# Solve a Holy Knight's tour

Solve a Holy Knight's tour
You are encouraged to solve this task according to the task description, using any language you may know.

Chess coaches have been known to inflict a kind of torture on beginners by taking a chess board, placing pennies on some squares and requiring that a Knight's tour be constructed that avoids the squares with pennies.

This kind of knight's tour puzzle is similar to   Hidato.

The present task is to produce a solution to such problems. At least demonstrate your program by solving the following:

Example
0 0 0
0   0 0
0 0 0 0 0 0 0
0 0 0     0   0
0   0     0 0 0
1 0 0 0 0 0 0
0 0   0
0 0 0

Note that the zeros represent the available squares, not the pennies.

Extra credit is available for other interesting examples.

## 11l

Translation of: Python
V moves = [
[-1, -2], [1, -2], [-1, 2], [1, 2],
[-2, -1], [-2, 1], [2, -1], [2, 1]
]

F solve(&pz, sz, sx, sy, idx, cnt)
I idx > cnt
R 1

L(i) 0 .< :moves.len
V x = sx + :moves[i][0]
V y = sy + :moves[i][1]
I sz > x & x > -1 & sz > y & y > -1 & pz[x][y] == 0
pz[x][y] = idx
I 1 == solve(&pz, sz, x, y, idx + 1, cnt)
R 1
pz[x][y] = 0
R 0

F find_solution(pz, sz)
V p = [[-1] * sz] * sz
V idx = 0
V x = 0
V y = 0
V cnt = 0
L(j) 0 .< sz
L(i) 0 .< sz
I pz[idx] == ‘x’
p[i][j] = 0
cnt++
E I pz[idx] == ‘s’
p[i][j] = 1
cnt++
x = i
y = j
idx++

I 1 == solve(&p, sz, x, y, 2, cnt)
L(j) 0 .< sz
L(i) 0 .< sz
I p[i][j] != -1
print(‘ #02’.format(p[i][j]), end' ‘’)
E
print(‘   ’, end' ‘’)
print()
E
print(‘Cannot solve this puzzle!’)

find_solution(‘.xxx.....x.xx....xxxxxxxxxx..x.xx.x..xxxsxxxxxx...xx.x.....xxx..’, 8)
print()
find_solution(‘.....s.x..........x.x.........xxxxx.........xxx.......x..x.x..x..xxxxx...xxxxx..xx.....xx..xxxxx...xxxxx..x..x.x..x.......xxx.........xxxxx.........x.x..........x.x.....’, 13)
Output:
17 14 29
28    18 15
13 16 27 30 19 32 07
25 02 11       06    20
12    26       31 08 33
01 24 03 10 05 34 21
36 23    09
04 35 22

01    05
10    12
02 13 04 09 06
08 11 14
36       03    07       16
35 42 33 44 37          15 20 27 22 25
38 41                17 24
39 34 43 32 45          19 28 21 26 23
40       31    29       18
46 51 56
52 55 30 47 50
48    54
53    49

This solution uses the package Knights_Tour from Knight's Tour#Ada. The board is quadratic, the size of the board is read from the command line and the board itself is read from the standard input. For the board itself, Space and Minus indicate a no-go (i.e., a coin on the board), all other characters represent places the knight must visit. A '1' represents the start point. Ill-formatted input will crash the program.

procedure Holy_Knight is

package KT is new Knights_Tour(Size => Size);
Board: KT.Tour := (others => (others => Natural'Last));

Start_X, Start_Y: KT.Index:= 1; -- default start place (1,1)
S: String(KT.Index);
I: Positive := KT.Index'First;
begin
-- read the board from standard input
while not Ada.Text_IO.End_Of_File and I <= Size loop
for J in KT.Index loop
if S(J) = ' ' or S(J) = '-' then
Board(I,J) := Natural'Last;
elsif S(J) = '1' then
Start_X := I; Start_Y := J;  Board(I,J) := 1;
else Board(I,J) := 0;
end if;
end loop;
I := I + 1;
end loop;

-- print the board
& Natural'Image(KT.Count_Moves(Board)) & "):");
KT.Tour_IO(Board, Width => 1);

-- search for the tour and print it
KT.Tour_IO(KT.Warnsdorff_Get_Tour(Start_X, Start_Y, Board));
end Holy_Knight;
Output:
>holy_knight 8 < standard_problem.txt
Start Configuration (Length: 36):
--000---
--0-00--
-0000000
000--0-0
0-0--000
1000000-
--00-0--
---000--

Tour:
-   -  30  15  20   -   -   -
-   -  21   -  29  16   -   -
-  33  14  31  22  19   6  17
13  36  23   -   -  28   -   8
34   -  32   -   -   7  18   5
1  12  35  24  27   4   9   -
-   -   2  11   -  25   -   -
-   -   -  26   3  10   -   -

### Extra Credit

The Holy_Knight program can immediately be used to tackle "more interesting" problems, such as those from New Knight's Tour Puzzles and Graphs. Here is one sample solution:

>holy_knight 13 < problem10.txt
Start Configuration (Length: 56):
-----1-0-----
-----0-0-----
----00000----
-----000-----
--0--0-0--0--
00000---00000
--00-----00--
00000---00000
--0--0-0--0--
-----000-----
----00000----
-----0-0-----
-----0-0-----

Tour:
-   -   -   -   -   1   -  27   -   -   -   -   -
-   -   -   -   -  56   -   2   -   -   -   -   -
-   -   -   -  24   3  28  55  26   -   -   -   -
-   -   -   -   -  54  25   4   -   -   -   -   -
-   -  50   -   -  23   -  29   -   -   6   -   -
51  20  47  22  53   -   -   -   5  30   9  32   7
-   -  52  49   -   -   -   -   -  33  36   -   -
19  48  21  46  17   -   -   -  37  10  31   8  35
-   -  18   -   -  45   -  11   -   -  34   -   -
-   -   -   -   -  16  41  38   -   -   -   -   -
-   -   -   -  42  39  44  15  12   -   -   -   -
-   -   -   -   -  14   -  40   -   -   -   -   -
-   -   -   -   -  43   -  13   -   -   -   -   -

## ALGOL 68

Uses a modified version of the Knight's Tour#ALGOL 68 solution.

# directions for moves #
INT nne = 1, ne  = 2, se = 3, sse = 4;
INT ssw = 5, sw  = 6, nw = 7, nnw = 8;

INT lowest move  = nne;
INT highest move = nnw;

# the vertical position changes of the moves                             #
[]INT offset v = ( -2, -1,  1,  2,  2,  1, -1, -2 );
# the horizontal position changes of the moves                           #
[]INT offset h = (  1,  2,  2,  1, -1, -2, -2, -1 );

MODE SQUARE = STRUCT( INT move      # the number of the move that caused #
# the knight to reach this square    #
, INT direction # the direction of the move that     #
# brought the knight here - one of   #
# nne, ne, se, sse, ssw, sw, nw or   #
# nnw                                #
);
# get the size of the board - must be between 4 and 8                    #
INT board size = 8;
# the board #
[ board size, board size ]SQUARE board;
# starting position #
INT start row := 1;
INT start col := 1;
# the tour will be complete when we have made as many moves              #
# as there are free squares in the initial board                         #
INT  final move    := 0;

# initialise the board setting the free squares from the supplied pttern #
# the pattern has the rows in revers order                               #
PROC initialise board = ( []STRING pattern )VOID:
BEGIN
INT pattern row := UPB board;
FOR row FROM 1 LWB board TO 1 UPB board
DO
FOR col FROM 2 LWB board TO 2 UPB board
DO
IF pattern[ pattern row ][ col ] = "-"
THEN
# can't use this square                              #
board[ row, col ] := ( -1, -1 )
ELSE
# available square                                   #
board[ row, col ] := ( 0, 0 );
final move +:= 1;
IF pattern[ pattern row ][ col ] = "1"
THEN
# have the start position                        #
start row := row;
start col := col
FI
FI
OD;
pattern row -:= 1
OD
END; # initialise board #
# statistics #
INT iterations := 0;
INT backtracks := 0;

# prints the board #
PROC print tour = VOID:
BEGIN
# format "number" into at least two characters #
PROC n2 = ( INT number )STRING:
IF   number < 0
THEN
" -"
ELIF number < 10 AND number >= 0
THEN
" " + whole( number, 0 )
ELSE
whole( number, 0 )
FI; # n2 #
print( ( "     a  b  c  d  e  f  g  h", newline ) );
print( ( "   ________________________", newline ) );
FOR row FROM 1 UPB board BY -1 TO 1 LWB board
DO
print( ( n2( row ) ) );
print( ( "|" ) );

FOR col FROM 2 LWB board TO 2 UPB board
DO
print( ( " " ) );
print( ( n2( move OF board[ row, col ] ) ) )
OD;
print( ( newline ) )
OD
END; # print tour #

# update the board to the first knight's tour found starting from       #
# "start row" and "start col".                                          #
# return TRUE if one was found, FALSE otherwise                         #
PROC find tour = BOOL:
BEGIN

BOOL result       := TRUE;
INT  move number  := 1;
INT  row          := start row;
INT  col          := start col;
INT  direction    := lowest move - 1;
# the first move is to place the knight on the starting square #
board[ row, col ] := ( move number, lowest move - 1 );
# attempt to find a sequence of moves that will reach each square once #
WHILE
move number < final move AND result
DO
IF direction < highest move
THEN
# try the next move from this position #
direction +:= 1;
INT new row = row + offset v[ direction ];
INT new col = col + offset h[ direction ];
IF  new row <= 1 UPB board
AND new row >= 1 LWB board
AND new col <= 2 UPB board
AND new col >= 2 LWB board
THEN
# the move is legal, check the new square is unused #
IF move OF board[ new row, new col ] = 0
THEN
# can move here #
iterations       +:= 1;
row               := new row;
col               := new col;
move number      +:= 1;
board[ row, col ] := ( move number, direction );
direction         := lowest move - 1
FI
FI
ELSE
# no more moves from this position - backtrack #
IF move number = 1
THEN
# at the starting position - no solution #
result := FALSE
ELSE
# not at the starting position - undo the latest move #
backtracks  +:= 1;
move number -:= 1;
INT curr row := row;
INT curr col := col;
row -:= offset v[ direction OF board[ curr row, curr col ] ];
col -:= offset h[ direction OF board[ curr row, curr col ] ];
# determine which direction to try next #
direction := direction OF board[ curr row, curr col ];
# reset the square we just backtracked from #
board[ curr row, curr col ] := ( 0, 0 )
FI
FI
OD;
result
END; # find tour #

main:(
initialise board( ( "-000----"
, "-0-00---"
, "-0000000"
, "000--0-0"
, "0-0--000"
, "1000000-"
, "--00-0--"
, "---000--"
)
);
IF find tour
THEN
# found a solution #
print tour
ELSE
# couldn't find a solution #
FI;
print( ( iterations, " iterations, ", backtracks, " backtracks", newline ) )
)
Output:
a  b  c  d  e  f  g  h
________________________
8|  - 21 34 25  -  -  -  -
7|  - 24  - 20  7  -  -  -
6|  - 35 22 33 26 11  6  9
5| 23 32 19  -  -  8  - 12
4| 36  - 16  -  - 27 10  5
3|  1 18 31 28 15  4 13  -
2|  -  -  2 17  - 29  -  -
1|  -  -  - 30  3 14  -  -
+578929 iterations,     +578894 backtracks

## Bracmat

This solution can handle different input formats: the widths of the first and the other columns are computed. The cell were to start from should have a unique value, but this value is not prescribed. Non-empty cells (such as the start cell) should contain a character that is different from '-', '.' or white space. The puzzle solver itself is only a few lines long.

( ( Holy-Knight
=     begin colWidth crumbs non-empty pairs path parseLine
, display isolateStartCell minDistance numberElementsAndSort
, parseBoard reverseList rightAlign solve strlen
.   "'non-empty' is a pattern that is used several times in bigger patterns."
& ( non-empty
=
=   %@
: ~( "."
| "-"
| " "
| \t
| \r
| \n
)
)
& ( reverseList
=   a L
.   :?L
& whl'(!arg:%?a ?arg&!a !L:?L)
& !L
)
& (strlen=e.@(!arg:? [?e)&!e)
& ( rightAlign
=   string width
.   !arg:(?width,?string)
& !width+-1*strlen\$!string:?width
&   whl
' ( !width+-1:~<0:?width
& " " !string:?string
)
& str\$!string
)
& ( minDistance
=   board pat1 pat2 minWidth pos1 pos2 pattern
.   !arg:(?board,(=?pat1),(=?pat2))
& -1:?minWidth
& "Construct a pattern using a template.
The pattern finds the smallest distance between any two columns in the input.
Assumption: all columns have the same width and columns are separated by one or
more spaces. The function can also be used to find the width of the first column
by letting pat1 match a new line."
&
' ( ?
(   \$pat1
[?pos1
(? " "|`)
()\$pat2
[?pos2
?
&   !pos2+-1*!pos1
: ( <!minWidth
| ?&!minWidth:<0
)
: ?minWidth
& ~
)
)
: (=?pattern)
& "'pattern', by design, always fails. The interesting part is a side effect:
the column width."
& (@(!board:!pattern)|!minWidth)
)
& ( numberElementsAndSort
=   a sum n
.   0:?sum:?n
& "An evaluated sum is always sorted. The terms are structured so the sorting
order is by row and then by column (both part of 'a')."
&   whl
' ( !arg:%?a ?arg
& 1+!n:?n
& (!a,!n)+!sum:?sum
)
& "return the sorted list (sum) and also the size of a field that can contain
the highest number."
& (!sum.strlen\$!n+1)
)
& ( parseLine
=     line row columnWidth width col
, bins val A M Z cell validPat
.   !arg:(?line,?row,?width,?columnWidth,?bins)
& 0:?col
& "Find the cells and create a pair [row,col] for each. Put each pair in a bin.
There are as many bins as there are different values in cells."
&   '(? (\$!non-empty:?val) ?)
: (=?validPat)
&   whl
' ( @(!line:?cell [!width ?line)
& (   @(!cell:!validPat)
&   (   !bins:?A (!val.?M) ?Z
& !A (!val.(!row.!col) !M) !Z
| (!val.!row.!col) !bins
)
: ?bins
|
)
& !columnWidth:?width
& 1+!col:?col
)
& !bins
)
& ( parseBoard
=   board firstColumnWidth columnWidth,row bins line
.   !arg:?board
&   (   minDistance
\$ (str\$(\r \n !arg),(=\n),!non-empty)
, minDistance\$(!arg,!non-empty,!non-empty)
)
: (?firstColumnWidth,?columnWidth)
& 0:?row
& :?bins
&   whl
' ( @(!board:?line \n ?board)
&     parseLine
\$ (!line,!row,!firstColumnWidth,!columnWidth,!bins)
: ?bins
& (!bins:|1+!row:?row)
)
&     parseLine
\$ (!board,!row,!firstColumnWidth,!columnWidth,!bins)
: ?bins
)
& "Find the first bin with only one pair. Return this pair and the combined pairs in
all remaining bins."
& ( isolateStartCell
=   A begin Z valuedPairs pairs
.   !arg:?A (?.? [1:?begin) ?Z
& !A !Z:?arg
& :?pairs
&   whl
' ( !arg:(?.?valuedPairs) ?arg
& !valuedPairs !pairs:?pairs
)
& (!begin.!pairs)
)
& ( display
=   board solution row col x y n colWidth
.   !arg:(?board,?solution,?colWidth)
& out\$!board
& 0:?row
& -1:?col
&   whl
' ( !solution:((?y.?x),?n)+?solution
&   whl
' ( !row:<!y
& 1+!row:?row
& -1:?col
& put\$\n
)
&   whl
' ( 1+!col:?col:<!x
& put\$(rightAlign\$(!colWidth,))
)
& put\$(rightAlign\$(!colWidth,!n))
)
& put\$\n
)
& ( solve
=   A Z x y crumbs pairs X Y solution
.   !arg:((?y.?x),?crumbs,?pairs)
& ( !pairs:&(!y.!x) !crumbs
|     !pairs
:   ?A
( (?Y.?X) ?Z
&   (!x+-1*!X)*(!y+-1*!Y)
: (2|-2)
&     solve
\$ ( (!Y.!X)
, (!y.!x) !crumbs
, !A !Z
)
: ?solution
)
& !solution
)
)
& ( isolateStartCell\$(parseBoard\$!arg):(?begin.?pairs)
| out\$"Sorry, I cannot identify a start cell."&~
)
& solve\$(!begin,,!pairs):?crumbs
&   numberElementsAndSort\$(reverseList\$!crumbs)
: (?path.?colWidth)
& display\$(!arg,!path,!colWidth)
)
&     "

0 0 0
0   0 0
0 0 0 0 0 0 0
0 0 0     0   0
0   0     0 0 0
1 0 0 0 0 0 0
0 0   0
0 0 0
"
"
-----1-0-----
-----0-0-----
----00000----
-----000-----
--0--0-0--0--
00000---00000
--00-----00--
00000---00000
--0--0-0--0--
-----000-----
----00000----
-----0-0-----
-----0-0-----"
: ?boards
& whl'(!boards:%?board ?boards&Holy-Knight\$!board)
& done
);

Output:

0 0 0
0   0 0
0 0 0 0 0 0 0
0 0 0     0   0
0   0     0 0 0
1 0 0 0 0 0 0
0 0   0
0 0 0

21 30 19
36    22 29
31 20 35 18 23 28 25
15 34 17       26     8
32    14        9 24 27
1 16 33 10 13  4  7
2  5    11
12  3  6

-----1-0-----
-----0-0-----
----00000----
-----000-----
--0--0-0--0--
00000---00000
--00-----00--
00000---00000
--0--0-0--0--
-----000-----
----00000----
-----0-0-----
-----0-0-----
1    27
26    56
30 55  2 25 28
24 29 54
36       31     3       50
37 34 39 32 23          53  4 47  6 51
22 35                49 52
21 38 33 40 19           9 46  5 48  7
20       41    45        8
18 43 10
42 11 14 17 44
16    12
13    15

## C#

The same solver can solve Hidato, Holy Knight's Tour, Hopido and Numbrix puzzles.
The input can be an array of strings if each cell is one character. The length of the first row must be the number of columns in the puzzle.
Any non-numeric value indicates a no-go.
If there are cells that require more characters, then a 2-dimensional array of ints must be used. Any number < 0 indicates a no-go.
The puzzle can be made circular (the end cell must connect to the start cell). In that case, no start cell needs to be given.

using System.Collections;
using System.Collections.Generic;
using static System.Console;
using static System.Math;
using static System.Linq.Enumerable;

public class Solver
{
private static readonly (int dx, int dy)[]
//other puzzle types elided
knightMoves = {(1,-2),(2,-1),(2,1),(1,2),(-1,2),(-2,1),(-2,-1),(-1,-2)};

private (int dx, int dy)[] moves;

public static void Main()
{
var knightSolver = new Solver(knightMoves);
Print(knightSolver.Solve(true,
".000....",
".0.00...",
".0000000",
"000..0.0",
"0.0..000",
"1000000.",
"..00.0..",
"...000.."));

Print(knightSolver.Solve(true,
".....0.0.....",
".....0.0.....",
"....00000....",
".....000.....",
"..0..0.0..0..",
"00000...00000",
"..00.....00..",
"00000...00000",
"..0..0.0..0..",
".....000.....",
"....00000....",
".....0.0.....",
".....0.0....."
));
}

public Solver(params (int dx, int dy)[] moves) => this.moves = moves;

public int[,] Solve(bool circular, params string[] puzzle)
{
var (board, given, count) = Parse(puzzle);
return Solve(board, given, count, circular);
}

public int[,] Solve(bool circular, int[,] puzzle)
{
var (board, given, count) = Parse(puzzle);
return Solve(board, given, count, circular);
}

private int[,] Solve(int[,] board, BitArray given, int count, bool circular)
{
var (height, width) = (board.GetLength(0), board.GetLength(1));
bool solved = false;
for (int x = 0; x < height && !solved; x++) {
solved = Range(0, width).Any(y => Solve(board, given, circular, (height, width), (x, y), count, (x, y), 1));
if (solved) return board;
}
return null;
}

private bool Solve(int[,] board, BitArray given, bool circular,
(int h, int w) size, (int x, int y) start, int last, (int x, int y) current, int n)
{
var (x, y) = current;
if (x < 0 || x >= size.h || y < 0 || y >= size.w) return false;
if (board[x, y] < 0) return false;
if (given[n - 1]) {
if (board[x, y] != n) return false;
} else if (board[x, y] > 0) return false;
board[x, y] = n;
if (n == last) {
if (!circular || AreNeighbors(start, current)) return true;
}
for (int i = 0; i < moves.Length; i++) {
var move = moves[i];
if (Solve(board, given, circular, size, start, last, (x + move.dx, y + move.dy), n + 1)) return true;
}
if (!given[n - 1]) board[x, y] = 0;
return false;

bool AreNeighbors((int x, int y) p1, (int x, int y) p2) => moves.Any(m => (p2.x + m.dx, p2.y + m.dy).Equals(p1));
}

private static (int[,] board, BitArray given, int count) Parse(string[] input)
{
(int height, int width) = (input.Length, input[0].Length);
int[,] board = new int[height, width];
int count = 0;
for (int x = 0; x < height; x++) {
string line = input[x];
for (int y = 0; y < width; y++) {
board[x, y] = y < line.Length && char.IsDigit(line[y]) ? line[y] - '0' : -1;
if (board[x, y] >= 0) count++;
}
}
BitArray given = Scan(board, count, height, width);
return (board, given, count);
}

private static (int[,] board, BitArray given, int count) Parse(int[,] input)
{
(int height, int width) = (input.GetLength(0), input.GetLength(1));
int[,] board = new int[height, width];
int count = 0;
for (int x = 0; x < height; x++)
for (int y = 0; y < width; y++)
if ((board[x, y] = input[x, y]) >= 0) count++;
BitArray given = Scan(board, count, height, width);
return (board, given, count);
}

private static BitArray Scan(int[,] board, int count, int height, int width)
{
var given = new BitArray(count + 1);
for (int x = 0; x < height; x++)
for (int y = 0; y < width; y++)
if (board[x, y] > 0) given[board[x, y] - 1] = true;
return given;
}

private static void Print(int[,] board)
{
if (board == null) {
WriteLine("No solution");
} else {
int w = board.Cast<int>().Where(i => i > 0).Max(i => (int?)Ceiling(Log10(i+1))) ?? 1;
string e = new string('-', w);
foreach (int x in Range(0, board.GetLength(0)))
WriteLine(string.Join(" ", Range(0, board.GetLength(1))
.Select(y => board[x, y] < 0 ? e : board[x, y].ToString().PadLeft(w, ' '))));
}
WriteLine();
}

}
Output:
-- 23 32 21 -- -- -- --
-- 16 -- 24 31 -- -- --
-- 33 22 15 20 25 30 27
17 36 19 -- -- 28 --  8
34 -- 14 -- --  9 26 29
1 18 35 10 13  4  7 --
-- --  2  5 -- 11 -- --
-- -- -- 12  3  6 -- --

-- -- -- -- --  1 -- 37 -- -- -- -- --
-- -- -- -- -- 34 -- 56 -- -- -- -- --
-- -- -- --  2 55 38 33 36 -- -- -- --
-- -- -- -- -- 32 35 54 -- -- -- -- --
-- -- 28 -- --  3 -- 39 -- -- 48 -- --
29  6 25  4 31 -- -- -- 53 40 51 42 47
-- -- 30 27 -- -- -- -- -- 49 46 -- --
7 26  5 24  9 -- -- -- 45 52 41 50 43
-- --  8 -- -- 23 -- 15 -- -- 44 -- --
-- -- -- -- -- 10 19 22 -- -- -- -- --
-- -- -- -- 18 21 16 11 14 -- -- -- --
-- -- -- -- -- 12 -- 20 -- -- -- -- --
-- -- -- -- -- 17 -- 13 -- -- -- -- --

## C++

#include <vector>
#include <sstream>
#include <iostream>
#include <iterator>
#include <stdlib.h>
#include <string.h>

using namespace std;

struct node
{
int val;
unsigned char neighbors;
};

class nSolver
{
public:
nSolver()
{
dx[0] = -1; dy[0] = -2; dx[1] = -1; dy[1] =  2;
dx[2] =  1; dy[2] = -2; dx[3] =  1; dy[3] =  2;
dx[4] = -2; dy[4] = -1; dx[5] = -2; dy[5] =  1;
dx[6] =  2; dy[6] = -1; dx[7] =  2; dy[7] =  1;
}

void solve( vector<string>& puzz, int max_wid )
{
if( puzz.size() < 1 ) return;
wid = max_wid; hei = static_cast<int>( puzz.size() ) / wid;
int len = wid * hei, c = 0; max = len;
arr = new node[len]; memset( arr, 0, len * sizeof( node ) );

for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ )
{
if( ( *i ) == "*" ) { max--; arr[c++].val = -1; continue; }
arr[c].val = atoi( ( *i ).c_str() );
c++;
}

solveIt(); c = 0;
for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ )
{
if( ( *i ) == "." )
{
ostringstream o; o << arr[c].val;
( *i ) = o.str();
}
c++;
}
delete [] arr;
}

private:
bool search( int x, int y, int w )
{
if( w > max ) return true;

node* n = &arr[x + y * wid];
n->neighbors = getNeighbors( x, y );

for( int d = 0; d < 8; d++ )
{
if( n->neighbors & ( 1 << d ) )
{
int a = x + dx[d], b = y + dy[d];
if( arr[a + b * wid].val == 0 )
{
arr[a + b * wid].val = w;
if( search( a, b, w + 1 ) ) return true;
arr[a + b * wid].val = 0;
}
}
}
return false;
}

unsigned char getNeighbors( int x, int y )
{
unsigned char c = 0; int a, b;
for( int xx = 0; xx < 8; xx++ )
{
a = x + dx[xx], b = y + dy[xx];
if( a < 0 || b < 0 || a >= wid || b >= hei ) continue;
if( arr[a + b * wid].val > -1 ) c |= ( 1 << xx );
}
return c;
}

void solveIt()
{
int x, y, z; findStart( x, y, z );
if( z == 99999 ) { cout << "\nCan't find start point!\n"; return; }
search( x, y, z + 1 );
}

void findStart( int& x, int& y, int& z )
{
z = 99999;
for( int b = 0; b < hei; b++ )
for( int a = 0; a < wid; a++ )
if( arr[a + wid * b].val > 0 && arr[a + wid * b].val < z )
{
x = a; y = b;
z = arr[a + wid * b].val;
}

}

int wid, hei, max, dx[8], dy[8];
node* arr;
};

int main( int argc, char* argv[] )
{
int wid; string p;
//p = "* . . . * * * * * . * . . * * * * . . . . . . . . . . * * . * . . * . * * . . . 1 . . . . . . * * * . . * . * * * * * . . . * *"; wid = 8;
p = "* * * * * 1 * . * * * * * * * * * * . * . * * * * * * * * * . . . . . * * * * * * * * * . . . * * * * * * * . * * . * . * * . * * . . . . . * * * . . . . . * * . . * * * * * . . * * . . . . . * * * . . . . . * * . * * . * . * * . * * * * * * * . . . * * * * * * * * * . . . . . * * * * * * * * * . * . * * * * * * * * * * . * . * * * * * "; wid = 13;
istringstream iss( p ); vector<string> puzz;
copy( istream_iterator<string>( iss ), istream_iterator<string>(), back_inserter<vector<string> >( puzz ) );
nSolver s; s.solve( puzz, wid );
int c = 0;
for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ )
{
if( ( *i ) != "*" && ( *i ) != "." )
{
if( atoi( ( *i ).c_str() ) < 10 ) cout << "0";
cout << ( *i ) << " ";
}
else cout << "   ";
if( ++c >= wid ) { cout << endl; c = 0; }
}
cout << endl << endl;
return system( "pause" );
}
Output:
17 14 29
28    18 15
13 16 27 30 19 32 07
25 02 11       06    20
12    26       31 08 33
01 24 03 10 05 34 21
36 23    09
04 35 22

01    05
10    12
02 13 04 09 06
08 11 14
34       03    07       16
7 30 39 28 35          15 56 49 54 51
36 33                17 52
1 38 29 40 27          19 48 55 50 53
32       41    47       18
26 23 20
42 21 44 25 46
24    22
43    45

## D

Translation of: C++

From the refactored C++ version with more precise typing, and some optimizations. The HolyKnightPuzzle struct is created at compile-time, so its pre-conditions can catch most malformed puzzles at compile-time.

import std.stdio, std.conv, std.string, std.range, std.algorithm,
std.typecons, std.typetuple;

struct HolyKnightPuzzle {
private alias InputCellBaseType = char;
private enum InputCell : InputCellBaseType { available = '#', unavailable = '.', start='1' }
private alias Cell = uint;
private enum : Cell { unknownCell = 0, unavailableCell = Cell.max, startCell=1 } // Special Cell values.

// Neighbors, [shift row, shift column].
static struct P { int x, y; }
alias shifts = TypeTuple!(P(-2, -1), P(2, -1), P(-2, 1), P(2, 1),
P(-1, -2), P(1, -2), P(-1, 2), P(1, 2));

immutable size_t gridWidth, gridHeight;
private immutable Cell nAvailableCells;
private /*immutable*/ const InputCell[] flatPuzzle;
private Cell[] grid; // Flattened mutable game grid.

@disable this();

this(in string[] rawPuzzle) pure @safe
in {
assert(!rawPuzzle.empty);
assert(!rawPuzzle[0].empty);
assert(rawPuzzle.all!(row => row.length == rawPuzzle[0].length)); // Is rectangular.
assert(rawPuzzle.join.count(InputCell.start) == 1); // Exactly one start point.
} body {
//immutable puzzle = rawPuzzle.to!(InputCell[][]);
immutable puzzle = rawPuzzle.map!representation.array.to!(InputCell[][]);

gridWidth = puzzle[0].length;
gridHeight = puzzle.length;
flatPuzzle = puzzle.join;

// This counts the start cell too.
nAvailableCells = flatPuzzle.representation.count!(ic => ic != InputCell.unavailable);

grid = flatPuzzle
.map!(ic => ic.predSwitch(InputCell.available,   unknownCell,
InputCell.unavailable, unavailableCell,
InputCell.start,       startCell))
.array;
}

Nullable!(string[][]) solve(size_t width)() pure /*nothrow*/ @safe
out(result) {
if (!result.isNull)
assert(!grid.canFind(unknownCell));
} body {
assert(width == gridWidth);

// Find start position.
foreach (immutable r; 0 ..  gridHeight)
foreach (immutable c; 0 .. width)
if (grid[r * width + c] == startCell &&
search!width(r, c, startCell + 1)) {
auto result = zip(flatPuzzle, grid) // Not nothrow.
//.map!({p, c} => ...
.map!(pc => (pc[0] == InputCell.available) ?
pc[1].text :
InputCellBaseType(pc[0]).text)
.array
.chunks(width)
.array;
return typeof(return)(result);
}

return typeof(return)();
}

private bool search(size_t width)(in size_t r, in size_t c, in Cell cell) pure nothrow @safe @nogc {
if (cell > nAvailableCells)
return true; // One solution found.

// This doesn't use the Warnsdorff rule.
foreach (immutable sh; shifts) {
immutable r2 = r + sh.x,
c2 = c + sh.y,
pos = r2 * width + c2;
// No need to test for >= 0 because uint wraps around.
if (c2 < width && r2 < gridHeight && grid[pos] == unknownCell) {
grid[pos] = cell;        // Try.
if (search!width(r2, c2, cell + 1))
return true;
grid[pos] = unknownCell; // Restore.
}
}

return false;
}
}

void main() @safe {
// Enum HolyKnightPuzzle to catch malformed puzzles at compile-time.
enum puzzle1 = ".###....
.#.##...
.#######
###..#.#
#.#..###
1######.
..##.#..
...###..".split.HolyKnightPuzzle;

enum puzzle2 = ".....1.#.....
.....#.#.....
....#####....
.....###.....
..#..#.#..#..
#####...#####
..##.....##..
#####...#####
..#..#.#..#..
.....###.....
....#####....
.....#.#.....
.....#.#.....".split.HolyKnightPuzzle;

foreach (/*enum*/ puzzle; TypeTuple!(puzzle1, puzzle2)) {
//immutable solution = puzzle.solve!(puzzle.gridWidth);
enum width = puzzle.gridWidth;
immutable solution = puzzle.solve!width; // Solved at run-time.
if (solution.isNull)
writeln("No solution found for puzzle.\n");
else
writefln("One solution:\n%(%-(%2s %)\n%)\n", solution);
}
}
Output:
One solution:
. 17 14 29  .  .  .  .
. 28  . 18 15  .  .  .
. 13 16 27 30 19 32  7
25  2 11  .  .  6  . 20
12  . 26  .  . 31  8 33
1 24  3 10  5 34 21  .
.  . 36 23  .  9  .  .
.  .  .  4 35 22  .  .

One solution:
.  .  .  .  .  1  .  5  .  .  .  .  .
.  .  .  .  . 10  . 12  .  .  .  .  .
.  .  .  .  2 13  4  9  6  .  .  .  .
.  .  .  .  .  8 11 14  .  .  .  .  .
.  . 34  .  .  3  .  7  .  . 16  .  .
37 30 39 28 35  .  .  . 15 56 49 54 51
.  . 36 33  .  .  .  .  . 17 52  .  .
31 38 29 40 27  .  .  . 19 48 55 50 53
.  . 32  .  . 41  . 47  .  . 18  .  .
.  .  .  .  . 26 23 20  .  .  .  .  .
.  .  .  . 42 21 44 25 46  .  .  .  .
.  .  .  .  . 24  . 22  .  .  .  .  .
.  .  .  .  . 43  . 45  .  .  .  .  .

Run-time about 0.58 seconds with ldc2 compiler (using a switch statement if you don't have the predSwitch yet in Phobos), about 23 times faster than the Haskell entry.

## Elixir

Translation of: Ruby

This solution uses HLPsolver from here

# require HLPsolver

"""
. . 0 0 0
. . 0 . 0 0
. 0 0 0 0 0 0 0
0 0 0 . . 0 . 0
0 . 0 . . 0 0 0
1 0 0 0 0 0 0
. . 0 0 . 0
. . . 0 0 0
"""

"""
_ _ _ _ _ 1 _ 0
_ _ _ _ _ 0 _ 0
_ _ _ _ 0 0 0 0 0
_ _ _ _ _ 0 0 0
_ _ 0 _ _ 0 _ 0 _ _ 0
0 0 0 0 0 _ _ _ 0 0 0 0 0
_ _ 0 0 _ _ _ _ _ 0 0
0 0 0 0 0 _ _ _ 0 0 0 0 0
_ _ 0 _ _ 0 _ 0 _ _ 0
_ _ _ _ _ 0 0 0
_ _ _ _ 0 0 0 0 0
_ _ _ _ _ 0 _ 0
_ _ _ _ _ 0 _ 0
"""
Output:
Problem:
0  0  0
0     0  0
0  0  0  0  0  0  0
0  0  0        0     0
0     0        0  0  0
1  0  0  0  0  0  0
0  0     0
0  0  0

Solution:
18 21 36
13    19 22
17 20 35 14 25  6 23
31 12 15       34    26
16    32        7 24  5
1 30 11  8 33  4 27
2 29     9
10  3 28

Problem:
1     0
0     0
0  0  0  0  0
0  0  0
0        0     0        0
0  0  0  0  0           0  0  0  0  0
0  0                 0  0
0  0  0  0  0           0  0  0  0  0
0        0     0        0
0  0  0
0  0  0  0  0
0     0
0     0

Solution:
1    55
34    36
2 37 56 33 54
32 35 38
28        3    53       44
29  6 25  4 31          39 52 41 50 43
30 27                45 48
7 26  5 24  9          47 40 51 42 49
8       23    15       46
10 19 22
18 21 16 11 14
12    20
17    13

## Go

Translation of: Python
package main

import "fmt"

var moves = [][2]int{
{-1, -2}, {1, -2}, {-1, 2}, {1, 2}, {-2, -1}, {-2, 1}, {2, -1}, {2, 1},
}

var board1 = " xxx    " +
" x xx   " +
" xxxxxxx" +
"xxx  x x" +
"x x  xxx" +
"sxxxxxx " +
"  xx x  " +
"   xxx  "

var board2 = ".....s.x....." +
".....x.x....." +
"....xxxxx...." +
".....xxx....." +
"..x..x.x..x.." +
"xxxxx...xxxxx" +
"..xx.....xx.." +
"xxxxx...xxxxx" +
"..x..x.x..x.." +
".....xxx....." +
"....xxxxx...." +
".....x.x....." +
".....x.x....."

func solve(pz [][]int, sz, sx, sy, idx, cnt int) bool {
if idx > cnt {
return true
}
for i := 0; i < len(moves); i++ {
x := sx + moves[i][0]
y := sy + moves[i][1]
if (x >= 0 && x < sz) && (y >= 0 && y < sz) && pz[x][y] == 0 {
pz[x][y] = idx
if solve(pz, sz, x, y, idx+1, cnt) {
return true
}
pz[x][y] = 0
}
}
return false
}

func findSolution(b string, sz int) {
pz := make([][]int, sz)
for i := 0; i < sz; i++ {
pz[i] = make([]int, sz)
for j := 0; j < sz; j++ {
pz[i][j] = -1
}
}
var x, y, idx, cnt int
for j := 0; j < sz; j++ {
for i := 0; i < sz; i++ {
switch b[idx] {
case 'x':
pz[i][j] = 0
cnt++
case 's':
pz[i][j] = 1
cnt++
x, y = i, j
}
idx++
}
}

if solve(pz, sz, x, y, 2, cnt) {
for j := 0; j < sz; j++ {
for i := 0; i < sz; i++ {
if pz[i][j] != -1 {
fmt.Printf("%02d  ", pz[i][j])
} else {
fmt.Print("--  ")
}
}
fmt.Println()
}
} else {
fmt.Println("Cannot solve this puzzle!")
}
}

func main() {
findSolution(board1, 8)
fmt.Println()
findSolution(board2, 13)
}
Output:
--  17  14  29  --  --  --  --
--  28  --  18  15  --  --  --
--  13  16  27  30  19  32  07
25  02  11  --  --  06  --  20
12  --  26  --  --  31  08  33
01  24  03  10  05  34  21  --
--  --  36  23  --  09  --  --
--  --  --  04  35  22  --  --

--  --  --  --  --  01  --  05  --  --  --  --  --
--  --  --  --  --  10  --  12  --  --  --  --  --
--  --  --  --  02  13  04  09  06  --  --  --  --
--  --  --  --  --  08  11  14  --  --  --  --  --
--  --  36  --  --  03  --  07  --  --  16  --  --
35  42  33  44  37  --  --  --  15  20  27  22  25
--  --  38  41  --  --  --  --  --  17  24  --  --
39  34  43  32  45  --  --  --  19  28  21  26  23
--  --  40  --  --  31  --  29  --  --  18  --  --
--  --  --  --  --  46  51  56  --  --  --  --  --
--  --  --  --  52  55  30  47  50  --  --  --  --
--  --  --  --  --  48  --  54  --  --  --  --  --
--  --  --  --  --  53  --  49  --  --  --  --  --

import Data.Array
(Array, (//), (!), assocs, elems, bounds, listArray)
import Data.Foldable (forM_)
import Data.List (intercalate, transpose)
import Data.Maybe

type Position = (Int, Int)

type KnightBoard = Array Position (Maybe Int)

toSlot :: Char -> Maybe Int
toSlot '0' = Just 0
toSlot '1' = Just 1
toSlot _ = Nothing

toString :: Maybe Int -> String
toString Nothing = replicate 3 ' '
toString (Just n) = replicate (3 - length nn) ' ' ++ nn
where
nn = show n

chunksOf :: Int -> [a] -> [[a]]
chunksOf _ [] = []
chunksOf n xs =
let (chunk, rest) = splitAt n xs
in chunk : chunksOf n rest

showBoard :: KnightBoard -> String
showBoard board =
intercalate "\n" . map concat . transpose . chunksOf (height + 1) . map toString \$
elems board
where
(_, (_, height)) = bounds board

toBoard :: [String] -> KnightBoard
toBoard strs = board
where
height = length strs
width = minimum (length <\$> strs)
board =
listArray ((0, 0), (width - 1, height - 1)) . map toSlot . concat . transpose \$
take width <\$> strs

:: Num a
=> (a, a) -> (a, a) -> (a, a)
add (a, b) (x, y) = (a + x, b + y)

within
:: Ord a
=> ((a, a), (a, a)) -> (a, a) -> Bool
within ((a, b), (c, d)) (x, y) = a <= x && x <= c && b <= y && y <= d

-- Enumerate valid moves given a board and a knight's position.
validMoves :: KnightBoard -> Position -> [Position]
validMoves board position = filter isValid plausible
where
bound = bounds board
plausible =
[(1, 2), (2, 1), (2, -1), (-1, 2), (-2, 1), (1, -2), (-1, -2), (-2, -1)]
isValid pos = within bound pos && maybe False (== 0) (board ! pos)

isSolved :: KnightBoard -> Bool
isSolved = all (maybe True (0 /=))

-- Solve the knight's tour with a simple Depth First Search.
solveKnightTour :: KnightBoard -> Maybe KnightBoard
solveKnightTour board = solve board 1 initPosition
where
initPosition = fst \$ head \$ filter ((== Just 1) . snd) \$ assocs board
solve boardA depth position =
let boardB = boardA // [(position, Just depth)]
in if isSolved boardB
then Just boardB
else listToMaybe \$
mapMaybe (solve boardB \$ depth + 1) \$ validMoves boardB position

tourExA :: [String]
tourExA =
[ " 000    "
, " 0 00   "
, " 0000000"
, "000  0 0"
, "0 0  000"
, "1000000 "
, "  00 0  "
, "   000  "
]

tourExB :: [String]
tourExB =
[ "-----1-0-----"
, "-----0-0-----"
, "----00000----"
, "-----000-----"
, "--0--0-0--0--"
, "00000---00000"
, "--00-----00--"
, "00000---00000"
, "--0--0-0--0--"
, "-----000-----"
, "----00000----"
, "-----0-0-----"
, "-----0-0-----"
]

main :: IO ()
main =
forM_
[tourExA, tourExB]
(\board ->
case solveKnightTour \$ toBoard board of
Nothing -> putStrLn "No solution.\n"
Just solution -> putStrLn \$ showBoard solution ++ "\n")
Output:
19 26 17
36    20 25
31 18 27 16 21  6 23
35 28 15       24     8
30    32        7 22  5
1 34 29 14 11  4  9
2 33    13
12  3 10

1    31
32    28
56 27  2 33 30
34 29 26
48       55     3       24
47 52 45 54 35          25  4 11  6 23
36 49                 9 22
51 46 53 44 37          21 12  5 10  7
50       43    13        8
38 41 20
42 19 16 39 14
40    18
17    15

As requested, in an attempt to make this solution faster, the following is a version that replaces the Array with an STUArray (unboxed and mutable), and yields a speedup of 4.2. No speedups were gained until move validation was inlined with the logic in `solve'. This seems to point to the list consing as the overhead for time and allocation, although profiling did show that about 25% of the time in the immutable version was spent creating arrays. Perhaps a more experienced Haskeller could provide insight on how to further optimize this or what optimizations were frivolous (barring a different algorithm or search heuristic, and jumping into C, unless those are the only way).

{-# LANGUAGE FlexibleContexts #-}

import qualified Data.Array.Unboxed as AU

import Data.Array.Base (unsafeFreeze)

import Data.List (intercalate, transpose)

import Data.Array.ST

type Position = (Int, Int)

type KnightBoard = AU.UArray Position Int

toSlot :: Char -> Int
toSlot '0' = 0
toSlot '1' = 1
toSlot _ = -1

toString :: Int -> String
toString (-1) = replicate 3 ' '
toString n = replicate (3 - length nn) ' ' ++ nn
where
nn = show n

chunksOf :: Int -> [a] -> [[a]]
chunksOf _ [] = []
chunksOf n xs = uncurry ((. chunksOf n) . (:)) (splitAt n xs)

showBoard :: KnightBoard -> String
showBoard board =
intercalate "\n" . map concat . transpose . chunksOf (height + 1) . map toString \$
AU.elems board
where
(_, (_, height)) = AU.bounds board

toBoard :: [String] -> KnightBoard
toBoard strs = board
where
height = length strs
width = minimum (length <\$> strs)
board =
AU.listArray ((0, 0), (width - 1, height - 1)) . map toSlot . concat . transpose \$
take width <\$> strs

:: Num a
=> (a, a) -> (a, a) -> (a, a)
add (a, b) (x, y) = (a + x, b + y)

within
:: Ord a
=> ((a, a), (a, a)) -> (a, a) -> Bool
within ((a, b), (c, d)) (x, y) = a <= x && x <= c && b <= y && y <= d

-- Solve the knight's tour with a simple Depth First Search.
solveKnightTour :: KnightBoard -> Maybe KnightBoard
solveKnightTour board =
runST \$
do let assocs = AU.assocs board
bounds = AU.bounds board
array <-
newListArray bounds (AU.elems board) :: ST s (STUArray s Position Int)
let initPosition = fst \$ head \$ filter ((== 1) . snd) assocs
maxDepth = fromIntegral \$ 1 + length (filter ((== 0) . snd) assocs)
offsets =
[ (1, 2)
, (2, 1)
, (2, -1)
, (-1, 2)
, (-2, 1)
, (1, -2)
, (-1, -2)
, (-2, -1)
]
solve depth position =
if within bounds position
then do
if oldValue == 0
then do
writeArray array position depth
if depth == maxDepth
then return True
-- This mapM-any combo can be reduced to a string of ||'s
-- with the goal of removing the allocation overhead due to consing
-- which the compiler may not be able to optimize out.
else do
results <- mapM (solve (depth + 1) . add position) offsets
if or results
then return True
else do
writeArray array position oldValue
return False
else return False
else return False
writeArray array initPosition 0
result <- solve 1 initPosition
farray <- unsafeFreeze array
return \$
if result
then Just farray
else Nothing

tourExA :: [String]
tourExA =
[ " 000    "
, " 0 00   "
, " 0000000"
, "000  0 0"
, "0 0  000"
, "1000000 "
, "  00 0  "
, "   000  "
]

tourExB :: [String]
tourExB =
[ "-----1-0-----"
, "-----0-0-----"
, "----00000----"
, "-----000-----"
, "--0--0-0--0--"
, "00000---00000"
, "--00-----00--"
, "00000---00000"
, "--0--0-0--0--"
, "-----000-----"
, "----00000----"
, "-----0-0-----"
, "-----0-0-----"
]

main :: IO ()
main =
forM_
[tourExA, tourExB]
(\board ->
case solveKnightTour \$ toBoard board of
Nothing -> putStrLn "No solution.\n"
Just solution -> putStrLn \$ showBoard solution ++ "\n")

This version is similar to the previous one but:

• the working code is cleaned up slightly with minor optimisations here and there
• only valid board fields are taken into consideration: previously a huge amount of time was wasted on constantly verifying if moves were valid rather than building only valid moves to start with
• vector is used instead of array to take advantage of any fusion

This results in 117x speedup over the very first version. This speed up comes from a smarter traversal rather than from minor code optimisations.

{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE BangPatterns #-}
{-# OPTIONS_GHC -ddump-simpl -ddump-to-file -ddump-stg -O2 -fforce-recomp #-}
module Main (main) where

import           Data.List (intercalate, transpose)
import qualified Data.Ix as Ix
import qualified Data.Vector as V
import           Data.Vector.Unboxed (Vector)
import qualified Data.Vector.Unboxed         as U
import qualified Data.Vector.Unboxed.Mutable as MU
import           Data.Foldable (for_)

type Position = ( Int, Int )

type Bounds = (Position, Position)

type KnightBoard = (Bounds, Vector Int)

toSlot :: Char -> Int
toSlot '0' = 0
toSlot '1' = 1
toSlot _ = -1

toString :: Int -> String
toString (-1) = replicate 3 ' '
toString n = replicate (3 - length nn) ' ' ++ nn
where
nn = show n

chunksOf :: Int -> [a] -> [[a]]
chunksOf _ [] = []
chunksOf n xs = uncurry ((. chunksOf n) . (:)) (splitAt n xs)

showBoard :: KnightBoard -> String
showBoard (bounds, board) =
intercalate "\n" . map concat . transpose . chunksOf (height + 1) . map toString \$
U.toList board
where
(_, (_, height)) = bounds

toBoard :: [String] -> KnightBoard
toBoard strs = (((0,0),(width-1,height-1)), board)
where
height = length strs
width = minimum (length <\$> strs)
board =
U.fromListN (width*height) . map toSlot . concat . transpose \$
take width <\$> strs

-- Solve the knight's tour with a simple Depth First Search.
solveKnightTour :: KnightBoard -> Maybe KnightBoard
solveKnightTour (bounds@(_,(_,yb)), board) = runST \$ do
array <- U.thaw board
let maxDepth = U.length \$ U.filter (/= (-1)) board
Just iniIdx = U.findIndex (==1) board
initPosition = mkPos iniIdx
!hops = V.generate  (U.length board) \$ \i ->
if board `U.unsafeIndex` i == -1
then U.empty
else mkHops (mkPos i)

solve !depth !position = MU.unsafeRead array position >>= \case
0 -> do
MU.unsafeWrite array position depth
case depth == maxDepth of
True -> return True
False -> do
results <- U.mapM (solve (depth + 1)) (hops `V.unsafeIndex` position)
if U.or results
then return True
else do
MU.unsafeWrite array position 0
return False
_ -> pure False

MU.unsafeWrite array (Ix.index bounds initPosition) 0
result <- solve 1 (Ix.index bounds initPosition)
farray <- U.unsafeFreeze array
return \$ if result then Just (bounds, farray) else Nothing
where
offsets = U.fromListN 8 [ (1, 2), (2, 1), (2, -1), (-1, 2), (-2, 1), (1, -2), (-1, -2), (-2, -1) ]
mkHops pos = U.filter (\i -> board `U.unsafeIndex` i == 0)
\$ U.map (Ix.index bounds)
\$ U.filter (Ix.inRange bounds)
add (x, y) (x', y') = (x + x', y + y')
mkPos idx = idx `quotRem` (yb+1)

tourExA :: [String]
tourExA =
[ " 000    "
, " 0 00   "
, " 0000000"
, "000  0 0"
, "0 0  000"
, "1000000 "
, "  00 0  "
, "   000  "
]

tourExB :: [String]
tourExB =
[ "-----1-0-----"
, "-----0-0-----"
, "----00000----"
, "-----000-----"
, "--0--0-0--0--"
, "00000---00000"
, "--00-----00--"
, "00000---00000"
, "--0--0-0--0--"
, "-----000-----"
, "----00000----"
, "-----0-0-----"
, "-----0-0-----"
]

main :: IO ()
main =
for_
[tourExA, tourExB]
(\board -> do
case solveKnightTour \$ toBoard board of
Nothing -> putStrLn "No solution.\n"
Just solution -> putStrLn \$ showBoard solution ++ "\n")

## Icon and Unicon

This is a Unicon-specific solution:

global nCells, cMap, best
record Pos(r,c)

procedure main(A)
QMouse(puzzle,findStart(puzzle),&null,0)
showPuzzle("Output", solvePuzzle(puzzle)) | write("No solution!")
end

p := [[-1],[-1]]
nCells := maxCols := 0
every line := !&input do {
put(p,[: -1 | -1 | gencells(line) | -1 | -1 :])
maxCols <:= *p[-1]
}
every put(p, [-1]|[-1])
# Now normalize all rows to the same length
every i := 1 to *p do p[i] := [: !p[i] | (|-1\(maxCols - *p[i])) :]
return p
end

procedure gencells(s)
static WS, NWS
initial {
NWS := ~(WS := " \t")
cMap := table()     # Map to/from internal model
cMap["#"] := -1;  cMap["_"] :=  0
cMap[-1]  := " "; cMap[0]   := "_"
}

s ? while not pos(0) do {
w := (tab(many(WS))|"", tab(many(NWS))) | break
w := numeric(\cMap[w]|w)
if -1 ~= w then nCells +:= 1
suspend w
}
end

procedure showPuzzle(label, p)
write(label," with ",nCells," cells:")
every r := !p do {
every c := !r do writes(right((\cMap[c]|c),*nCells+1))
write()
}
return p
end

procedure findStart(p)
if \p[r := !*p][c := !*p[r]] = 1 then return Pos(r,c)
end

procedure solvePuzzle(puzzle)
if path := \best then {
repeat {
loc := path.getLoc()
puzzle[loc.r][loc.c] := path.getVal()
path := \path.getParent() | break
}
return puzzle
}
end

class QMouse(puzzle, loc, parent, val)

method getVal(); return val; end
method getLoc(); return loc; end
method getParent(); return parent; end
method atEnd(); return nCells = val; end

method visit(r,c)
if /best & validPos(r,c) then return Pos(r,c)
end

method validPos(r,c)
v := val+1
xv := (0 <= puzzle[r][c]) | fail
if xv = (v|0) then {  # make sure this path hasn't already gone there
ancestor := self
while xl := (ancestor := \ancestor.getParent()).getLoc() do
if (xl.r = r) & (xl.c = c) then fail
return
}
end

initially
val := val+1
if atEnd() then return best := self
QMouse(puzzle, visit(loc.r-2,loc.c-1), self, val)
QMouse(puzzle, visit(loc.r-2,loc.c+1), self, val)
QMouse(puzzle, visit(loc.r-1,loc.c+2), self, val)
QMouse(puzzle, visit(loc.r+1,loc.c+2), self, val)
QMouse(puzzle, visit(loc.r+2,loc.c+1), self, val)
QMouse(puzzle, visit(loc.r+2,loc.c-1), self, val)
QMouse(puzzle, visit(loc.r+1,loc.c-2), self, val)
QMouse(puzzle, visit(loc.r-1,loc.c-2), self, val)
end

Sample run:

->hkt <hkt.in
Input with 36 cells:

_  _  _
_     _  _
_  _  _  _  _  _  _
_  _  _        _     _
_     _        _  _  _
1  _  _  _  _  _  _
_  _     _
_  _  _

Output with 36 cells:

19  4 13
12    18  5
25 20  3 14 17  6 31
21  2 11       32    16
26    24       15 30  7
1 22 27 10 35  8 33
36 23    29
28  9 34

->

## J

The simplest J implementation here uses a breadth first search - but that can be memory inefficient so it's worth representing the boards as characters (several orders of magnitude space improvement) and it's worth capping how much memory we allow J to use (2^34 is 16GB):

9!:21]2^34

unpack=:verb define
board=. (255 0 1{a.) {~ {.@:>:@:"."0 mask#"1 y
)

ex1=:unpack ];._2]0 :0
0 0 0
0   0 0
0 0 0 0 0 0 0
0 0 0     0   0
0   0     0 0 0
1 0 0 0 0 0 0
0 0   0
0 0 0
)

solve=:verb define
board=.,:y
for_move.1+i.+/({.a.)=,y do.
board=. ;move <@knight"2 board
end.
)

kmoves=: ,/(2 1,:1 2)*"1/_1^#:i.4

pos=. (\$y)#:(,y)i.x{a.
moves=. <"1(#~ 0&<: */"1@:* (\$y)&>"1)pos+"1 kmoves
moves=. (#~ (0{a.)={&y) moves
moves y adverb def (':';'y x} m')"0 (x+1){a.
)

Letting that cook:

\$~.sol
48422 8 8

That's 48422 solutions. Here's one of them:

(a.i.{.sol){(i.255),__
__ 11 28 13 __ __ __ __
__ 22 __ 10 29 __ __ __
__ 27 12 21 14  9 16 31
23  2 25 __ __ 30 __  8
26 __ 20 __ __ 15 32 17
1 24  3 34  5 18  7 __
__ __ 36 19 __ 33 __ __
__ __ __  4 35  6 __ __

and here's a couple more:

(a.i.{:sol){(i.255),__
__  5  8 31 __ __ __ __
__ 32 __  6  9 __ __ __
__  7  4 33 30 23 10 21
3 34 29 __ __ 20 __ 24
36 __  2 __ __ 11 22 19
1 28 35 12 15 18 25 __
__ __ 16 27 __ 13 __ __
__ __ __ 14 17 26 __ __
(a.i.24211{sol){(i.255),__
__ 11 14 33 __ __ __ __
__ 34 __ 10 13 __ __ __
__ 19 12 15 32  9  6 25
35 16 31 __ __ 24 __  8
18 __ 20 __ __  7 26  5
1 36 17 30 27  4 23 __
__ __  2 21 __ 29 __ __
__ __ __ 28  3 22 __ __

This is something of a problem, however, because finding all those solutions is slow. And even having to be concerned about a 16GB memory limit for this small of a problem is troubling (and using 64 bit integers, instead of 8 bit characters, to represent board squares, would exceed that limit). Also, you'd get bored, inspecting 48422 boards.

So, let's just find one solution:

unpack=:verb define
board=. __ 0 1 {~ {.@:>:@:"."0 mask#"1 y
)

ex1=:unpack ];._2]0 :0
0 0 0
0   0 0
0 0 0 0 0 0 0
0 0 0     0   0
0   0     0 0 0
1 0 0 0 0 0 0
0 0   0
0 0 0
)

solve1=:verb define
(1,+/0=,y) solve1 ,:y
:
for_block._10 <\ y do.
board=. ;({.x) <@knight"2 ;block
if. #board do.
if. =/x do.
{.board return.
else.
board=. (1 0+x) solve1 board
if. #board do.
board return.
end.
end.
end.
end.
i.0 0
)

kmoves=: ,/(2 1,:1 2)*"1/_1^#:i.4

pos=. (\$y)#:(,y)i.x
moves=. <"1(#~ 0&<: */"1@:* (\$y)&>"1)pos+"1 kmoves
moves=. (#~ 0={&y) moves
moves y adverb def (':';'y x} m')"0 x+1
)

Here, we break our problem space up into blocks of no more than 10 boards each, and use recursion to investigate each batch of boards. When we find a solution, we stop there (for each iteration at each level of recursion):

solve1 ex1
__ 11 28 13 __ __ __ __
__ 22 __ 10 29 __ __ __
__ 27 12 21 14  9 16 31
23  2 25 __ __ 30 __  8
26 __ 20 __ __ 15 32 17
1 24  3 34  5 18  7 __
__ __ 36 19 __ 33 __ __
__ __ __  4 35  6 __ __

[Why ten boards and not just one board? Because 10 is a nice compromise between amortizing the overhead of each attempt and not trying too much at one time. Most individual attempts will fail, but by splitting up the workload after exceeding 10 possibilities, instead of investigating each possibility individually, we increase the chances that we are investigating something useful. Also, J implementations penalize the performance of algorithms which are overly serial in structure.]

With this tool in hand, we can now attempt bigger problems:

ex2=:unpack ];._2]0 :0
1   0
0   0
0 0 0 0 0
0 0 0
0     0   0     0
0 0 0 0 0       0 0 0 0 0
0 0           0 0
0 0 0 0 0       0 0 0 0 0
0     0   0     0
0 0 0
0 0 0 0 0
0   0
0   0
)

Finding a solution for this looks like:

solve1 ex2
__ __ __ __ __  1 __  5 __ __ __ __ __
__ __ __ __ __  6 __ 46 __ __ __ __ __
__ __ __ __ 48 45  2  7  4 __ __ __ __
__ __ __ __ __  8 47 44 __ __ __ __ __
__ __ 56 __ __ 49 __  3 __ __ 42 __ __
13 52 11 50  9 __ __ __ 43 38 31 36 33
__ __ 14 55 __ __ __ __ __ 41 34 __ __
53 12 51 10 15 __ __ __ 39 30 37 32 35
__ __ 54 __ __ 23 __ 29 __ __ 40 __ __
__ __ __ __ __ 16 19 22 __ __ __ __ __
__ __ __ __ 24 21 26 17 28 __ __ __ __
__ __ __ __ __ 18 __ 20 __ __ __ __ __
__ __ __ __ __ 25 __ 27 __ __ __ __ __

## Java

Works with: Java version 8
import java.util.*;

public class HolyKnightsTour {

final static String[] board = {
" xxx    ",
" x xx   ",
" xxxxxxx",
"xxx  x x",
"x x  xxx",
"1xxxxxx ",
"  xx x  ",
"   xxx  "};

private final static int base = 12;
private final static int[][] moves = {{1, -2}, {2, -1}, {2, 1}, {1, 2},
{-1, 2}, {-2, 1}, {-2, -1}, {-1, -2}};
private static int[][] grid;
private static int total = 2;

public static void main(String[] args) {
int row = 0, col = 0;

grid = new int[base][base];

for (int r = 0; r < base; r++) {
Arrays.fill(grid[r], -1);
for (int c = 2; c < base - 2; c++) {
if (r >= 2 && r < base - 2) {
if (board[r - 2].charAt(c - 2) == 'x') {
grid[r][c] = 0;
total++;
}
if (board[r - 2].charAt(c - 2) == '1') {
row = r;
col = c;
}
}
}
}

grid[row][col] = 1;

if (solve(row, col, 2))
printResult();
}

private static boolean solve(int r, int c, int count) {
if (count == total)
return true;

List<int[]> nbrs = neighbors(r, c);

if (nbrs.isEmpty() && count != total)
return false;

Collections.sort(nbrs, (a, b) -> a[2] - b[2]);

for (int[] nb : nbrs) {
r = nb[0];
c = nb[1];
grid[r][c] = count;
if (solve(r, c, count + 1))
return true;
grid[r][c] = 0;
}

return false;
}

private static List<int[]> neighbors(int r, int c) {
List<int[]> nbrs = new ArrayList<>();

for (int[] m : moves) {
int x = m[0];
int y = m[1];
if (grid[r + y][c + x] == 0) {
int num = countNeighbors(r + y, c + x) - 1;
nbrs.add(new int[]{r + y, c + x, num});
}
}
return nbrs;
}

private static int countNeighbors(int r, int c) {
int num = 0;
for (int[] m : moves)
if (grid[r + m[1]][c + m[0]] == 0)
num++;
return num;
}

private static void printResult() {
for (int[] row : grid) {
for (int i : row) {
if (i == -1)
System.out.printf("%2s ", ' ');
else
System.out.printf("%2d ", i);
}
System.out.println();
}
}
}
19 26 21
28    18 25
33 20 27 22 17 24  7
29  2 35        8    16
34    32       23  6  9
1 30  3 36 13 10 15
12 31     5
4 11 14

## JavaScript

### ES6

By composition of generic functions, cacheing degree-sorted moves for each node.

(() => {
'use strict';

// problems :: [[String]]
const problems = [
[
" 000    " //
, " 0 00   " //
, " 0000000" //
, "000  0 0" //
, "0 0  000" //
, "1000000 " //
, "  00 0  " //
, "   000  " //
],
[
"-----1-0-----" //
, "-----0-0-----" //
, "----00000----" //
, "-----000-----" //
, "--0--0-0--0--" //
, "00000---00000" //
, "--00-----00--" //
, "00000---00000" //
, "--0--0-0--0--" //
, "-----000-----" //
, "----00000----" //
, "-----0-0-----" //
, "-----0-0-----" //
]
];

// GENERIC FUNCTIONS ------------------------------------------------------

// comparing :: (a -> b) -> (a -> a -> Ordering)
const comparing = f =>
(x, y) => {
const
a = f(x),
b = f(y);
return a < b ? -1 : a > b ? 1 : 0
};

// concat :: [[a]] -> [a] | [String] -> String
const concat = xs =>
xs.length > 0 ? (() => {
const unit = typeof xs[0] === 'string' ? '' : [];
return unit.concat.apply(unit, xs);
})() : [];

// charColRow :: Char -> [String] -> Maybe (Int, Int)
const charColRow = (c, rows) =>
foldr((a, xs, iRow) =>
a.nothing ? (() => {
const mbiCol = elemIndex(c, xs);
return mbiCol.nothing ? mbiCol : {
just: [mbiCol.just, iRow],
nothing: false
};
})() : a, {
nothing: true
}, rows);

// 2 or more arguments
// curry :: Function -> Function
const curry = (f, ...args) => {
const go = xs => xs.length >= f.length ? (f.apply(null, xs)) :
function () {
return go(xs.concat(Array.from(arguments)));
};
return go([].slice.call(args, 1));
};

// elem :: Eq a => a -> [a] -> Bool
const elem = (x, xs) => xs.indexOf(x) !== -1;

// elemIndex :: Eq a => a -> [a] -> Maybe Int
const elemIndex = (x, xs) => {
const i = xs.indexOf(x);
return {
nothing: i === -1,
just: i
};
};

// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = (m, n) =>
Array.from({
length: Math.floor(n - m) + 1
}, (_, i) => m + i);

// filter :: (a -> Bool) -> [a] -> [a]
const filter = (f, xs) => xs.filter(f);

// findIndex :: (a -> Bool) -> [a] -> Maybe Int
const findIndex = (f, xs) => {
for (var i = 0, lng = xs.length; i < lng; i++) {
if (f(xs[i])) return {
nothing: false,
just: i
};
}
return {
nothing: true
};
};

// foldl :: (b -> a -> b) -> b -> [a] -> b
const foldl = (f, a, xs) => xs.reduce(f, a);

// foldr (a -> b -> b) -> b -> [a] -> b
const foldr = (f, a, xs) => xs.reduceRight(f, a);

// groupBy :: (a -> a -> Bool) -> [a] -> [[a]]
const groupBy = (f, xs) => {
const dct = xs.slice(1)
.reduce((a, x) => {
const
h = a.active.length > 0 ? a.active[0] : undefined,
blnGroup = h !== undefined && f(h, x);
return {
active: blnGroup ? a.active.concat([x]) : [x],
sofar: blnGroup ? a.sofar : a.sofar.concat([a.active])
};
}, {
active: xs.length > 0 ? [xs[0]] : [],
sofar: []
});
return dct.sofar.concat(dct.active.length > 0 ? [dct.active] : []);
};

// intercalate :: String -> [a] -> String
const intercalate = (s, xs) => xs.join(s);

// intersectBy::(a - > a - > Bool) - > [a] - > [a] - > [a]
const intersectBy = (eq, xs, ys) =>
(xs.length > 0 && ys.length > 0) ?
xs.filter(x => ys.some(curry(eq)(x))) : [];

// justifyRight :: Int -> Char -> Text -> Text
const justifyRight = (n, cFiller, strText) =>
n > strText.length ? (
(cFiller.repeat(n) + strText)
.slice(-n)
) : strText;

// length :: [a] -> Int
const length = xs => xs.length;

// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) => xs.map(f);

// mappendComparing :: [(a -> b)] -> (a -> a -> Ordering)
const mappendComparing = fs => (x, y) =>
fs.reduce((ord, f) => {
if (ord !== 0) return ord;
const
a = f(x),
b = f(y);
return a < b ? -1 : a > b ? 1 : 0
}, 0);

// maximumBy :: (a -> a -> Ordering) -> [a] -> a
const maximumBy = (f, xs) =>
xs.reduce((a, x) => a === undefined ? x : (
f(x, a) > 0 ? x : a
), undefined);

// min :: Ord a => a -> a -> a
const min = (a, b) => b < a ? b : a;

// replicate :: Int -> a -> [a]
const replicate = (n, a) => {
let v = [a],
o = [];
if (n < 1) return o;
while (n > 1) {
if (n & 1) o = o.concat(v);
n >>= 1;
v = v.concat(v);
}
return o.concat(v);
};

// sortBy :: (a -> a -> Ordering) -> [a] -> [a]
const sortBy = (f, xs) => xs.slice()
.sort(f);

// splitOn :: String -> String -> [String]
const splitOn = (s, xs) => xs.split(s);

// take :: Int -> [a] -> [a]
const take = (n, xs) => xs.slice(0, n);

// unlines :: [String] -> String
const unlines = xs => xs.join('\n');

// until :: (a -> Bool) -> (a -> a) -> a -> a
const until = (p, f, x) => {
let v = x;
while (!p(v)) v = f(v);
return v;
};

// zip :: [a] -> [b] -> [(a,b)]
const zip = (xs, ys) =>
xs.slice(0, Math.min(xs.length, ys.length))
.map((x, i) => [x, ys[i]]);

// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = (f, xs, ys) =>
Array.from({
length: min(xs.length, ys.length)
}, (_, i) => f(xs[i], ys[i]));

// HOLY KNIGHT's TOUR FUNCTIONS -------------------------------------------

// kmoves :: (Int, Int) -> [(Int, Int)]
const kmoves = ([x, y]) => map(
([a, b]) => [a + x, b + y], [
[1, 2],
[1, -2],
[-1, 2],
[-1, -2],
[2, 1],
[2, -1],
[-2, 1],
[-2, -1]
]);

// rowPosns :: Int -> String -> [(Int, Int)]
const rowPosns = (iRow, s) => {
return foldl((a, x, i) => (elem(x, ['0', '1']) ? (
a.concat([
[i, iRow]
])
) : a), [], splitOn('', s));
};

// hash :: (Int, Int) -> String
const hash = ([col, row]) => col.toString() + '.' + row.toString();

// Start node, and degree-sorted cache of moves from each node
// All node references are hash strings (for this cache)

// problemModel :: [[String]] -> {cache: {nodeKey: [nodeKey], start:String}}
const problemModel = boardLines => {
const
steps = foldl((a, xs, i) =>
a.concat(rowPosns(i, xs)), [], boardLines),
courseMoves = (xs, [x, y]) => intersectBy(
([a, b], [c, d]) => a === c && b === d, kmoves([x, y]), xs
),
maybeStart = charColRow('1', boardLines);
return {
start: maybeStart.nothing ? '' : hash(maybeStart.just),
boardWidth: boardLines.length > 0 ? boardLines[0].length : 0,
stepCount: steps.length,
cache: (() => {
const moveCache = foldl((a, xy) => (
a[hash(xy)] = map(hash, courseMoves(steps, xy)),
a
), {}, steps),
lstMoves = Object.keys(moveCache),
dctDegree = foldl((a, k) =>
(a[k] = moveCache[k].length,
a), {}, lstMoves);

return foldl((a, k) => (
a[k] = sortBy(comparing(x => dctDegree[x]), moveCache[k]),
a
), {}, lstMoves);
})()
};
};

// firstSolution :: {nodeKey: [nodeKey]} -> Int ->
//      nodeKey -> nodeKey -> [nodeKey] ->
//      -> {path::[nodeKey], pathLen::Int, found::Bool}
const firstSolution = (dctMoves, intTarget, strStart, strNodeKey, path) => {
const
intPath = path.length,
moves = dctMoves[strNodeKey];

if ((intTarget - intPath) < 2 && elem(strStart, moves)) {
return {
nothing: false,
just: [strStart, strNodeKey].concat(path),
pathLen: intTarget
};
}

const
nexts = filter(k => !elem(k, path), moves),
intNexts = nexts.length,
lstFullPath = [strNodeKey].concat(path);

// Until we find a full path back to start
return until(
x => (x.nothing === false || x.i >= intNexts),
x => {
const
idx = x.i,
dctSoln = firstSolution(
dctMoves, intTarget, strStart, nexts[idx], lstFullPath
);
return {
i: idx + 1,
nothing: dctSoln.nothing,
just: dctSoln.just,
pathLen: dctSoln.pathLen
};
}, {
nothing: true,
just: [],
i: 0
}
);
};

// maybeTour :: [String] -> {
//    nothing::Bool, Just::[nodeHash], i::Int: pathLen::Int }
const maybeTour = trackLines => {
const
dctModel = problemModel(trackLines),
strStart = dctModel.start;
return strStart !== '' ? firstSolution(
dctModel.cache, dctModel.stepCount, strStart, strStart, []
) : {
nothing: true
};
};

// showLine :: Int -> Int -> String -> Maybe (Int, Int) ->
//              [(Int, Int, String)] -> String
const showLine = curry((intCell, strFiller, maybeStart, xs) => {
const
blnSoln = maybeStart.nothing,
[startCol, startRow] = blnSoln ? [0, 0] : maybeStart.just;
return foldl((a, [iCol, iRow, sVal], i, xs) => ({
col: iCol + 1,
txt: a.txt +
concat(replicate((iCol - a.col) * intCell, strFiller)) +
justifyRight(
intCell, strFiller,
(blnSoln ? sVal : (
iRow === startRow &&
iCol === startCol ? '1' : '0')
)
)
}), {
col: 0,
txt: ''
},
xs
)
.txt
});

// solutionString :: [String] -> Int -> String
const solutionString = (boardLines, iProblem) => {
const
dtePre = Date.now(),
intCols = boardLines.length > 0 ? boardLines[0].length : 0,
soln = maybeTour(boardLines),
intMSeconds = Date.now() - dtePre;

if (soln.nothing) return 'No solution found …';

const
kCol = 0,
kRow = 1,
kSeq = 2,
steps = soln.just,
lstTriples = zipWith((h, n) => {
const [col, row] = map(
x => parseInt(x, 10), splitOn('.', h)
);
return [col, row, n.toString()];
},
steps,
enumFromTo(1, soln.pathLen)),
cellWidth = length(maximumBy(
comparing(x => length(x[kSeq])), lstTriples
)[kSeq]) + 1,
lstGroups = groupBy(
(a, b) => a[kRow] === b[kRow],
sortBy(
mappendComparing([x => x[kRow], x => x[kCol]]),
lstTriples
)),
startXY = take(2, lstTriples[0]),
strMap = 'PROBLEM ' + (parseInt(iProblem, 10) + 1) + '.\n\n' +
unlines(map(showLine(cellWidth, ' ', {
nothing: false,
just: startXY
}), lstGroups)),
strSoln = 'First solution found in c. ' +
intMSeconds + ' milliseconds:\n\n' +
unlines(map(showLine(cellWidth, ' ', {
nothing: true,
just: startXY
}), lstGroups)) + '\n\n';

console.log(strSoln);
return strMap + '\n\n' + strSoln;
};

// TEST -------------------------------------------------------------------
return unlines(map(solutionString, problems));
})();
Output:

(Executed in Atom editor, using 'Script' package).

PROBLEM 1.

0  0  0
0     0  0
0  0  0  0  0  0  0
0  0  0        0     0
0     0        0  0  0
1  0  0  0  0  0  0
0  0     0
0  0  0

First solution found in c. 21 milliseconds:

25 14 23
8    26 15
13 24  7 22 27 16 31
9 36 11       30    28
12     6       21 32 17
1 10 35 20  3 18 29
2  5    33
34 19  4

PROBLEM 2.

1     0
0     0
0  0  0  0  0
0  0  0
0        0     0        0
0  0  0  0  0           0  0  0  0  0
0  0                 0  0
0  0  0  0  0           0  0  0  0  0
0        0     0        0
0  0  0
0  0  0  0  0
0     0
0     0

First solution found in c. 7084 milliseconds:

1     3
50    52
56 53  2 49  4
48 51 54
46       55     5       10
45 42 35 40 47          11  6 13  8 15
44 37                 9 16
43 36 41 34 39          19 12  7 14 17
38       33    27       18
26 23 20
32 21 30 25 28
24    22
31    29

[Finished in 7.2s]

## jq

Works with jq, the C implementation of jq

Works with jaq, the Rust implementation of jq

Works with gojq, the Go implementation of jq (*)

(*) The current implementations of jq are not well-suited to solving this kind of problem efficiently, but the C and Rust implementations were able to run the following program to completion, producing the results shown below, without requiring excessive amounts of memory.

The Go implementation was able to produce the first solution but required about 3GB of RAM.

def lpad(\$len): tostring | (\$len - length) as \$l | (" " * \$l) + .;

def moves: [ [-1, -2], [1, -2], [-1, 2], [1, 2], [-2, -1], [-2, 1], [2, -1], [2, 1] ];

def board1:
" xxx    " +
" x xx   " +
" xxxxxxx" +
"xxx  x x" +
"x x  xxx" +
"sxxxxxx " +
"  xx x  " +
"   xxx  "
;
def board2:
".....s.x....." +
".....x.x....." +
"....xxxxx...." +
".....xxx....." +
"..x..x.x..x.." +
"xxxxx...xxxxx" +
"..xx.....xx.." +
"xxxxx...xxxxx" +
"..x..x.x..x.." +
".....xxx....." +
"....xxxxx...." +
".....x.x....." +
".....x.x....."
;

# Input: {pz}
def solve(\$sz; \$sx; \$sy; \$idx; \$cnt):
# debug( "solve(\(\$sz); \(\$sx); \(\$sy); \(\$idx); \(\$cnt))" ) |
if \$idx > \$cnt then .
else first(
range(0;moves|length) as \$i
| (\$sx + moves[\$i][0]) as \$x
| (\$sy + moves[\$i][1]) as \$y
| if \$x >= 0 and \$x < \$sz and \$y >= 0 and \$y < \$sz and .pz[\$x][\$y] == 0
then .pz[\$x][\$y] = \$idx
| solve(\$sz; \$x; \$y; \$idx + 1; \$cnt)
else empty
end )
end;

def printSolution(\$sz):
range(0; \$sz) as \$j
|  .emit = ""
| reduce range(0; \$sz) as \$i (.;
if .pz[\$i][\$j] != -1
then .emit += (.pz[\$i][\$j] | lpad(3))
else .emit += " --"
end )
| .emit;

# \$b should be a board of size \$sz
def findSolution(\$b; \$sz):
[range(0; \$sz) | -1] as \$minus
| { pz: [range(0;\$sz) | \$minus],
x: 0,
y: 0,
idx: 0,
cnt: 0
}
| reduce range(0; \$sz) as \$j (.;
reduce range(0; \$sz) as \$i (.;
if \$b[.idx: .idx+1] == "x"
then .pz[\$i][\$j] = 0
| .cnt +=  1
elif \$b[.idx: .idx+1] == "s"
then
.pz[\$i][\$j] = 1
| .cnt += 1
| .x = \$i
| .y = \$j
end
| .idx += 1 ))
| (solve(\$sz; .x; .y; 2; .cnt) | printSolution(\$sz))
// "Whoops!" ;

findSolution(board1;  8),
"",
findSolution(board2; 13)
Output:
-- 17 14 29 -- -- -- --
-- 28 -- 18 15 -- -- --
-- 13 16 27 30 19 32  7
25  2 11 -- --  6 -- 20
12 -- 26 -- -- 31  8 33
1 24  3 10  5 34 21 --
-- -- 36 23 --  9 -- --
-- -- --  4 35 22 -- --

-- -- -- -- --  1 --  5 -- -- -- -- --
-- -- -- -- -- 10 -- 12 -- -- -- -- --
-- -- -- --  2 13  4  9  6 -- -- -- --
-- -- -- -- --  8 11 14 -- -- -- -- --
-- -- 36 -- --  3 --  7 -- -- 16 -- --
35 42 33 44 37 -- -- -- 15 20 27 22 25
-- -- 38 41 -- -- -- -- -- 17 24 -- --
39 34 43 32 45 -- -- -- 19 28 21 26 23
-- -- 40 -- -- 31 -- 29 -- -- 18 -- --
-- -- -- -- -- 46 51 56 -- -- -- -- --
-- -- -- -- 52 55 30 47 50 -- -- -- --
-- -- -- -- -- 48 -- 54 -- -- -- -- --
-- -- -- -- -- 53 -- 49 -- -- -- -- --

## Julia

Uses the Hidato puzzle solver module, which has its source code listed here in the Hadato task.

using .Hidato       # Note that the . here means to look locally for the module rather than in the libraries

const holyknight = """
. 0 0 0 . . . .
. 0 . 0 0 . . .
. 0 0 0 0 0 0 0
0 0 0 . . 0 . 0
0 . 0 . . 0 0 0
1 0 0 0 0 0 0 .
. . 0 0 . 0 . .
. . . 0 0 0 . . """

const knightmoves = [[-2, -1], [-2, 1], [-1, -2], [-1, 2], [1, -2], [1, 2], [2, -1], [2, 1]]

board, maxmoves, fixed, starts = hidatoconfigure(holyknight)
printboard(board, " 0", "  ")
hidatosolve(board, maxmoves, knightmoves, fixed, starts[1][1], starts[1][2], 1)
printboard(board)
Output:

0 0 0
0   0 0
0 0 0 0 0 0 0
0 0 0     0   0
0   0     0 0 0
1  0 0 0 0 0 0
0 0   0
0 0 0
7  4 17
16     8  5
9  6  3 18 25 20 23
31  2 15       22    26
10    30       19 24 21
1 32 11 14 29 34 27
36 33    13
12 35 28

## Kotlin

Translation of: Python
// version 1.1.3

val moves = arrayOf(
intArrayOf(-1, -2), intArrayOf( 1, -2), intArrayOf(-1,  2), intArrayOf(1, 2),
intArrayOf(-2, -1), intArrayOf(-2,  1), intArrayOf( 2, -1), intArrayOf(2, 1)
)

val board1 =
" xxx    " +
" x xx   " +
" xxxxxxx" +
"xxx  x x" +
"x x  xxx" +
"sxxxxxx " +
"  xx x  " +
"   xxx  "

val board2 =
".....s.x....." +
".....x.x....." +
"....xxxxx...." +
".....xxx....." +
"..x..x.x..x.." +
"xxxxx...xxxxx" +
"..xx.....xx.." +
"xxxxx...xxxxx" +
"..x..x.x..x.." +
".....xxx....." +
"....xxxxx...." +
".....x.x....." +
".....x.x....."

fun solve(pz: Array<IntArray>, sz: Int, sx: Int, sy: Int, idx: Int, cnt: Int): Boolean {
if (idx > cnt) return true
for (i in 0 until moves.size) {
val x = sx + moves[i][0]
val y = sy + moves[i][1]
if ((x in 0 until sz) && (y in 0 until sz) && pz[x][y] == 0) {
pz[x][y] = idx
if (solve(pz, sz, x, y, idx + 1, cnt)) return true
pz[x][y] = 0
}
}
return false
}

fun findSolution(b: String, sz: Int) {
val pz = Array(sz) { IntArray(sz) { -1 } }
var x = 0
var y = 0
var idx = 0
var cnt = 0
for (j in 0 until sz) {
for (i in 0 until sz) {
if (b[idx] == 'x') {
pz[i][j] = 0
cnt++
}
else if (b[idx] == 's') {
pz[i][j] = 1
cnt++
x = i
y = j
}
idx++
}
}

if (solve(pz, sz, x, y, 2, cnt)) {
for (j in 0 until sz) {
for (i in 0 until sz) {
if (pz[i][j] != -1)
print("%02d  ".format(pz[i][j]))
else
print("--  ")
}
println()
}
}
else println("Cannot solve this puzzle!")
}

fun main(args: Array<String>) {
findSolution(board1,  8)
println()
findSolution(board2, 13)
}
Output:
--  17  14  29  --  --  --  --
--  28  --  18  15  --  --  --
--  13  16  27  30  19  32  07
25  02  11  --  --  06  --  20
12  --  26  --  --  31  08  33
01  24  03  10  05  34  21  --
--  --  36  23  --  09  --  --
--  --  --  04  35  22  --  --

--  --  --  --  --  01  --  05  --  --  --  --  --
--  --  --  --  --  10  --  12  --  --  --  --  --
--  --  --  --  02  13  04  09  06  --  --  --  --
--  --  --  --  --  08  11  14  --  --  --  --  --
--  --  36  --  --  03  --  07  --  --  16  --  --
35  42  33  44  37  --  --  --  15  20  27  22  25
--  --  38  41  --  --  --  --  --  17  24  --  --
39  34  43  32  45  --  --  --  19  28  21  26  23
--  --  40  --  --  31  --  29  --  --  18  --  --
--  --  --  --  --  46  51  56  --  --  --  --  --
--  --  --  --  52  55  30  47  50  --  --  --  --
--  --  --  --  --  48  --  54  --  --  --  --  --
--  --  --  --  --  53  --  49  --  --  --  --  --

## Lua

local p1, p1W = ".xxx.....x.xx....xxxxxxxxxx..x.xx.x..xxxsxxxxxx...xx.x.....xxx..", 8
local p2, p2W = ".....s.x..........x.x.........xxxxx.........xxx.......x..x.x..x..xxxxx...xxxxx..xx.....xx..xxxxx...xxxxx..x..x.x..x.......xxx.........xxxxx.........x.x..........x.x.....", 13
local puzzle, movesCnt, wid = {}, 0, 0
local moves = { { -1, -2 }, {  1, -2 }, { -1,  2 }, {  1,  2 },
{ -2, -1 }, { -2,  1 }, {  2, -1 }, {  2,  1 } }

function isValid( x, y )
return( x > 0 and x <= wid and y > 0 and y <= wid and puzzle[x + y * wid - wid] == 0 )
end
function solve( x, y, s )
if s > movesCnt then return true end
local test, a, b
for i = 1, #moves do
test = false
a = x + moves[i][1]; b = y + moves[i][2]
if isValid( a, b ) then
puzzle[a + b * wid - wid] = s
if solve( a, b, s + 1 ) then return true end
puzzle[a + b * wid - wid] = 0
end
end
return false
end
function printSolution()
local lp
for j = 1, wid do
for i = 1, wid do
lp = puzzle[i + j * wid - wid]
if lp == -1 then io.write( "   " )
else io.write( string.format( " %.2d", lp ) )
end
end
print()
end
print( "\n" )
end
local sx, sy
function fill( pz, w )
puzzle = {}; wid = w; movesCnt = #pz
local lp
for i = 1, #pz do
lp = pz:sub( i, i )
if lp == "x" then
table.insert( puzzle, 0 )
elseif lp == "." then
table.insert( puzzle, -1 ); movesCnt = movesCnt - 1
else
table.insert( puzzle, 1 )
sx = 1 + ( i - 1 ) % wid; sy = math.floor( ( i + wid - 1 ) / wid )
end
end
end
-- [[ entry point ]] --
print( "\n\n" ); fill( p1, p1W );
if solve( sx, sy, 2 ) then printSolution() end
print( "\n\n" ); fill( p2, p2W );
if solve( sx, sy, 2 ) then printSolution() end
Output:

17 14 29
28    18 15
13 16 27 30 19 32 07
25 02 11       06    20
12    26       31 08 33
01 24 03 10 05 34 21
36 23    09
04 35 22

01    05
10    12
02 13 04 09 06
08 11 14
36       03    07       16
35 42 33 44 37          15 20 27 22 25
38 41                17 24
39 34 43 32 45          19 28 21 26 23
40       31    29       18
46 51 56
52 55 30 47 50
48    54
53    49

## Mathematica /Wolfram Language

Outputs coordinates and a visualization of the tour:

puzzle = "  0 0 0
0   0 0
0 0 0 0 0 0 0
0 0 0     0   0
0   0     0 0 0
1 0 0 0 0 0 0
0 0   0
0 0 0";
puzzle = StringSplit[puzzle, "\n"];
puzzle = StringTake[#, {1, -1, 2}] & /@ puzzle;
pos0 = Join @@ Table[{i, #} & /@ StringPosition[puzzle[[i]], "0"][[All, 1]], {i, Length@puzzle}];
pos1 = Join @@ Table[{i, #} & /@ StringPosition[puzzle[[i]], "1"][[All, 1]], {i, Length@puzzle}];

allpoints = Join[pos1, pos0];
validmoves = Select[Subsets[allpoints, {2}], Differences /* Norm /* EqualTo[Sqrt[5]]];
g = Graph[UndirectedEdge @@@ validmoves];
e = VertexList[g];
order = FindShortestTour[g][[2]]
Graphics[{Red, Disk[#, 0.2] & /@ e, Black, BlockMap[Arrow, e[[order]], 2, 1]}]
Output:
{{6,1},{4,2},{6,3},{8,4},{7,6},{6,4},{5,6},{3,5},{1,4},{2,2},{4,3},{5,1},{3,2},{2,4},{1,2},{3,3},{4,1},{6,2},{7,4},{8,6},{6,7},{4,8},{3,6},{5,7},{3,8},{4,6},{6,5},{5,3},{3,4},{1,3},{2,5},{3,7},{5,8},{6,6},{8,5},{7,3},{6,1}}
[Visualization of the tour]

## Nim

Translation of: Go

In this version, the board is described as an array of strings rather than a string in the Go version (so, we don’t have to specify the size). The way to initialize is also different and even if the Moves where in the same order, the solution would be different. Changing the order of the Moves may have a great impact on performance, but there is no best order. The order we have chosen provides excellent performance with the two examples: less than 20 ms on our laptop. But with another order, it took more than 2 seconds!

import sequtils, strformat

const Moves = [[-1, -2], [1, -2], [-2, -1], [2, -1], [-2, 1], [2, 1], [-1, 2], [1, 2]]

proc solve(pz: var seq[seq[int]]; sx, sy, idx, count: Natural): bool =

if idx > count: return true

var x, y: int
for move in Moves:
x = sx + move[0]
y = sy + move[1]
if x in 0..pz.high and y in 0..pz.high and pz[x][y] == 0:
pz[x][y] = idx
if pz.solve(x, y, idx + 1, count): return true
pz[x][y] = 0

proc findSolution(board: openArray[string]) =
let sz = board.len
var pz = newSeqWith(sz, repeat(-1, sz))

var count = 0
var x, y: int
for i in 0..<sz:
for j in 0..<sz:
case board[i][j]
of 'x':
pz[i][j] = 0
inc count
of 's':
pz[i][j] = 1
inc count
(x, y) = (i, j)
else:

if pz.solve(x, y, 2, count):
for i in 0..<sz:
for j in 0..<sz:
if pz[i][j] != -1:
stdout.write &"{pz[i][j]:02}  "
else:
stdout.write "--  "
stdout.write '\n'

when isMainModule:

const

Board1 = [" xxx    ",
" x xx   ",
" xxxxxxx",
"xxx  x x",
"x x  xxx",
"sxxxxxx ",
"  xx x  ",
"   xxx  "]

Board2 = [".....s.x.....",
".....x.x.....",
"....xxxxx....",
".....xxx.....",
"..x..x.x..x..",
"xxxxx...xxxxx",
"..xx.....xx..",
"xxxxx...xxxxx",
"..x..x.x..x..",
".....xxx.....",
"....xxxxx....",
".....x.x.....",
".....x.x....."]

Board1.findSolution()
echo()
Board2.findSolution()
Output:
--  13  06  15  --  --  --  --
--  08  --  12  31  --  --  --
--  05  14  07  16  27  32  29
09  02  11  --  --  30  --  26
04  --  22  --  --  17  28  33
01  10  03  18  21  34  25  --
--  --  36  23  --  19  --  --
--  --  --  20  35  24  --  --

--  --  --  --  --  01  --  55  --  --  --  --  --
--  --  --  --  --  50  --  48  --  --  --  --  --
--  --  --  --  02  47  54  51  56  --  --  --  --
--  --  --  --  --  52  49  46  --  --  --  --  --
--  --  14  --  --  03  --  53  --  --  44  --  --
09  06  11  04  13  --  --  --  45  38  33  40  43
--  --  08  15  --  --  --  --  --  35  42  --  --
07  10  05  12  17  --  --  --  37  32  39  34  41
--  --  16  --  --  23  --  31  --  --  36  --  --
--  --  --  --  --  18  21  24  --  --  --  --  --
--  --  --  --  22  25  28  19  30  --  --  --  --
--  --  --  --  --  20  --  26  --  --  --  --  --
--  --  --  --  --  27  --  29  --  --  --  --  --

## Perl

We perform a brute-force search. As an enhancement, we unroll the search by one level and use Parallel::ForkManager to search the top-level sub-trees concurrently, subject to the number of cores of course. We implement the search with explicit recursion, which impacts performance but improves readability and provides a use case for the "local" keyword.

package KT_Locations;
# A sequence of locations on a 2-D board whose order might or might not
# matter. Suitable for representing a partial tour, a complete tour, or the
# required locations to visit.
use strict;
use English;
# 'locations' must be a reference to an array of 2-element array references,
# where the first element is the rank index and the second is the file index.
use Class::Tiny qw(N locations);
use List::Util qw(all);

sub BUILD {
my \$self = shift;
\$self->{N} //= 8;
\$self->{N} >= 3 or die "N must be at least 3";
all {ref(\$ARG) eq 'ARRAY' && scalar(@{\$ARG}) == 2} @{\$self->{locations}}
or die "At least one element of 'locations' is invalid";
return;
}

sub as_string {
my \$self = shift;
my %idxs;
my \$idx = 1;
foreach my \$loc (@{\$self->locations}) {
\$idxs{join(q{K},@{\$loc})} = \$idx++;
}
my \$str;
{
my \$w = int(log(scalar(@{\$self->locations}))/log(10.)) + 2;
my \$fmt = "%\${w}d";
my \$N = \$self->N;
my \$non_tour = q{ } x (\$w-1) . q{-};
for (my \$r=0; \$r<\$N; \$r++) {
for (my \$f=0; \$f<\$N; \$f++) {
my \$k = join(q{K}, \$r, \$f);
\$str .= exists(\$idxs{\$k}) ? sprintf(\$fmt, \$idxs{\$k}) : \$non_tour;
}
\$str .= "\n";
}
}
return \$str;
}

sub as_idx_hash {
my \$self = shift;
my \$N = \$self->N;
my \$result;
foreach my \$pair (@{\$self->locations}) {
my (\$r, \$f) = @{\$pair};
\$result->{\$r * \$N + \$f}++;
}
return \$result;
}

package KnightsTour;
use strict;
# If supplied, 'str' is parsed to set 'N', 'start_location', and
# 'locations_to_visit'.  'legal_move_idxs' is for improving performance.
use Class::Tiny qw( N start_location locations_to_visit str legal_move_idxs );
use English;
use Parallel::ForkManager;
use Time::HiRes qw( gettimeofday tv_interval );

sub BUILD {
my \$self = shift;
if (\$self->{str}) {
my (\$n, \$sl, \$ltv) = _parse_input_string(\$self->{str});
\$self->{N} = \$n;
\$self->{start_location} = \$sl;
\$self->{locations_to_visit} = \$ltv;
}
\$self->{N} //= 8;
\$self->{N} >= 3 or die "N must be at least 3";
exists(\$self->{start_location}) or die "Must supply start_location";
die "start_location is invalid"
if ref(\$self->{start_location}) ne 'ARRAY' ||
scalar(@{\$self->{start_location}}) != 2;
exists(\$self->{locations_to_visit}) or die "Must supply locations_to_visit";
ref(\$self->{locations_to_visit}) eq 'KT_Locations'
or die "locations_to_visit must be a KT_Locations instance";
\$self->{N} == \$self->{locations_to_visit}->N
or die "locations_to_visit has mismatched board size";
\$self->precompute_legal_moves();
return;
}

sub _parse_input_string {
my @rows = split(/[\r\n]+/s, shift);
my \$N = scalar(@rows);
my (\$start_location, @to_visit);
for (my \$r=0; \$r<\$N; \$r++) {
my \$row_r = \$rows[\$r];
for (my \$f=0; \$f<\$N; \$f++) {
my \$c = substr(\$row_r, \$f, 1);
if (\$c eq '1') { \$start_location = [\$r, \$f]; }
elsif (\$c eq '0') { push @to_visit, [\$r, \$f]; }
}
}
\$start_location or die "No starting location provided";
return (\$N,
\$start_location,
KT_Locations->new(N => \$N, locations => \@to_visit));
}

sub precompute_legal_moves {
my \$self = shift;
my \$N = \$self->{N};
my \$ktl_ixs = \$self->{locations_to_visit}->as_idx_hash();
for (my \$r=0; \$r<\$N; \$r++) {
for (my \$f=0; \$f<\$N; \$f++) {
my \$k = \$r * \$N + \$f;
\$self->{legal_move_idxs}->{\$k} =
_precompute_legal_move_idxs(\$r, \$f, \$N, \$ktl_ixs);
}
}
return;
}

sub _precompute_legal_move_idxs {
my (\$r, \$f, \$N, \$ktl_ixs) = @ARG;
my \$r_plus_1  = \$r + 1;  my \$r_plus_2 = \$r + 2;
my \$r_minus_1 = \$r - 1;  my \$r_minus_2 = \$r - 2;
my \$f_plus_1  = \$f + 1;  my \$f_plus_2 = \$f + 2;
my \$f_minus_1 = \$f - 1;  my \$f_minus_2 = \$f - 2;
my @result = grep { exists(\$ktl_ixs->{\$ARG}) }
map { \$ARG->[0] * \$N + \$ARG->[1] }
grep {\$ARG->[0] >= 0 && \$ARG->[0] < \$N &&
\$ARG->[1] >= 0 && \$ARG->[1] < \$N}
([\$r_plus_2,  \$f_minus_1], [\$r_plus_2,  \$f_plus_1],
[\$r_minus_2, \$f_minus_1], [\$r_minus_2, \$f_plus_1],
[\$r_plus_1,  \$f_plus_2],  [\$r_plus_1,  \$f_minus_2],
[\$r_minus_1, \$f_plus_2],  [\$r_minus_1, \$f_minus_2]);
return \@result;
}

sub find_tour {
my \$self = shift;
my \$num_to_visit = scalar(@{\$self->locations_to_visit->locations});
my \$N = \$self->N;
my \$start_loc_idx =
\$self->start_location->[0] * \$N + \$self->start_location->[1];
my \$visited;  for (my \$i=0; \$i<\$N*\$N; \$i++) { vec(\$visited, \$i, 1) = 0; }
vec(\$visited, \$start_loc_idx, 1) = 1;
# We unwind the search by one level and use Parallel::ForkManager to search
# the top-level sub-trees concurrently, assuming there are enough cores.
my @next_loc_idxs = @{\$self->legal_move_idxs->{\$start_loc_idx}};
my \$pm = new Parallel::ForkManager(scalar(@next_loc_idxs));
foreach my \$next_loc_idx (@next_loc_idxs) {
\$pm->start and next;  # Do the fork
my \$t0 = [gettimeofday];
vec(\$visited, \$next_loc_idx, 1) = 1;  # (The fork cloned \$visited.)
my \$tour = _find_tour_helper(\$N,
\$num_to_visit - 1,
\$next_loc_idx,
\$visited,
\$self->legal_move_idxs);
my \$elapsed = tv_interval(\$t0);
my (\$r, \$f) = _idx_to_rank_and_file(\$next_loc_idx, \$N);
if (defined \$tour) {
my @tour_locs =
map { [_idx_to_rank_and_file(\$ARG, \$N)] }
(\$start_loc_idx, \$next_loc_idx, split(/\s+/s, \$tour));
my \$kt_locs = KT_Locations->new(N => \$N, locations => \@tour_locs);
print "Found a tour after first move (\$r, \$f) ",
"in \$elapsed seconds:\n", \$kt_locs, "\n";
}
else {
print "No tour found after first move (\$r, \$f). ",
"Took \$elapsed seconds.\n";
}
\$pm->finish; # Do the exit in the child process
}
\$pm->wait_all_children;
return;
}

sub _idx_to_rank_and_file {
my (\$idx, \$N) = @ARG;
my \$f = \$idx % \$N;
my \$r = (\$idx - \$f) / \$N;
return (\$r, \$f);
}

sub _find_tour_helper {
my (\$N, \$num_to_visit, \$current_loc_idx, \$visited, \$legal_move_idxs) = @ARG;

# The performance hot spot.
local *inner_helper = sub {
my (\$num_to_visit, \$current_loc_idx, \$visited) = @ARG;
if (\$num_to_visit == 0) {
return q{ };  # Solution found.
}
my @next_loc_idxs = @{\$legal_move_idxs->{\$current_loc_idx}};
my \$num_to_visit2 = \$num_to_visit - 1;
foreach my \$loc_idx2 (@next_loc_idxs) {
next if vec(\$visited, \$loc_idx2, 1);
my \$visited2 = \$visited;
vec(\$visited2, \$loc_idx2, 1) = 1;
my \$recursion = inner_helper(\$num_to_visit2, \$loc_idx2, \$visited2);
return \$loc_idx2 . q{ } . \$recursion if defined \$recursion;
}
return;
};

return inner_helper(\$num_to_visit, \$current_loc_idx, \$visited);
}

package main;
use strict;

solve_size_8_problem();
solve_size_13_problem();
exit 0;

sub solve_size_8_problem {
my \$problem = <<"END_SIZE_8_PROBLEM";
--000---
--0-00--
-0000000
000--0-0
0-0--000
1000000-
--00-0--
---000--
END_SIZE_8_PROBLEM
my \$kt = KnightsTour->new(str => \$problem);
print "Finding a tour for an 8x8 problem...\n";
\$kt->find_tour();
return;
}

sub solve_size_13_problem {
my \$problem = <<"END_SIZE_13_PROBLEM";
-----1-0-----
-----0-0-----
----00000----
-----000-----
--0--0-0--0--
00000---00000
--00-----00--
00000---00000
--0--0-0--0--
-----000-----
----00000----
-----0-0-----
-----0-0-----
END_SIZE_13_PROBLEM
my \$kt = KnightsTour->new(str => \$problem);
print "Finding a tour for a 13x13 problem...\n";
\$kt->find_tour();
return;
}
Output:

The timings shown below were obtained on a Dell Optiplex 9020 with 4 cores.

...>holy_knights_tour.pl
Finding a tour for an 8x8 problem...
Found a tour after first move (6, 2) in 0.018372 seconds:
-  - 18 31 16  -  -  -
-  - 23  - 33 30  -  -
- 19 32 17 24 15 34 29
7 22  5  -  - 28  - 26
20  -  8  -  - 25 14 35
1  6 21  4 11 36 27  -
-  -  2  9  - 13  -  -
-  -  - 12  3 10  -  -

Found a tour after first move (4, 2) in 0.010491 seconds:
-  - 30 23 20  -  -  -
-  -  9  - 31 22  -  -
- 29 24 21 10 19 32 15
25  8 27  -  - 16  - 18
28  -  2  -  - 11 14 33
1 26  7 12  5 34 17  -
-  - 36  3  - 13  -  -
-  -  -  6 35  4  -  -

Found a tour after first move (3, 1) in 0.048164 seconds:
-  - 28 11 14  -  -  -
-  - 13  -  9 30  -  -
- 27 10 29 12 15 18 31
23  2 25  -  -  8  - 16
26  - 22  -  - 17 32 19
1 24  3 34  5 20  7  -
-  - 36 21  - 33  -  -
-  -  -  4 35  6  -  -

Finding a tour for a 13x13 problem...
Found a tour after first move (2, 6) in 78.827185 seconds:
-  -  -  -  -  1  - 21  -  -  -  -  -
-  -  -  -  - 22  -  6  -  -  -  -  -
-  -  -  -  4  7  2 23 20  -  -  -  -
-  -  -  -  - 24  5  8  -  -  -  -  -
-  - 34  -  -  3  - 19  -  - 56  -  -
35 30 37 28 25  -  -  -  9 18 11 16 13
-  - 26 33  -  -  -  -  - 55 14  -  -
31 36 29 38 27  -  -  - 53 10 17 12 15
-  - 32  -  - 39  - 45  -  - 54  -  -
-  -  -  -  - 46 49 52  -  -  -  -  -
-  -  -  - 40 51 42 47 44  -  -  -  -
-  -  -  -  - 48  - 50  -  -  -  -  -
-  -  -  -  - 41  - 43  -  -  -  -  -

Found a tour after first move (2, 4) in 100.327934 seconds:
-  -  -  -  -  1  - 23  -  -  -  -  -
-  -  -  -  - 24  - 20  -  -  -  -  -
-  -  -  -  2 19  4 25 22  -  -  -  -
-  -  -  -  - 26 21 18  -  -  -  -  -
-  - 36  -  -  3  -  5  -  - 12  -  -
37 32 39 30 27  -  -  - 17  6 15  8 13
-  - 28 35  -  -  -  -  - 11 56  -  -
33 38 31 40 29  -  -  - 55 16  7 14  9
-  - 34  -  - 41  - 47  -  - 10  -  -
-  -  -  -  - 48 51 54  -  -  -  -  -
-  -  -  - 42 53 44 49 46  -  -  -  -
-  -  -  -  - 50  - 52  -  -  -  -  -
-  -  -  -  - 43  - 45  -  -  -  -  -

Found a tour after first move (1, 7) in 1443.340089 seconds:
-  -  -  -  -  1  - 21  -  -  -  -  -
-  -  -  -  - 22  -  2  -  -  -  -  -
-  -  -  - 18  3 16 23 20  -  -  -  -
-  -  -  -  - 24 19  4  -  -  -  -  -
-  - 34  -  - 17  - 15  -  - 56  -  -
35 30 37 28 25  -  -  -  5 14  7 12  9
-  - 26 33  -  -  -  -  - 55 10  -  -
31 36 29 38 27  -  -  - 53  6 13  8 11
-  - 32  -  - 39  - 45  -  - 54  -  -
-  -  -  -  - 46 49 52  -  -  -  -  -
-  -  -  - 40 51 42 47 44  -  -  -  -
-  -  -  -  - 48  - 50  -  -  -  -  -
-  -  -  -  - 41  - 43  -  -  -  -  -

## Phix

Tweaked the knights tour algorithm (to use a limit variable rather than size*size). Bit slow on the second one... (hence omitted under pwa/p2js)

with javascript_semantics
sequence board, warnsdorffs

integer size, limit, nchars
string fmt, blank

constant ROW = 1, COL = 2,
moves = {{-1,-2},{-2,-1},{-2,1},{-1,2},{1,2},{2,1},{2,-1},{1,-2}}

function onboard(integer row, integer col)
return row>=1 and row<=size and col>=nchars and col<=nchars*size
end function

procedure init_warnsdorffs()
for row=1 to size do
for col=nchars to nchars*size by nchars do
for move=1 to length(moves) do
integer nrow = row+moves[move][ROW],
ncol = col+moves[move][COL]*nchars
if onboard(nrow,ncol) then
warnsdorffs[nrow][ncol] += 1
end if
end for
end for
end for
end procedure

atom t0 = time(),
t1 = time()+1
integer tries = 0, backtracks = 0
function solve(integer row, integer col, integer n)
if time()>t1 and platform()!=JS then
?{row,floor(col/nchars),n,tries}
puts(1,join(board,"\n")&"\n")
t1 = time()+1
end if
tries+= 1
if n>limit then return 1 end if
sequence wmoves = {}
integer nrow, ncol
for move=1 to length(moves) do
nrow = row+moves[move][ROW]
ncol = col+moves[move][COL]*nchars
if onboard(nrow,ncol)
and board[nrow][ncol]=' ' then
wmoves = append(wmoves,{warnsdorffs[nrow][ncol],nrow,ncol})
end if
end for
wmoves = sort(wmoves)
-- avoid creating orphans
if length(wmoves)<2 or wmoves[2][1]>1 then
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
warnsdorffs[nrow][ncol] -= 1
end for
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
integer scol = ncol-nchars+1
board[nrow][scol..ncol] = sprintf(fmt,n)
if solve(nrow,ncol,n+1) then return 1 end if
backtracks += 1
board[nrow][scol..ncol] = blank
end for
for m=1 to length(wmoves) do
{?,nrow,ncol} = wmoves[m]
warnsdorffs[nrow][ncol] += 1
end for
end if
return 0
end function

procedure holyknight(sequence s)
s = split(s,'\n')
size = length(s)
nchars = length(sprintf(" %d",size*size))
fmt = sprintf(" %%%dd",nchars-1)
blank = repeat(' ',nchars)
board = repeat(repeat(' ',size*nchars),size)
limit = 1
integer sx, sy
for x=1 to size do
integer y = nchars
for j=1 to size do
integer ch = iff(j>length(s[x])?'-':s[x][j])
if ch=' ' then
ch = '-'
elsif ch='0' then
ch = ' '
limit += 1
elsif ch='1' then
sx = x
sy = y
end if
board[x][y] = ch
y += nchars
end for
end for
warnsdorffs = repeat(repeat(0,size*nchars),size)
init_warnsdorffs()
t0 = time()
tries = 0
backtracks = 0
t1 = time()+1
if solve(sx,sy,2) then
puts(1,join(board,"\n"))
printf(1,"\nsolution found in %d tries, %d backtracks (%3.2fs)\n",{tries,backtracks,time()-t0})
else
puts(1,"no solutions found\n")
end if
end procedure

constant board1 = """
000
0 00
0000000
000  0 0
0 0  000
1000000
00 0
000"""

holyknight(board1)

constant board2 = """
-----1-0-----
-----0-0-----
----00000----
-----000-----
--0--0-0--0--
00000---00000
--00-----00--
00000---00000
--0--0-0--0--
-----000-----
----00000----
-----0-0-----
-----0-0-----"""

if platform()!=JS then
holyknight(board2)
end if

{} = wait_key()
Output:
- 21  4 19  -  -  -  -
- 18  - 22  5  -  -  -
- 15 20  3 32 23  6  9
17  2 33  -  -  8  - 24
14  - 16  -  - 31 10  7
1 34 13 30 27 36 25  -
-  - 28 35  - 11  -  -
-  -  - 12 29 26  -  -
solution found in 31718 tries, 31682 backtracks (0.11s)

-   -   -   -   -   1   -  55   -   -   -   -   -
-   -   -   -   -   8   -   2   -   -   -   -   -
-   -   -   -   6   3  54   9  56   -   -   -   -
-   -   -   -   -  10   7   4   -   -   -   -   -
-   -  12   -   -   5   -  53   -   -  46   -   -
13  16  23  18  11   -   -   -  45  52  43  50  41
-   -  14  21   -   -   -   -   -  47  40   -   -
15  22  17  24  19   -   -   -  39  44  51  42  49
-   -  20   -   -  25   -  33   -   -  48   -   -
-   -   -   -   -  32  35  38   -   -   -   -   -
-   -   -   -  26  37  28  31  34   -   -   -   -
-   -   -   -   -  30   -  36   -   -   -   -   -
-   -   -   -   -  27   -  29   -   -   -   -   -
solution found in 61341542 tries, 61341486 backtracks (180.56s)

## Picat

import sat.

main =>
S = {".000....",
".0.00...",
".0000000",
"000..0.0",
"0.0..000",
"1000000.",
"..00.0..",
"...000.."},
MaxR = len(S),
MaxC = len(S[1]),
M = new_array(MaxR,MaxC),
StartR = _,    % make StartR and StartC global
StartC = _,
foreach (R in 1..MaxR)
fill_row(R, S[R], M[R], 1, StartR, StartC)
end,
NZeros = len([1 : R in 1..MaxR, C in 1..MaxC, M[R,C] == 0]),
M :: 0..MaxR*MaxC-NZeros,
Vs = [{(R,C),1} : R in 1..MaxR, C in 1..MaxC, M[R,C] !== 0],
Es = [{(R,C),(R1,C1),_} : R in 1..MaxR, C in 1..MaxC, M[R,C] !== 0,
neibs(M,MaxR,MaxC,R,C,Neibs),
(R1,C1) in Neibs, M[R1,C1] !== 0],
hcp(Vs,Es),
foreach ({(R,C),(R1,C1),B} in Es)
B #/\ (R1 #!= StartR #\/ C1 #!= StartC) #=> M[R1,C1] #= M[R,C]+1
end,
solve(M),
foreach (R in 1..MaxR)
foreach (C in 1..MaxC)
if M[R,C] == 0 then
printf("%4c", '.')
else
printf("%4d", M[R,C])
end
end,
nl
end.

neibs(M,MaxR,MaxC,R,C,Neibs) =>
Connected = [(R+1, C+2),
(R+1, C-2),
(R-1, C+2),
(R-1, C-2),
(R+2, C+1),
(R+2, C-1),
(R-2, C+1),
(R-2, C-1)],
Neibs = [(R1,C1) : (R1,C1) in Connected,
R1 >= 1, R1 =< MaxR, C1 >= 1, C1 =< MaxC,
M[R1,C1] !== 0].

fill_row(_R, [], _Row, _C, _StartR, _StartC) => true.
fill_row(R, ['.'|Str], Row, C, StartR, StartC) =>
Row[C] = 0,
fill_row(R,Str, Row, C+1, StartR, StartC).
fill_row(R, ['0'|Str], Row, C, StartR, StartC) =>
fill_row(R, Str, Row, C+1, StartR, StartC).
fill_row(R, ['1'|Str], Row, C, StartR, StartC) =>
Row[C] = 1,
StartR = R,
StartC = C,
fill_row(R, Str, Row, C+1, StartR, StartC).
Output:
.   7   4  25   .   .   .   .
.  26   .   8   5   .   .   .
.  29   6   3  24   9  18  11
27   2  23   .   .  12   .  16
30   .  28   .   .  17  10  19
1  22  31  34  13  20  15   .
.   .  36  21   .  33   .   .
.   .   .  32  35  14   .   .

## Python

from sys import stdout
moves = [
[-1, -2], [1, -2], [-1, 2], [1, 2],
[-2, -1], [-2, 1], [2, -1], [2, 1]
]

def solve(pz, sz, sx, sy, idx, cnt):
if idx > cnt:
return 1

for i in range(len(moves)):
x = sx + moves[i][0]
y = sy + moves[i][1]
if sz > x > -1 and sz > y > -1 and pz[x][y] == 0:
pz[x][y] = idx
if 1 == solve(pz, sz, x, y, idx + 1, cnt):
return 1
pz[x][y] = 0

return 0

def find_solution(pz, sz):
p = [[-1 for j in range(sz)] for i in range(sz)]
idx = x = y = cnt = 0
for j in range(sz):
for i in range(sz):
if pz[idx] == "x":
p[i][j] = 0
cnt += 1
elif pz[idx] == "s":
p[i][j] = 1
cnt += 1
x = i
y = j
idx += 1

if 1 == solve(p, sz, x, y, 2, cnt):
for j in range(sz):
for i in range(sz):
if p[i][j] != -1:
stdout.write(" {:0{}d}".format(p[i][j], 2))
else:
stdout.write("   ")
print()
else:
print("Cannot solve this puzzle!")

# entry point
find_solution(".xxx.....x.xx....xxxxxxxxxx..x.xx.x..xxxsxxxxxx...xx.x.....xxx..", 8)
print()
find_solution(".....s.x..........x.x.........xxxxx.........xxx.......x..x.x..x..xxxxx...xxxxx..xx.....xx..xxxxx...xxxxx..x..x.x..x.......xxx.........xxxxx.........x.x..........x.x.....", 13)
Output:

17 14 29
28    18 15
13 16 27 30 19 32 07
25 02 11       06    20
12    26       31 08 33
01 24 03 10 05 34 21
36 23    09
04 35 22

01    05
10    12
02 13 04 09 06
08 11 14
36       03    07       16
35 42 33 44 37          15 20 27 22 25
38 41                17 24
39 34 43 32 45          19 28 21 26 23
40       31    29       18
46 51 56
52 55 30 47 50
48    54
53    49

## Racket

This solution uses the module "hidato-family-solver.rkt" from Solve a Numbrix puzzle#Racket. The difference between the two is essentially the neighbourhood function.

It solves the tasked problem, as well as the "extra credit" from #Ada.

#lang racket
(require "hidato-family-solver.rkt")

(define knights-neighbour-offsets
'((+1 +2) (-1 +2) (+1 -2) (-1 -2) (+2 +1) (-2 +1) (+2 -1) (-2 -1)))

(define solve-a-knights-tour (solve-hidato-family knights-neighbour-offsets))

(displayln
(puzzle->string
(solve-a-knights-tour
#(#(_ 0 0 0 _ _ _ _)
#(_ 0 _ 0 0 _ _ _)
#(_ 0 0 0 0 0 0 0)
#(0 0 0 _ _ 0 _ 0)
#(0 _ 0 _ _ 0 0 0)
#(1 0 0 0 0 0 0 _)
#(_ _ 0 0 _ 0 _ _)
#(_ _ _ 0 0 0 _ _)))))

(newline)

(displayln
(puzzle->string
(solve-a-knights-tour
#(#(- - - - - 1 - 0 - - - - -)
#(- - - - - 0 - 0 - - - - -)
#(- - - - 0 0 0 0 0 - - - -)
#(- - - - - 0 0 0 - - - - -)
#(- - 0 - - 0 - 0 - - 0 - -)
#(0 0 0 0 0 - - - 0 0 0 0 0)
#(- - 0 0 - - - - - 0 0 - -)
#(0 0 0 0 0 - - - 0 0 0 0 0)
#(- - 0 - - 0 - 0 - - 0 - -)
#(- - - - - 0 0 0 - - - - -)
#(- - - - 0 0 0 0 0 - - - -)
#(- - - - - 0 - 0 - - - - -)
#(- - - - - 0 - 0 - - - - -)))))
Output:
_ 13 30 23  _  _  _  _
_ 24  _ 14 31  _  _  _
_ 29 12 25 22 15 32  7
11 26 21  _  _  6  _ 16
28  _ 10  _  _ 33  8  5
1 20 27 34  9  4 17  _
_  _  2 19  _ 35  _  _
_  _  _ 36  3 18  _  _

_   _   _   _   _   1   _  51   _   _   _   _   _
_   _   _   _   _  50   _   2   _   _   _   _   _
_   _   _   _  56   3  52  49  54   _   _   _   _
_   _   _   _   _  48  55   4   _   _   _   _   _
_   _  46   _   _   5   _  53   _   _  24   _   _
45   8  11   6  47   _   _   _  23  30  19  28  21
_   _  44   9   _   _   _   _   _  25  22   _   _
43  10   7  12  41   _   _   _  31  18  29  20  27
_   _  42   _   _  13   _  17   _   _  26   _   _
_   _   _   _   _  40  37  32   _   _   _   _   _
_   _   _   _  36  33  14  39  16   _   _   _   _
_   _   _   _   _  38   _  34   _   _   _   _   _
_   _   _   _   _  35   _  15   _   _   _   _   _

## Raku

(formerly Perl 6)

This uses a Warnsdorff solver, which cuts down the number of tries by more than a factor of six over the brute force approach. This same solver is used in:

[ -2, -1],  [ -2, 1],
[-1,-2],                       [-1,+2],
[+1,-2],                       [+1,+2],
[ +2, -1],  [ +2, 1];

put "\n" xx 60;

solveboard q:to/END/;
. 0 0 0
. 0 . 0 0
. 0 0 0 0 0 0 0
0 0 0 . . 0 . 0
0 . 0 . . 0 0 0
1 0 0 0 0 0 0
. . 0 0 . 0
. . . 0 0 0
END

sub solveboard(\$board) {
my \$max = +\$board.comb(/\w+/);
my \$width = \$max.chars;

my @grid;
my @known;
my @neigh;
my @degree;

@grid = \$board.lines.map: -> \$line {
[ \$line.words.map: { /^_/ ?? 0 !! /^\./ ?? Rat !! \$_ } ]
}

sub neighbors(\$y,\$x --> List) {
my \$y1 = \$y + .[0];
my \$x1 = \$x + .[1];
take [\$y1,\$x1] if defined @grid[\$y1][\$x1];
}
}

for ^@grid -> \$y {
for ^@grid[\$y] -> \$x {
if @grid[\$y][\$x] -> \$v {
@known[\$v] = [\$y,\$x];
}
if @grid[\$y][\$x].defined {
@neigh[\$y][\$x] = neighbors(\$y,\$x);
@degree[\$y][\$x] = +@neigh[\$y][\$x];
}
}
}
print "\e[0H\e[0J";

my \$tries = 0;

try_fill 1, @known[1];

sub try_fill(\$v, \$coord [\$y,\$x] --> Bool) {
return True if \$v > \$max;
\$tries++;

my \$old = @grid[\$y][\$x];

return False if +\$old and \$old != \$v;
return False if @known[\$v] and @known[\$v] !eqv \$coord;

@grid[\$y][\$x] = \$v;               # conjecture grid value

print "\e[0H";                    # show conjectured board
for @grid -> \$r {
say do for @\$r {
when Rat { ' ' x \$width }
when 0   { '_' x \$width }
default  { .fmt("%{\$width}d") }
}
}

my @neighbors = @neigh[\$y][\$x][];

my @degrees;
for @neighbors -> \n [\$yy,\$xx] {
my \$d = --@degree[\$yy][\$xx];  # conjecture new degrees
push @degrees[\$d], n;         # and categorize by degree
}

for @degrees.grep(*.defined) -> @ties {
for @ties.reverse {           # reverse works better for this hidato anyway
return True if try_fill \$v + 1, \$_;
}
}

for @neighbors -> [\$yy,\$xx] {
++@degree[\$yy][\$xx];          # undo degree conjectures
}

@grid[\$y][\$x] = \$old;             # undo grid value conjecture
return False;
}

say "\$tries tries";
}
Output:
25 14 27
36    24 15
31 26 13 28 23  6 17
35 12 29       16    22
30    32        7 18  5
1 34 11  8 19  4 21
2 33     9
10  3 20
84 tries

## REXX

This REXX program is essentially a modified   knight's tour   REXX program with support to place pennies on the chessboard.

Also supported is the specification of the size of the chessboard and the placement of the knight (initial position).

This is an   open tour   solution.   (See this task's   discussion   page for an explanation   [in the first section]).

/*REXX program solves the  holy knight's tour  problem for a (general)  NxN  chessboard.*/
blank=pos('//', space(arg(1), 0))\==0            /*see if the pennies are to be shown.  */
parse arg  ops   '/'   cent                      /*obtain the options and the pennies.  */
parse var  ops  N  sRank  sFile .                /*boardsize, starting position, pennys.*/
if     N=='' |     N==","  then     N=8          /*no boardsize specified?  Use default.*/
if sRank=='' | sRank==","  then sRank=N          /*starting rank given?      "      "   */
if sFile=='' | sFile==","  then sFile=1          /*    "    file   "         "      "   */
NN=N**2;  NxN='a ' N"x"N ' chessboard'           /*file  [↓]          [↓]   r=rank      */
@.=;              do r=1  for N;  do f=1  for N;   @.r.f=.;    end /*f*/;        end /*r*/
/*[↑]  create an empty  NxN chessboard.*/
cent=space( translate( cent, , ',') )            /*allow use of comma (,) for separater.*/
cents=0                                          /*number of pennies on the chessboard. */
do  while  cent\=''                       /* [↓]  possibly place the pennies.    */
parse var  cent   cr  cf  x  '/'  cent    /*extract where to place the pennies.  */
if x=''   then x=1                        /*if number not specified, use 1 penny.*/
if cr=''  then iterate                    /*support the  "blanking"  option.     */
do cf=cf  for x   /*now, place  X  pennies on chessboard.*/
@.cr.cf= '¢'      /*mark chessboard position with a penny*/
end   /*cf*/      /* [↑]  places X pennies on chessboard.*/
end   /*while*/                           /* [↑]  allows of the placing of  X  ¢s*/
/* [↓]  traipse through the chessboard.*/
do r=1  for N;  do f=1  for N;  cents=cents + (@.r.f=='¢');   end /*f*/;     end /*r*/
/* [↑]  count the number of pennies.   */
if cents\==0  then say cents   'pennies placed on chessboard.'
target=NN - cents                                /*use this as the number of moves left.*/
Kr = '2 1 -1 -2 -2 -1  1  2'  /*the legal "rank"  moves for a knight.*/
Kf = '1 2  2  1 -1 -2 -2 -1'  /* "    "   "file"    "    "  "    "   */
kr.M=words(Kr)                                   /*number of possible moves for a Knight*/
parse var Kr  Kr.1 Kr.2 Kr.3 Kr.4 Kr.5 Kr.6 Kr.7 Kr.8   /*parse the legal moves by hand.*/
parse var Kf  Kf.1 Kf.2 Kf.3 Kf.4 Kf.5 Kf.6 Kf.7 Kf.8   /*  "    "    "     "    "   "  */
beg= '-1-'                                       /* [↑]   create the  NxN  chessboard.  */
if @.sRank.sFile ==.    then @.sRank.sFile=beg   /*the knight's starting position.      */
if @.sRank.sFile\==beg  then do    sRank=1  for N   /*find starting rank for the knight.*/
do sFile=1  for N   /*  "     "     file  "   "     "   */
if @.sRank.sFile\==.  then iterate
@.sRank.sFile=beg   /*the knight's starting position.   */
leave sRank         /*we have a spot, so leave all this.*/
end   /*sFile*/
end      /*sRank*/
@hkt= "holy knight's tour"                       /*a handy─dandy literal for the  SAYs. */
if \move(2,sRank,sFile)  &  \(N==1)  then say 'No'    @hkt    "solution for"        NxN'.'
else say 'A solution for the'   @hkt   "on"    NxN':'

/*show chessboard with moves and coins.*/
!=left('', 9 * (n<18) );           say           /*used for indentation of chessboard.  */
_=substr( copies("┼───", N), 2);   say  ! translate('┌'_"┐", '┬', "┼")
do   r=N  for N  by -1;       if r\==N      then say ! '├'_"┤";     L=@.
do f=1  for N; ?=@.r.f;     if ?==target  then ?='end';           L=L'│'center(?,3)
end      /*f*/
if blank then L=translate(L,,'¢')           /*blank out the pennies on chessboard ?*/
say !  translate(L'│', , .)                 /*display  a  rank of the  chessboard. */
end        /*r*/                            /*19x19 chessboard can be shown 80 cols*/
say  !  translate('└'_"┘", '┴', "┼")             /*display the last rank of chessboard. */
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
move: procedure expose @. Kr. Kf. target; parse arg #,rank,file /*obtain move,rank,file.*/
do t=1  for Kr.M;   nr=rank+Kr.t;         nf=file+Kf.t  /*position of the knight*/
if @.nr.nf==.  then do;                   @.nr.nf=#     /*Empty? Knight can move*/
if #==target       then return 1 /*is this the last move?*/
if move(#+1,nr,nf) then return 1 /* "   "   "    "    "  */
@.nr.nf=.                        /*undo the above move.  */
end                                 /*try a different move. */
end   /*t*/                                             /* [↑]  all moves tried.*/
return 0                                                   /*tour isn't possible.  */

output   when the following is used for input:
, 3 1 /1,1 3 /1,7 2 /2,1 2 /2,5 /2,7 2 /3,8 /4,2 /4,4 2 /5,4 2 /5,7 /6,1 /7,1 /7,3 /7,6 3 /8,1 /8,5 4

28 pennies placed on chessboard.
A solution for the holy knight's tour on a  8x8  chessboard:

┌───┬───┬───┬───┬───┬───┬───┬───┐
│ ¢ │25 │12 │27 │ ¢ │ ¢ │ ¢ │ ¢ │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ ¢ │end│ ¢ │24 │13 │ ¢ │ ¢ │ ¢ │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ ¢ │11 │26 │ 3 │28 │23 │14 │ 5 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│35 │ 2 │31 │ ¢ │ ¢ │ 4 │ ¢ │22 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│10 │ ¢ │34 │ ¢ │ ¢ │29 │ 6 │15 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│-1-│32 │ 9 │30 │19 │16 │21 │ ¢ │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ ¢ │ ¢ │18 │33 │ ¢ │ 7 │ ¢ │ ¢ │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ ¢ │ ¢ │ ¢ │ 8 │17 │20 │ ¢ │ ¢ │
└───┴───┴───┴───┴───┴───┴───┴───┘

output   when the following   (for a "cleaner" chessboard, no pennies are shown)   is used for input:
, 3 1 /1,1 3 /1,7 2 /2,1 2 /2,5 /2,7 2 /3,8 /4,2 /4,4 2 /5,4 2 /5,7 /6,1 /7,1 /7,3 /7,6 3 /8,1 /8,5 4 //

28 pennies placed on chessboard.
A solution for the holy knight's tour on a  8x8  chessboard:

┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │25 │12 │27 │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │end│   │24 │13 │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │11 │26 │ 3 │28 │23 │14 │ 5 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│35 │ 2 │31 │   │   │ 4 │   │22 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│10 │   │34 │   │   │29 │ 6 │15 │
├───┼───┼───┼───┼───┼───┼───┼───┤
│-1-│32 │ 9 │30 │19 │16 │21 │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │18 │33 │   │ 7 │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │ 8 │17 │20 │   │   │
└───┴───┴───┴───┴───┴───┴───┴───┘

## Ruby

This solution uses HLPsolver from here

require 'HLPsolver'

boardy = <<EOS
. . 0 0 0
. . 0 . 0 0
. 0 0 0 0 0 0 0
0 0 0 . . 0 . 0
0 . 0 . . 0 0 0
1 0 0 0 0 0 0
. . 0 0 . 0
. . . 0 0 0
EOS
t0 = Time.now
HLPsolver.new(boardy).solve
puts " #{Time.now - t0} sec"

Which produces:

Problem:
0  0  0
0     0  0
0  0  0  0  0  0  0
0  0  0        0     0
0     0        0  0  0
1  0  0  0  0  0  0
0  0     0
0  0  0

Solution:
8 33 14
13     7 32
9 34 31 22 15  6 29
35 12 21       30    16
10    36       23 28  5
1 20 11 24 27  4 17
2 19    25
26  3 18

0.005 sec

## Tcl

Works with: Tcl version 8.6
package require Tcl 8.6

oo::class create HKTSolver {
variable grid start limit
constructor {puzzle} {
set grid \$puzzle
for {set y 0} {\$y < [llength \$grid]} {incr y} {
for {set x 0} {\$x < [llength [lindex \$grid \$y]]} {incr x} {
if {[set cell [lindex \$grid \$y \$x]] == 1} {
set start [list \$y \$x]
}
incr limit [expr {\$cell>=0}]
}
}
if {![info exist start]} {
return -code error "no starting position found"
}
}
method moves {} {
return {
-1 -2   1 -2
-2 -1          2 -1
-2  1          2 1
-1 2    1 2
}
}
method Moves {g r c} {
set valid {}
foreach {dr dc} [my moves] {
set R [expr {\$r + \$dr}]
set C [expr {\$c + \$dc}]
if {[lindex \$g \$R \$C] == 0} {
lappend valid \$R \$C
}
}
return \$valid
}

method Solve {g r c v} {
lset g \$r \$c [incr v]
if {\$v >= \$limit} {return \$g}
foreach {r c} [my Moves \$g \$r \$c] {
return [my Solve \$g \$r \$c \$v]
}
return -code continue
}

method solve {} {
while {[incr i]==1} {
set grid [my Solve \$grid {*}\$start 0]
return
}
return -code error "solution not possible"
}
method solution {} {return \$grid}
}

proc parsePuzzle {str} {
foreach line [split \$str "\n"] {
if {[string trim \$line] eq ""} continue
lappend rows [lmap {- c} [regexp -all -inline {(.)\s?} \$line] {
string map {" " -1} \$c
}]
}
set len [tcl::mathfunc::max {*}[lmap r \$rows {llength \$r}]]
for {set i 0} {\$i < [llength \$rows]} {incr i} {
while {[llength [lindex \$rows \$i]] < \$len} {
lset rows \$i end+1 -1
}
}
return \$rows
}
proc showPuzzle {grid name} {
foreach row \$grid {foreach cell \$row {incr c [expr {\$cell>=0}]}}
set len [string length \$c]
set u [string repeat "_" \$len]
puts "\$name with \$c cells"
foreach row \$grid {
puts [format "  %s" [join [lmap c \$row {
format "%*s" \$len [if {\$c==-1} list elseif {\$c==0} {set u} {set c}]
}]]]
}
}

set puzzle [parsePuzzle {
0 0 0
0   0 0
0 0 0 0 0 0 0
0 0 0     0   0
0   0     0 0 0
1 0 0 0 0 0 0
0 0   0
0 0 0
}]
showPuzzle \$puzzle "Input"
HKTSolver create hkt \$puzzle
hkt solve
showPuzzle [hkt solution] "Output"
Output:
Input with 36 cells
__ __ __
__    __ __
__ __ __ __ __ __ __
__ __ __       __    __
__    __       __ __ __
1 __ __ __ __ __ __
__ __    __
__ __ __
Output with 36 cells
13  6 15
8    12 31
5 14  7 16 27 32 29
9  2 11       30    26
4    22       17 28 33
1 10  3 18 21 34 25
36 23    19
20 35 24

## Wren

Translation of: Kotlin
Library: Wren-fmt
import "./fmt" for Fmt

var moves = [ [-1, -2], [1, -2], [-1, 2], [1, 2], [-2, -1], [-2, 1], [2, -1], [2, 1] ]

var board1 =
" xxx    " +
" x xx   " +
" xxxxxxx" +
"xxx  x x" +
"x x  xxx" +
"sxxxxxx " +
"  xx x  " +
"   xxx  "

var board2 =
".....s.x....." +
".....x.x....." +
"....xxxxx...." +
".....xxx....." +
"..x..x.x..x.." +
"xxxxx...xxxxx" +
"..xx.....xx.." +
"xxxxx...xxxxx" +
"..x..x.x..x.." +
".....xxx....." +
"....xxxxx...." +
".....x.x....." +
".....x.x....."

var solve // recursive
solve = Fn.new { |pz, sz, sx, sy, idx, cnt|
if (idx > cnt) return true
for (i in 0...moves.count) {
var x = sx + moves[i][0]
var y = sy + moves[i][1]
if ((x >= 0 && x < sz) && (y >= 0 && y < sz) && pz[x][y] == 0) {
pz[x][y] = idx
if (solve.call(pz, sz, x, y, idx + 1, cnt)) return true
pz[x][y] = 0
}
}
return false
}

var findSolution = Fn.new { |b, sz|
var pz = List.filled(sz, null)
for (i in 0...sz) pz[i] = List.filled(sz, -1)
var x = 0
var y = 0
var idx = 0
var cnt = 0
for (j in 0...sz) {
for (i in 0...sz) {
if (b[idx] == "x") {
pz[i][j] = 0
cnt = cnt + 1
} else if (b[idx] == "s") {
pz[i][j] = 1
cnt = cnt + 1
x = i
y = j
}
idx = idx + 1
}
}

if (solve.call(pz, sz, x, y, 2, cnt)) {
for (j in 0...sz) {
for (i in 0...sz) {
if (pz[i][j] != -1) {
Fmt.write("\$02d  ", pz[i][j])
} else {
System.write("--  ")
}
}
System.print()
}
} else {
System.print("Cannot solve this puzzle!")
}
}

findSolution.call(board1,  8)
System.print()
findSolution.call(board2, 13)
Output:
--  17  14  29  --  --  --  --
--  28  --  18  15  --  --  --
--  13  16  27  30  19  32  07
25  02  11  --  --  06  --  20
12  --  26  --  --  31  08  33
01  24  03  10  05  34  21  --
--  --  36  23  --  09  --  --
--  --  --  04  35  22  --  --

--  --  --  --  --  01  --  05  --  --  --  --  --
--  --  --  --  --  10  --  12  --  --  --  --  --
--  --  --  --  02  13  04  09  06  --  --  --  --
--  --  --  --  --  08  11  14  --  --  --  --  --
--  --  36  --  --  03  --  07  --  --  16  --  --
35  42  33  44  37  --  --  --  15  20  27  22  25
--  --  38  41  --  --  --  --  --  17  24  --  --
39  34  43  32  45  --  --  --  19  28  21  26  23
--  --  40  --  --  31  --  29  --  --  18  --  --
--  --  --  --  --  46  51  56  --  --  --  --  --
--  --  --  --  52  55  30  47  50  --  --  --  --
--  --  --  --  --  48  --  54  --  --  --  --  --
--  --  --  --  --  53  --  49  --  --  --  --  --