject(nCols int) *dlx {
ch := make([]y, nCols+1)
h := &ch[0]
d := &dlx{ch, h, nil}
h.c = h
h.l = &ch[nCols].x
ch[nCols].r = &h.x
nh := ch[1:]
for i := range ch[1:] {
hi := &nh[i]
ix := &hi.x
hi.n = i
hi.c = hi
hi.u = ix
hi.d = ix
hi.l = &h.x
h.r = ix
h = hi
}
return d
}

// rows define constraints
func (d *dlx) addRow(nr []int) {
if len(nr) == 0 {
return
}
r := make([]x, len(nr))
x0 := &r[0]
for x, j := range nr {
ch := &d.ch[j+1]
ch.s++
np := &r[x]
np.c = ch
np.u = ch.u
np.d = &ch.x
np.l = &r[(x+len(r)-1)%len(r)]
np.r = &r[(x+1)%len(r)]
np.u.d, np.d.u, np.l.r, np.r.l = np, np, np, np
np.x0 = x0
}
}

// extracts 81 character sudoku string
func (d *dlx) text() string {
b := make([]byte, len(d.o))
for _, r := range d.o {
x0 := r.x0
b[x0.c.n] = byte(x0.r.c.n%9) + '1'
}
return string(b)
}

// the dlx algorithm
func (d *dlx) search() bool {
h := d.h
j := h.r.c
if j == h {
return true
}
c := j
for minS := j.s; ; {
j = j.r.c
if j == h {
break
}
if j.s < minS {
c, minS = j, j.s
}
}

cover(c)
k := len(d.o)
d.o = append(d.o, nil)
for r := c.d; r != &c.x; r = r.d {
d.o[k] = r
for j := r.r; j != r; j = j.r {
cover(j.c)
}
if d.search() {
return true
}
r = d.o[k]
c = r.c
for j := r.l; j != r; j = j.l {
uncover(j.c)
}
}
d.o = d.o[:len(d.o)-1]
uncover(c)
return false
}

func cover(c *y) {
c.r.l, c.l.r = c.l, c.r
for i := c.d; i != &c.x; i = i.d {
for j := i.r; j != i; j = j.r {
j.d.u, j.u.d = j.u, j.d
j.c.s--
}
}
}

func uncover(c *y) {
for i := c.u; i != &c.x; i = i.u {
for j := i.l; j != i; j = j.l {
j.c.s++
j.d.u, j.u.d = j, j
}
}
c.r.l, c.l.r = &c.x, &c.x
}
```
Output:
```puzzle:
3 9 4 |     2 | 6 7
| 3     | 4
5     | 6 9   |   2
------+-------+------
4 5 |       | 9
6     |       |     7
7 |       | 5 8
------+-------+------
1   |   6 7 |     8
9 |     8 |
2 6 | 4     | 7 3 5
solved:
3 9 4 | 8 5 2 | 6 7 1
2 6 8 | 3 7 1 | 4 5 9
5 7 1 | 6 9 4 | 8 2 3
------+-------+------
1 4 5 | 7 8 3 | 9 6 2
6 8 2 | 9 4 5 | 3 1 7
9 3 7 | 1 2 6 | 5 8 4
------+-------+------
4 1 3 | 5 6 7 | 2 9 8
7 5 9 | 2 3 8 | 1 4 6
8 2 6 | 4 1 9 | 7 3 5
```

Golfscript

Imprime todas las soluciones posibles, sale con un error, pero funciona.

```'Solution:'
;'2 8 4 3 7 5 1 6 9
0 0 9 2 0 0 0 0 7
0 0 1 0 0 4 0 0 2
0 5 0 0 0 0 8 0 0
0 0 8 0 0 0 9 0 0
0 0 6 0 0 0 0 4 0
9 0 0 1 0 0 5 0 0
8 0 0 0 0 7 6 0 4
4 2 5 6 8 9 7 3 1'
{9/[n]*puts}:p; #optional formatting

~]{:@0?:^~!{@p}*10,@9/^9/=-@^9%>9%-@3/^9%3/>3%3/^27/={+}*-{@^<\+@1^+>+}/1}do```

Groovy

Non-guessing part is iterative. Guessing part is recursive. Implementation uses exception handling to back out of bad guesses.

I consider this a "brute force" solution of sorts, in that it is the same method I use when solving Sudokus manually.

```final CELL_VALUES = ('1'..'9')

class GridException extends Exception {
GridException(String message) { super(message) }
}

def string2grid = { string ->
assert string.size() == 81
(0..8).collect { i -> (0..8).collect { j -> string[9*i+j] } }
}

def gridRow = { grid, slot -> grid[slot.i] as Set }

def gridCol = { grid, slot -> grid.collect { it[slot.j] } as Set }

def gridBox = { grid, slot ->
def t, l; (t, l) = [slot.i.intdiv(3)*3, slot.j.intdiv(3)*3]
(0..2).collect { row -> (0..2).collect { col -> grid[t+row][l+col] } }.flatten() as Set
}

def slotList = { grid ->
def slots = (0..8).collect { i -> (0..8).findAll { j -> grid[i][j] == '.' } \
.collect {j -> [i: i, j: j] } }.flatten()
}

def assignCandidates = { grid, slots = slotList(grid) ->
slots.each { slot ->
def unavailable = [gridRow, gridCol, gridBox].collect { it(grid, slot) }.sum() as Set
slot.candidates = CELL_VALUES - unavailable
}
slots.sort { - it.candidates.size() }
if (slots && ! slots[-1].candidates) {
throw new GridException('Invalid Sudoku Grid, overdetermined slot: ' + slots[-1])
}
slots
}

def isSolved = { grid -> ! (grid.flatten().find { it == '.' }) }

def solve
solve = { grid ->
def slots = assignCandidates(grid)
if (! slots) { return grid }
while (slots[-1].candidates.size() == 1) {
def slot = slots.pop()
grid[slot.i][slot.j] = slot.candidates[0]
if (! slots) { return grid }
slots = assignCandidates(grid, slots)
}
if (! slots) { return grid }
def slot = slots.pop()
slot.candidates.each {
if (! isSolved(grid)) {
try {
def sGrid = grid.collect { row -> row.collect { cell -> cell } }
sGrid[slot.i][slot.j] = it
grid = solve(sGrid)
} catch (GridException ge) {
grid[slot.i][slot.j] = '.'
}
}
}
if (!isSolved(grid)) {
slots = assignCandidates(grid)
throw new GridException('Invalid Sudoku Grid, underdetermined slots: ' + slots)
}
grid
}
```

Test/Benchmark Cases

Mentions of "exceptionally difficult" example in Wikipedia refer to this (former) page: [Exceptionally difficult Sudokus]

```def sudokus = [
//Used in Curry solution:                             ~ 0.1 seconds
'819..5.....2...75..371.4.6.4..59.1..7..3.8..2..3.62..7.5.7.921..64...9.....2..438',

//Used in Perl and PicoLisp solutions:                ~ 0.1 seconds
'53..247....2...8..1..7.39.2..8.72.49.2.98..7.79.....8.....3.5.696..1.3...5.69..1.',

//Used in Fortran solution:                           ~ 0.1 seconds
'..3.2.6..9..3.5..1..18.64....81.29..7.......8..67.82....26.95..8..2.3..9..5.1.3..',

//Used in many other solutions, notably Algol 68:     ~ 0.1 seconds
'394..267....3..4..5..69..2..45...9..6.......7..7...58..1..67..8..9..8....264..735',

//Used in C# solution:                                ~ 0.2 seconds
'97.3...6..6.75.........8.5.......67.....3.....539..2..7...25.....2.1...8.4...73..',

//Used in Oz solution:                                ~ 0.2 seconds
'4......6.5...8.9..3....1....2.7....1.9.....4.8....3.5....2....7..6.5...8.1......6',

//Used in many other solutions, notably C++:          ~ 0.3 seconds
'85...24..72......9..4.........1.7..23.5...9...4...........8..7..17..........36.4.',

//Used in VBA solution:                               ~ 0.3 seconds
'..1..5.7.92.6.......8...6...9..2.4.1.........3.4.8..9...7...3.......7.69.1.8..7..',

//Used in Forth solution:                             ~ 0.8 seconds
'.9...4..7.....79..8........4.58.....3.......2.....97.6........4..35.....2..6...8.',

//3rd "exceptionally difficult" example in Wikipedia: ~ 2.3 seconds
'12.3....435....1....4........54..2..6...7.........8.9...31..5.......9.7.....6...8',

//Used in Curry solution:                             ~ 2.4 seconds
'9..2..5...4..6..3...3.....6...9..2......5..8...7..4..37.....1...5..2..4...1..6..9',

//"AL Escargot", so-called "hardest sudoku" (HA!):    ~ 3.0 seconds
'1....7.9..3..2...8..96..5....53..9...1..8...26....4...3......1..4......7..7...3..',

//1st "exceptionally difficult" example in Wikipedia: ~ 6.5 seconds
'12.4..3..3...1..5...6...1..7...9.....4.6.3.....3..2...5...8.7....7.....5.......98',

//Used in Bracmat and Scala solutions:                ~ 6.7 seconds
'..............3.85..1.2.......5.7.....4...1...9.......5......73..2.1........4...9',

//2nd "exceptionally difficult" example in Wikipedia: ~ 8.8 seconds
'.......39.....1..5..3.5.8....8.9...6.7...2...1..4.......9.8..5..2....6..4..7.....',

//Used in MATLAB solution:                            ~15   seconds
'....839..1......3...4....7..42.3....6.......4....7..1..2........8...92.....25...6',

//4th "exceptionally difficult" example in Wikipedia: ~29   seconds
'..3......4...8..36..8...1...4..6..73...9..........2..5..4.7..686........7..6..5..']

sudokus.each { sudoku ->
def grid = string2grid(sudoku)
println '\nPUZZLE'
grid.each { println it }

println '\nSOLUTION'
def start = System.currentTimeMillis()
def solution = solve(grid)
def elapsed = (System.currentTimeMillis() - start)/1000
solution.each { println it }
println "\nELAPSED: \${elapsed} seconds"
}
```
Output:
(last only)
```PUZZLE
[., ., 3, ., ., ., ., ., .]
[4, ., ., ., 8, ., ., 3, 6]
[., ., 8, ., ., ., 1, ., .]
[., 4, ., ., 6, ., ., 7, 3]
[., ., ., 9, ., ., ., ., .]
[., ., ., ., ., 2, ., ., 5]
[., ., 4, ., 7, ., ., 6, 8]
[6, ., ., ., ., ., ., ., .]
[7, ., ., 6, ., ., 5, ., .]

SOLUTION
[1, 2, 3, 4, 5, 6, 7, 8, 9]
[4, 5, 7, 1, 8, 9, 2, 3, 6]
[9, 6, 8, 3, 2, 7, 1, 5, 4]
[2, 4, 9, 5, 6, 1, 8, 7, 3]
[5, 7, 6, 9, 3, 8, 4, 1, 2]
[8, 3, 1, 7, 4, 2, 6, 9, 5]
[3, 1, 4, 2, 7, 5, 9, 6, 8]
[6, 9, 5, 8, 1, 4, 3, 2, 7]
[7, 8, 2, 6, 9, 3, 5, 4, 1]

ELAPSED: 28.978 seconds```

Java

```public class Sudoku
{
private int mBoard[][];
private int mBoardSize;
private int mBoxSize;
private boolean mRowSubset[][];
private boolean mColSubset[][];
private boolean mBoxSubset[][];

public Sudoku(int board[][]) {
mBoard = board;
mBoardSize = mBoard.length;
mBoxSize = (int)Math.sqrt(mBoardSize);
initSubsets();
}

public void initSubsets() {
mRowSubset = new boolean[mBoardSize][mBoardSize];
mColSubset = new boolean[mBoardSize][mBoardSize];
mBoxSubset = new boolean[mBoardSize][mBoardSize];
for(int i = 0; i < mBoard.length; i++) {
for(int j = 0; j < mBoard.length; j++) {
int value = mBoard[i][j];
if(value != 0) {
setSubsetValue(i, j, value, true);
}
}
}
}

private void setSubsetValue(int i, int j, int value, boolean present) {
mRowSubset[i][value - 1] = present;
mColSubset[j][value - 1] = present;
mBoxSubset[computeBoxNo(i, j)][value - 1] = present;
}

public boolean solve() {
return solve(0, 0);
}

public boolean solve(int i, int j) {
if(i == mBoardSize) {
i = 0;
if(++j == mBoardSize) {
return true;
}
}
if(mBoard[i][j] != 0) {
return solve(i + 1, j);
}
for(int value = 1; value <= mBoardSize; value++) {
if(isValid(i, j, value)) {
mBoard[i][j] = value;
setSubsetValue(i, j, value, true);
if(solve(i + 1, j)) {
return true;
}
setSubsetValue(i, j, value, false);
}
}

mBoard[i][j] = 0;
return false;
}

private boolean isValid(int i, int j, int val) {
val--;
boolean isPresent = mRowSubset[i][val] || mColSubset[j][val] || mBoxSubset[computeBoxNo(i, j)][val];
return !isPresent;
}

private int computeBoxNo(int i, int j) {
int boxRow = i / mBoxSize;
int boxCol = j / mBoxSize;
return boxRow * mBoxSize + boxCol;
}

public void print() {
for(int i = 0; i < mBoardSize; i++) {
if(i % mBoxSize == 0) {
System.out.println(" -----------------------");
}
for(int j = 0; j < mBoardSize; j++) {
if(j % mBoxSize == 0) {
System.out.print("| ");
}
System.out.print(mBoard[i][j] != 0 ? ((Object) (Integer.valueOf(mBoard[i][j]))) : "-");
System.out.print(' ');
}

System.out.println("|");
}

System.out.println(" -----------------------");
}

public static void main(String[] args) {
int[][] board = {
{8, 5, 0, 0, 0, 2, 4, 0, 0},
{7, 2, 0, 0, 0, 0, 0, 0, 9},
{0, 0, 4, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 1, 0, 7, 0, 0, 2},
{3, 0, 5, 0, 0, 0, 9, 0, 0},
{0, 4, 0, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 8, 0, 0, 7, 0},
{0, 1, 7, 0, 0, 0, 0, 0, 0},
{0, 0, 0, 0, 3, 6, 0, 4, 0}
};
Sudoku s = new Sudoku(board);
System.out.print("Starting grid:\n");
s.print();
if (s.solve()) {
System.out.print("\nSolution:\n");
s.print();
} else {
System.out.println("\nUnsolvable!");
}
}
}
```
Output:
```Starting grid:
-----------------------
| 8 5 - | - - 2 | 4 - - |
| 7 2 - | - - - | - - 9 |
| - - 4 | - - - | - - - |
-----------------------
| - - - | 1 - 7 | - - 2 |
| 3 - 5 | - - - | 9 - - |
| - 4 - | - - - | - - - |
-----------------------
| - - - | - 8 - | - 7 - |
| - 1 7 | - - - | - - - |
| - - - | - 3 6 | - 4 - |
-----------------------

Solution:
-----------------------
| 8 5 9 | 6 1 2 | 4 3 7 |
| 7 2 3 | 8 5 4 | 1 6 9 |
| 1 6 4 | 3 7 9 | 5 2 8 |
-----------------------
| 9 8 6 | 1 4 7 | 3 5 2 |
| 3 7 5 | 2 6 8 | 9 1 4 |
| 2 4 1 | 5 9 3 | 7 8 6 |
-----------------------
| 4 3 2 | 9 8 1 | 6 7 5 |
| 6 1 7 | 4 2 5 | 8 9 3 |
| 5 9 8 | 7 3 6 | 2 4 1 |
-----------------------
```

JavaScript

ES6

```//-------------------------------------------[ Dancing Links and Algorithm X ]--
/**
* The doubly-doubly circularly linked data object.
* Data object X
*/
class DoX {
/**
* @param {string} V
* @param {!DoX=} H
*/
constructor(V, H) {
this.V = V;
this.L = this;
this.R = this;
this.U = this;
this.D = this;
this.S = 1;
this.H = H || this;
H && (H.S += 1);
}
}

/**
* Helper function to help build a horizontal doubly linked list.
* @param {!DoX} e An existing node in the list.
* @param {!DoX} n A new node to add to the right of the existing node.
* @return {!DoX}
*/
const addRight = (e, n) => {
n.R = e.R;
n.L = e;
e.R.L = n;
return e.R = n;
};

/**
* Helper function to help build a vertical doubly linked list.
* @param {!DoX} e An existing node in the list.
* @param {!DoX} n A new node to add below the existing node.
*/
const addBelow = (e, n) => {
n.D = e.D;
n.U = e;
e.D.U = n;
return e.D = n;
};

/**
* Verbatim copy of DK's search algorithm. The meat of the DLX algorithm.
* @param {!DoX} h The root node.
* @param {!Array<!DoX>} s The solution array.
*/
const search = function(h, s) {
if (h.R == h) {
printSol(s);
} else {
let c = chooseColumn(h);
cover(c);
for (let r = c.D; r != c; r = r.D) {
s.push(r);
for (let j = r.R; r !=j; j = j.R) {
cover(j.H);
}
search(h, s);
r = s.pop();
for (let j = r.R; j != r; j = j.R) {
uncover(j.H);
}
}
uncover(c);
}
};

/**
* Verbatim copy of DK's algorithm for choosing the next column object.
* @param {!DoX} h
* @return {!DoX}
*/
const chooseColumn = h => {
let s = Number.POSITIVE_INFINITY;
let c = h;
for(let j = h.R; j != h; j = j.R) {
if (j.S < s) {
c = j;
s = j.S;
}
}
return c;
};

/**
* Verbatim copy of DK's cover algorithm
* @param {!DoX} c
*/
const cover = c => {
c.L.R = c.R;
c.R.L = c.L;
for (let i = c.D; i != c; i = i.D) {
for (let j = i.R; j != i; j = j.R) {
j.U.D = j.D;
j.D.U = j.U;
j.H.S = j.H.S - 1;
}
}
};

/**
* Verbatim copy of DK's cover algorithm
* @param {!DoX} c
*/
const uncover = c => {
for (let i = c.U; i != c; i = i.U) {
for (let j = i.L; i != j; j = j.L) {
j.H.S = j.H.S + 1;
j.U.D = j;
j.D.U = j;
}
}
c.L.R = c;
c.R.L = c;
};

//-----------------------------------------------------------[ Print Helpers ]--
/**
* Given the standard string format of a grid, print a formatted view of it.
* @param {!string|!Array} a
*/
const printGrid = function(a) {

const getChar = c => {
let r = Number(c);
if (isNaN(r)) { return c }

let o = 48;
if (r > 9 && r < 36) { o = 55 }
if (r >= 36) { o = 61 }
return String.fromCharCode(r + o)
};

a = 'string' == typeof a ? a.split('') : a;

let U = Math.sqrt(a.length);
let N = Math.sqrt(U);
let line = new Array(N).fill('+').reduce((p, c) => {
p.push(... Array.from(new Array(1 + N*2).fill('-')));
p.push(c);
return p;
}, ['\n+']).join('') + '\n';

a = a.reduce(function(p, c, i) {
let d = i && !(i % U), G = i && !(i % N);
i = !(i % (U * N));
d && !i && (p += '|\n| ');
d && i && (p += '|');
i && (p = '' + p + line + '| ');
return '' + p + (G && !d ? '| ' : '') + getChar(c) + ' ';
}, '') + '|' + line;
console.log(a);

};

/**
* Given a search solution, print the resultant grid.
* @param {!Array<!DoX>} a An array of data objects
*/
const printSol = a => {
printGrid(a.reduce((p, c) => {
let [i, v] = c.V.split(':');
p[i * 1] = v;
return p;
}, new Array(a.length).fill('.')));
};

//----------------------------------------------[ Grid to Exact cover Matrix ]--
/**
* Helper to get some meta about the grid.
* @param {!string} s The standard string representation of a grid.
* @return {!Array}
*/
const gridMeta = s => {
const g = s.split('');
const cellCount = g.length;
const tokenCount = Math.sqrt(cellCount);
const N = Math.sqrt(tokenCount);
const g2D = g.map(e => isNaN(e * 1) ?
new Array(tokenCount).fill(1).map((_, i) => i + 1) :
[e * 1]);
return [cellCount, N, tokenCount, g2D];
};

/**
* Given a cell grid index, return the row, column and box indexes.
* @param {!number} n The n-value of the grid. 3 for a 9x9 sudoku.
* @return {!function(!number): !Array<!number>}
*/
const indexesN = n => i => {
let c = Math.floor(i / (n * n));
i %= n * n;
return [c, i, Math.floor(c / n) * n + Math.floor(i / n)];
};

/**
* Given a puzzle string, reduce it to an exact-cover matrix and use
* Donald Knuth's DLX algorithm to solve it.
* @param puzString
*/
const reduceGrid = puzString => {

printGrid(puzString);
const [
numCells,   // The total number of cells in a grid (81 for a 9x9 grid)
N,          // the 'n' value of the grid. (3 for a 9x9 grid)
U,          // The total number of unique tokens to be placed.
g2D         // A 2D array representation of the grid, with each element
// being an array of candidates for a cell. Known cells are
// single element arrays.
] = gridMeta(puzString);

const getIndex = indexesN(N);

/**
* Its length is 4 times the grid's size. This is to be able to encode
* each of the 4 Sudoku constrains, onto each of the cells of the grid.
* The array is initialised with unlinked DoX nodes, but in the next step
* those nodes are all linked.
* @type {!Array.<!DoX>}
*/
const headRow = new Array(4 * numCells)
.fill('')
.map((_, i) => new DoX(`H\${i}`));

/**
* The header row root object. This is circularly linked to be to the left
* It is used as the entry point into the DLX algorithm.
* @type {!DoX}
*/
let H = new DoX('ROOT');

/**
* Transposed the sudoku puzzle into a exact cover matrix, so it can be passed
* to the DLX algorithm to solve.
*/
for (let i = 0; i < numCells; i++) {
const [ri, ci, bi] = getIndex(i);
g2D[i].forEach(num => {
let id = `\${i}:\${num}`;
let candIdx = num - 1;

// The 4 columns that we will populate.