# Solve a Numbrix puzzle

Solve a Numbrix puzzle
You are encouraged to solve this task according to the task description, using any language you may know.

Numbrix puzzles are similar to Hidato. The most important difference is that it is only possible to move 1 node left, right, up, or down (sometimes referred to as the Von Neumann neighborhood). Published puzzles also tend not to have holes in the grid and may not always indicate the end node. Two examples follow:

Example 1

Problem.

``` 0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0
```

Solution.

``` 49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5
```
Example 2

Problem.

``` 0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0
```

Solution.

```  9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

Write a program to solve puzzles of this ilk, demonstrating your program by solving the above examples. Extra credit for other interesting examples.

## 11l

 This example is incorrect. Please fix the code and remove this message.Details: 3rd solution has "00 00" in it where "02 01" shd be (as Python)
Translation of: Python
```V neighbours = [[-1, 0], [0, -1], [1, 0], [0, 1]]
[Int] exists
V lastNumber = 0
V wid = 0
V hei = 0

F find_next(pa, x, y, z)
L(i) 4
V a = x + :neighbours[i][0]
V b = y + :neighbours[i][1]
I a C -1 <.< :wid & b C -1 <.< :hei
I pa[a][b] == z
R (a, b)
R (-1, -1)

F find_solution(&pa, x, y, z)
I z > :lastNumber
R 1
I :exists[z] == 1
V s = find_next(pa, x, y, z)
I s[0] < 0
R 0
R find_solution(&pa, s[0], s[1], z + 1)

L(i) 4
V a = x + :neighbours[i][0]
V b = y + :neighbours[i][1]
I a C -1 <.< :wid & b C -1 <.< :hei
I pa[a][b] == 0
pa[a][b] = z
V r = find_solution(&pa, a, b, z + 1)
I r == 1
R 1
pa[a][b] = 0
R 0

F solve(pz, w, h)
:lastNumber = w * h
:wid = w
:hei = h
:exists = [0] * (:lastNumber + 1)

V pa = [[0] * h] * w
V st = pz.split(‘ ’)
V idx = 0

L(j) 0 .< h
L(i) 0 .< w
I st[idx] == ‘.’
idx++
E
pa[i][j] = Int(st[idx])
:exists[pa[i][j]] = 1
idx++

V x = 0
V y = 0
V t = w * h + 1
L(j) 0 .< h
L(i) 0 .< w
I pa[i][j] != 0 & pa[i][j] < t
t = pa[i][j]
x = i
y = j

R (find_solution(&pa, x, y, t + 1), pa)

F show_result(r)
I r[0] == 1
L(j) 0 .< :hei
L(i) 0 .< :wid
print(‘ #02’.format(r[1][i][j]), end' ‘’)
print()
E
print(‘No Solution!’)

print()

V r = solve(‘. . . . . . . . . . . 46 45 . 55 74 . . . 38 . . 43 . . 78 . . 35 . . . . . 71 . . . 33 . . . 59 . . . 17’""
‘ . . . . . 67 . . 18 . . 11 . . 64 . . . 24 21 . 1 2 . . . . . . . . . . .’, 9, 9)
show_result(r)
r = solve(‘. . . . . . . . . . 11 12 15 18 21 62 61 . . 6 . . . . . 60 . . 33 . . . . . 57 . . 32 . . . . . 56 . . 37’""
‘ . 1 . . . 73 . . 38 . . . . . 72 . . 43 44 47 48 51 76 77 . . . . . . . . . .’, 9, 9)
show_result(r)
r = solve(‘17 . . . 11 . . . 59 . 15 . . 6 . . 61 . . . 3 . . . 63 . . . . . . 66 . . . . 23 24 . 68 67 78 . 54 55’""
‘ . . . . 72 . . . . . . 35 . . . 49 . . . 29 . . 40 . . 47 . 31 . . . 39 . . . 45’, 9, 9)
show_result(r)```
Output:
``` 49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10 01 02 03 04
27 26 23 22 09 08 07 06 05

09 10 13 14 19 20 63 64 65
08 11 12 15 18 21 62 61 66
07 06 05 16 17 22 59 60 67
34 33 04 03 24 23 58 57 68
35 32 31 02 25 54 55 56 69
36 37 30 01 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79

17 16 13 12 11 10 09 60 59
18 15 14 05 06 07 08 61 58
19 20 03 04 65 64 63 62 57
22 21 00 00 66 79 80 81 56
23 24 69 68 67 78 77 54 55
26 25 70 71 72 75 76 53 52
27 28 35 36 73 74 49 50 51
30 29 34 37 40 41 48 47 46
31 32 33 38 39 42 43 44 45
```

## AutoHotkey

```SolveNumbrix(Grid, Locked, Max, row, col, num:=1, R:="", C:=""){
if (R&&C)							; if neighbors (not first iteration)
{
Grid[R, C] := ">" num 					; place num in current neighbor and mark it visited ">"
row:=R, col:=C						; move to current neighbor
}

num++								; increment num
if (num=max)							; if reached end
return map(Grid)					; return solution

if locked[num]							; if current num is a locked value
{
row := StrSplit((StrSplit(locked[num], ",").1) , ":").1	; find row of num
col := StrSplit((StrSplit(locked[num], ",").1) , ":").2	; find col of num
if SolveNumbrix(Grid, Locked, Max, row, col, num)	; solve for current location and value
return map(Grid)				; if solved, return solution
}
else
{
for each, value in StrSplit(Neighbor(row,col), ",")
{
R := StrSplit(value, ":").1
C := StrSplit(value, ":").2

if (Grid[R,C] = "")				; a hole or out of bounds
|| InStr(Grid[R, C], ">")			; visited
|| Locked[num+1] && !(Locked[num+1]~= "\b" R ":" C "\b") ; not neighbor of locked[num+1]
|| Locked[num-1] && !(Locked[num-1]~= "\b" R ":" C "\b") ; not neighbor of locked[num-1]
|| Locked[num]					; locked value
|| Locked[Grid[R, C]]				; locked cell
continue

if SolveNumbrix(Grid, Locked, Max, row, col, num, R, C)	; solve for current location, neighbor and value
return map(Grid)			; if solved, return solution
}
}
num--								; step back
for i, line in Grid
for j, element in line
if InStr(element, ">") && (StrReplace(element, ">") >= num)
Grid[i, j] := 0
}
;--------------------------------
;--------------------------------
;--------------------------------
Neighbor(row,col){
return row-1 ":" col
. "," row+1 ":" col
. "," row ":" col+1
. "," row ":" col-1
}
;--------------------------------
map(Grid){
for i, row in Grid
{
for j, element in row
line .= (A_Index > 1 ? "`t" : "") . element
map .= (map<>""?"`n":"") line
line := ""
}
return StrReplace(map, ">")
}
```

Examples:

```;--------------------------------
Grid := [[0,	0,	0,	0,	0,	0,	0,	0,	0]
,[0,	0,	46,	45,	0,	55,	74,	0,	0]
,[0,	38,	0,	0,	43,	0,	0,	78,	0]
,[0,	35,	0,	0,	0,	0,	0,	71,	0]
,[0,	0,	33,	0,	0,	0,	59,	0,	0]
,[0,	17,	0,	0,	0,	0,	0,	67,	0]
,[0,	18,	0,	0,	11,	0,	0,	64,	0]
,[0,	0,	24,	21,	0,	1,	2,	0,	0]
,[0,	0,	0,	0,	0,	0,	0,	0,	0]]
;--------------------------------
; find locked cells, find row and col of first value "1" and max value
Locked := []
max := 1
for i, line in Grid
for j, element in line
{
max ++
if element = 1
row :=i , col := j
if (element > 0)
Locked[element] := i ":" j "," Neighbor(i, j)	; save locked elements position and neighbors

}
;--------------------------------
MsgBox, 262144, ,% SolveNumbrix(Grid, Locked, Max, row, col)
return
```

Outputs:

```49	50	51	52	53	54	75	76	81
48	47	46	45	44	55	74	77	80
37	38	39	40	43	56	73	78	79
36	35	34	41	42	57	72	71	70
31	32	33	14	13	58	59	68	69
30	17	16	15	12	61	60	67	66
29	18	19	20	11	62	63	64	65
28	25	24	21	10	1	2	3	4
27	26	23	22	9	8	7	6	5```

## 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.

```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
numbrixMoves = {(1,0),(0,1),(-1,0),(0,-1)};

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

public static void Main()
{
var numbrixSolver = new Solver(numbrixMoves);
Print(numbrixSolver.Solve(false, new [,] {
{  0,  0,  0,  0,  0,  0,  0,  0,  0 },
{  0,  0, 46, 45,  0, 55, 74,  0,  0 },
{  0, 38,  0,  0, 43,  0,  0, 78,  0 },
{  0, 35,  0,  0,  0,  0,  0, 71,  0 },
{  0,  0, 33,  0,  0,  0, 59,  0,  0 },
{  0, 17,  0,  0,  0,  0,  0, 67,  0 },
{  0, 18,  0,  0, 11,  0,  0, 64,  0 },
{  0,  0, 24, 21,  0,  1,  2,  0,  0 },
{  0,  0,  0,  0,  0,  0,  0,  0,  0 },
}));

Print(numbrixSolver.Solve(false, new [,] {
{  0,  0,  0,  0,  0,  0,  0,  0,  0 },
{  0, 11, 12, 15, 18, 21, 62, 61,  0 },
{  0,  6,  0,  0,  0,  0,  0, 60,  0 },
{  0, 33,  0,  0,  0,  0,  0, 57,  0 },
{  0, 32,  0,  0,  0,  0,  0, 56,  0 },
{  0, 37,  0,  1,  0,  0,  0, 73,  0 },
{  0, 38,  0,  0,  0,  0,  0, 72,  0 },
{  0, 43, 44, 47, 48, 51, 76, 77,  0 },
{  0,  0,  0,  0,  0,  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:
```49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

## C++

```#include <vector>
#include <sstream>
#include <iostream>
#include <iterator>
#include <cstdlib>
#include <string>
#include <bitset>

using namespace std;
typedef bitset<4> hood_t;

struct node
{
int val;
hood_t neighbors;
};

class nSolver
{
public:

void solve(vector<string>& puzz, int max_wid)
{
if (puzz.size() < 1) return;
wid = max_wid;
hei = static_cast<int>(puzz.size()) / wid;
max = wid * hei;
int len = max, c = 0;
arr = vector<node>(len, node({ 0, 0 }));
weHave = vector<bool>(len + 1, false);

for (const auto& s : puzz)
{
if (s == "*") { max--; arr[c++].val = -1; continue; }
arr[c].val = atoi(s.c_str());
if (arr[c].val > 0) weHave[arr[c].val] = true;
c++;
}

solveIt(); c = 0;
for (auto&& s : puzz)
{
if (s == ".")
s = std::to_string(arr[c].val);
c++;
}
}

private:
bool search(int x, int y, int w, int dr)
{
if ((w > max && dr > 0) || (w < 1 && dr < 0) || (w == max && weHave[w])) return true;

node& n = arr[x + y * wid];
n.neighbors = getNeighbors(x, y);
if (weHave[w])
{
for (int d = 0; d < 4; d++)
{
if (n.neighbors[d])
{
int a = x + dx[d], b = y + dy[d];
if (arr[a + b * wid].val == w)
if (search(a, b, w + dr, dr))
return true;
}
}
return false;
}

for (int d = 0; d < 4; d++)
{
if (n.neighbors[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 + dr, dr))
return true;
arr[a + b * wid].val = 0;
}
}
}
return false;
}

hood_t getNeighbors(int x, int y)
{
hood_t retval;
for (int xx = 0; xx < 4; xx++)
{
int 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)
retval.set(xx);
}
return retval;
}

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, 1);
if (z > 1) search(x, y, z - 1, -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;
}

}

vector<int> dx = vector<int>({ -1, 1, 0, 0 });
vector<int> dy = vector<int>({ 0, 0, -1, 1 });
int wid, hei, max;
vector<node> arr;
vector<bool> weHave;
};

//------------------------------------------------------------------------------
int main(int argc, char* argv[])
{
int wid; string p;
//p = ". . . . . . . . . . . 46 45 . 55 74 . . . 38 . . 43 . . 78 . . 35 . . . . . 71 . . . 33 . . . 59 . . . 17 . . . . . 67 . . 18 . . 11 . . 64 . . . 24 21 . 1  2 . . . . . . . . . . ."; wid = 9;
//p = ". . . . . . . . . . 11 12 15 18 21 62 61 . .  6 . . . . . 60 . . 33 . . . . . 57 . . 32 . . . . . 56 . . 37 .  1 . . . 73 . . 38 . . . . . 72 . . 43 44 47 48 51 76 77 . . . . . . . . . ."; wid = 9;
p = "17 . . . 11 . . . 59 . 15 . . 6 . . 61 . . . 3 . . .  63 . . . . . . 66 . . . . 23 24 . 68 67 78 . 54 55 . . . . 72 . . . . . . 35 . . . 49 . . . 29 . . 40 . . 47 . 31 . . . 39 . . . 45"; wid = 9;

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 (const auto& s : puzz)
{
if (s != "*" && s != ".")
{
if (atoi(s.c_str()) < 10) cout << "0";
cout << s << " ";
}
else cout << "   ";
if (++c >= wid) { cout << endl; c = 0; }
}
cout << endl << endl;
return system("pause");
}
```
Output:
```49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10 01 02 03 04
27 26 23 22 09 08 07 06 05

09 10 13 14 19 20 63 64 65
08 11 12 15 18 21 62 61 66
07 06 05 16 17 22 59 60 67
34 33 04 03 24 23 58 57 68
35 32 31 02 25 54 55 56 69
36 37 30 01 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79

17 16 13 12 11 10 09 60 59
18 15 14 05 06 07 08 61 58
19 20 03 04 65 64 63 62 57
22 21 02 01 66 79 80 81 56
23 24 69 68 67 78 77 54 55
26 25 70 71 72 75 76 53 52
27 28 35 36 73 74 49 50 51
30 29 34 37 40 41 48 47 46
31 32 33 38 39 42 43 44 45
```

## D

From the refactored C++ version with more precise typing. The NumbrixPuzzle struct is created at compile-time, so its asserts and exceptions can catch most malformed puzzles at compile-time.

Translation of: C++
```import std.stdio, std.conv, std.string, std.range, std.array, std.typecons, std.algorithm;

struct  {
alias BitSet8 = ubyte; // A set of 8 bits.
alias Cell = uint;
enum : string { unavailableInCell = "#", availableInCell = "." }
enum : Cell { unavailableCell = Cell.max, availableCell = 0 }

this(in string inPuzzle) pure @safe {
const rawPuzzle = inPuzzle.splitLines.map!(row => row.split).array;
assert(!rawPuzzle.empty);
assert(!rawPuzzle[0].empty);
assert(rawPuzzle.all!(row => row.length == rawPuzzle[0].length)); // Is rectangular.

gridWidth = rawPuzzle[0].length;
gridHeight = rawPuzzle.length;
immutable nMaxCells = gridWidth * gridHeight;
grid = new Cell[nMaxCells];
auto knownMutable = new bool[nMaxCells + 1];
uint nAvailableMutable = nMaxCells;
bool[Cell] seenCells; // To avoid duplicate input numbers.

uint i = 0;
foreach (const piece; rawPuzzle.join) {
if (piece == unavailableInCell) {
nAvailableMutable--;
grid[i++] = unavailableCell;
continue;
} else if (piece == availableInCell) {
grid[i] = availableCell;
} else {
immutable cell = piece.to!Cell;
assert(cell > 0 && cell <= nMaxCells);
assert(cell !in seenCells);
seenCells[cell] = true;
knownMutable[cell] = true;
grid[i] = cell;
}

i++;
}

known = knownMutable.idup;
nAvailable = nAvailableMutable;
}

@disable this();

auto solve() pure nothrow @safe @nogc
out(result) {
if (!result.isNull) {
// Can't verify 'result' here because it's const.
// assert(!result.get.join.canFind(availableCell.text));

assert(!grid.canFind(availableCell));
auto values = grid.filter!(c => c != unavailableCell);
auto interval = iota(reduce!min(values.front, values.dropOne),
reduce!max(values.front, values.dropOne) + 1);
assert(values.walkLength == interval.length);
assert(interval.all!(c => values.count(c) == 1)); // Quadratic.
}
} body {
auto result = grid
.map!(c => (c == unavailableCell) ? unavailableInCell : c.text)
.chunks(gridWidth);
alias OutRange = Nullable!(typeof(result));

const start = findStart;
if (start.isNull)
return OutRange();

search(start.r, start.c, start.cell + 1, 1);
if (start.cell > 1) {
immutable direction = -1;
search(start.r, start.c, start.cell + direction, direction);
}

if (grid.any!(c => c == availableCell))
return OutRange();
else
return OutRange(result);
}

private:

bool search(in uint r, in uint c, in Cell cell, in int direction)
pure nothrow @safe @nogc {
if ((cell > nAvailable && direction > 0) || (cell == 0 && direction < 0) ||
(cell == nAvailable && known[cell]))
return true; // One solution found.

immutable neighbors = getNeighbors(r, c);

if (known[cell]) {
foreach (immutable i, immutable rc; shifts) {
if (neighbors & (1u << i)) {
immutable c2 = c + rc[0],
r2 = r + rc[1];
if (grid[r2 * gridWidth + c2] == cell)
if (search(r2, c2, cell + direction, direction))
return true;
}
}
return false;
}

foreach (immutable i, immutable rc; shifts) {
if (neighbors & (1u << i)) {
immutable c2 = c + rc[0],
r2 = r + rc[1],
pos = r2 * gridWidth + c2;
if (grid[pos] == availableCell) {
grid[pos] = cell;          // Try.
if (search(r2, c2, cell + direction, direction))
return true;
grid[pos] = availableCell; // Restore.
}
}
}
return false;
}

BitSet8 getNeighbors(in uint r, in uint c) const pure nothrow @safe @nogc {
typeof(return) usable = 0;

foreach (immutable i, immutable rc; shifts) {
immutable c2 = c + rc[0],
r2 = r + rc[1];
if (c2 >= gridWidth || r2 >= gridHeight)
continue;
if (grid[r2 * gridWidth + c2] != unavailableCell)
usable |= (1u << i);
}

return usable;
}

auto findStart() const pure nothrow @safe @nogc {
alias Triple = Tuple!(uint,"r", uint,"c", Cell,"cell");
Nullable!Triple result;

auto cell = Cell.max;
foreach (immutable r; 0 .. gridHeight) {
foreach (immutable c; 0 .. gridWidth) {
immutable pos = gridWidth * r + c;
if (grid[pos] != availableCell &&
grid[pos] != unavailableCell && grid[pos] < cell) {
cell = grid[pos];
result = Triple(r, c, cell);
}
}
}

return result;
}

static immutable int[2][4] shifts = [[0, -1], [0, 1], [-1, 0], [1, 0]];
immutable uint gridWidth, gridHeight;
immutable int nAvailable;
immutable bool[] known; // Given known cells of the puzzle.
Cell[] grid;  // Flattened mutable game grid.
}

void main() {
// enum NumbrixPuzzle to catch malformed puzzles at compile-time.
enum puzzle1 = ".  .  .  .  .  .  .  .  .
.  . 46 45  . 55 74  .  .
. 38  .  . 43  .  . 78  .
. 35  .  .  .  .  . 71  .
.  . 33  .  .  . 59  .  .
. 17  .  .  .  .  . 67  .
. 18  .  . 11  .  . 64  .
.  . 24 21  .  1  2  .  .
.  .  .  .  .  .  .  .  .".NumbrixPuzzle;

enum puzzle2 = ".  .  .  .  .  .  .  .  .
. 11 12 15 18 21 62 61  .
.  6  .  .  .  .  . 60  .
. 33  .  .  .  .  . 57  .
. 32  .  .  .  .  . 56  .
. 37  .  1  .  .  . 73  .
. 38  .  .  .  .  . 72  .
. 43 44 47 48 51 76 77  .
.  .  .  .  .  .  .  .  .".NumbrixPuzzle;

enum puzzle3 = "17  .  .  . 11  .  .  . 59
. 15  .  .  6  .  . 61  .
.  .  3  .  .  . 63  .  .
.  .  .  . 66  .  .  .  .
23 24  . 68 67 78  . 54 55
.  .  .  . 72  .  .  .  .
.  . 35  .  .  . 49  .  .
. 29  .  . 40  .  . 47  .
31  .  .  . 39  .  .  . 45".NumbrixPuzzle;

foreach (puzzle; [puzzle1, puzzle2, puzzle3]) {
auto solution = puzzle.solve; // 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:
49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

One solution:
9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79

One solution:
17 16 13 12 11 10  9 60 59
18 15 14  5  6  7  8 61 58
19 20  3  4 65 64 63 62 57
22 21  2  1 66 79 80 81 56
23 24 69 68 67 78 77 54 55
26 25 70 71 72 75 76 53 52
27 28 35 36 73 74 49 50 51
30 29 34 37 40 41 48 47 46
31 32 33 38 39 42 43 44 45```

## Elixir

Translation of: Ruby

This solution uses HLPsolver from here

```# require HLPsolver

adjacent = [{-1, 0}, {0, -1}, {0, 1}, {1, 0}]

board1 = """
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0
"""

board2 = """
0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0
"""
```
Output:
```Problem:
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0

Solution:
49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

Problem:
0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0

Solution:
9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

## Go

Translation of: Kotlin
```package main

import (
"fmt"
"sort"
"strconv"
"strings"
)

var example1 = []string{
"00,00,00,00,00,00,00,00,00",
"00,00,46,45,00,55,74,00,00",
"00,38,00,00,43,00,00,78,00",
"00,35,00,00,00,00,00,71,00",
"00,00,33,00,00,00,59,00,00",
"00,17,00,00,00,00,00,67,00",
"00,18,00,00,11,00,00,64,00",
"00,00,24,21,00,01,02,00,00",
"00,00,00,00,00,00,00,00,00",
}

var example2 = []string{
"00,00,00,00,00,00,00,00,00",
"00,11,12,15,18,21,62,61,00",
"00,06,00,00,00,00,00,60,00",
"00,33,00,00,00,00,00,57,00",
"00,32,00,00,00,00,00,56,00",
"00,37,00,01,00,00,00,73,00",
"00,38,00,00,00,00,00,72,00",
"00,43,44,47,48,51,76,77,00",
"00,00,00,00,00,00,00,00,00",
}

var moves = [][2]int{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}

var (
grid        [][]int
clues       []int
totalToFill = 0
)

func solve(r, c, count, nextClue int) bool {
if count > totalToFill {
return true
}

back := grid[r][c]

if back != 0 && back != count {
return false
}

if back == 0 && nextClue < len(clues) && clues[nextClue] == count {
return false
}

if back == count {
nextClue++
}

grid[r][c] = count
for _, move := range moves {
if solve(r+move[1], c+move[0], count+1, nextClue) {
return true
}
}
grid[r][c] = back
return false
}

func printResult(n int) {
fmt.Println("Solution for example", n, "\b:")
for _, row := range grid {
for _, i := range row {
if i == -1 {
continue
}
fmt.Printf("%2d ", i)
}
fmt.Println()
}
}

func main() {
for n, board := range [2][]string{example1, example2} {
nRows := len(board) + 2
nCols := len(strings.Split(board[0], ",")) + 2
startRow, startCol := 0, 0
grid = make([][]int, nRows)
totalToFill = (nRows - 2) * (nCols - 2)
var lst []int

for r := 0; r < nRows; r++ {
grid[r] = make([]int, nCols)
for c := 0; c < nCols; c++ {
grid[r][c] = -1
}
if r >= 1 && r < nRows-1 {
row := strings.Split(board[r-1], ",")
for c := 1; c < nCols-1; c++ {
val, _ := strconv.Atoi(row[c-1])
if val > 0 {
lst = append(lst, val)
}
if val == 1 {
startRow, startCol = r, c
}
grid[r][c] = val
}
}
}

sort.Ints(lst)
clues = lst
if solve(startRow, startCol, 1, 0) {
printResult(n + 1)
}
}
}
```
Output:
```Solution for example 1:

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

Solution for example 2:

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

## Icon and Unicon

This is a Unicon-specific solution, based on the Unicon Hidato problem solver:

```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 := []
nCells := maxCols := 0
every line := !&input do {
put(p,[: gencells(line) :])
maxCols <:= *p[-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["_"] :=  0; 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, puzzle[loc.r,loc.c] = (val|0)); end
method visit(r,c); return (/best, validPos(r,c), Pos(r,c)); end

method validPos(r,c)
v := val+1      # number we're looking for
xv := puzzle[r,c] | fail
if (xv ~= 0) & (xv != v) then fail
if xv = (0|v) then {
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-1,loc.c)  , self, val)   # North
QMouse(puzzle, visit(loc.r,  loc.c+1), self, val)   # East
QMouse(puzzle, visit(loc.r+1,loc.c),   self, val)   # South
QMouse(puzzle, visit(loc.r,  loc.c-1), self, val)   # West
end
```
Output:

Sample runs

```->numbrix <numbrix1.in
Input with 81 cells:

_  _  _  _  _  _  _  _  _
_  _ 46 45  _ 55 74  _  _
_ 38  _  _ 43  _  _ 78  _
_ 35  _  _  _  _  _ 71  _
_  _ 33  _  _  _ 59  _  _
_ 17  _  _  _  _  _ 67  _
_ 18  _  _ 11  _  _ 64  _
_  _ 24 21  _  1  2  _  _
_  _  _  _  _  _  _  _  _

Output with 81 cells:

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

->numbrix <numbrix2.in
Input with 81 cells:

_  _  _  _  _  _  _  _  _
_ 11 12 15 18 21 62 61  _
_  6  _  _  _  _  _ 60  _
_ 33  _  _  _  _  _ 57  _
_ 32  _  _  _  _  _ 56  _
_ 37  _  1  _  _  _ 73  _
_ 38  _  _  _  _  _ 72  _
_ 43 44 47 48 51 76 77  _
_  _  _  _  _  _  _  _  _

Output with 81 cells:

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79

->
```

## Java

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

public class Numbrix {

final static String[] board = {
"00,00,00,00,00,00,00,00,00",
"00,00,46,45,00,55,74,00,00",
"00,38,00,00,43,00,00,78,00",
"00,35,00,00,00,00,00,71,00",
"00,00,33,00,00,00,59,00,00",
"00,17,00,00,00,00,00,67,00",
"00,18,00,00,11,00,00,64,00",
"00,00,24,21,00,01,02,00,00",
"00,00,00,00,00,00,00,00,00"};

final static int[][] moves = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};

static int[][] grid;
static int[] clues;
static int totalToFill;

public static void main(String[] args) {
int nRows = board.length + 2;
int nCols = board[0].split(",").length + 2;
int startRow = 0, startCol = 0;

grid = new int[nRows][nCols];
totalToFill = (nRows - 2) * (nCols - 2);
List<Integer> lst = new ArrayList<>();

for (int r = 0; r < nRows; r++) {
Arrays.fill(grid[r], -1);

if (r >= 1 && r < nRows - 1) {

String[] row = board[r - 1].split(",");

for (int c = 1; c < nCols - 1; c++) {
int val = Integer.parseInt(row[c - 1]);
if (val > 0)
if (val == 1) {
startRow = r;
startCol = c;
}
grid[r][c] = val;
}
}
}

clues = lst.stream().sorted().mapToInt(i -> i).toArray();

if (solve(startRow, startCol, 1, 0))
printResult();
}

static boolean solve(int r, int c, int count, int nextClue) {
if (count > totalToFill)
return true;

if (grid[r][c] != 0 && grid[r][c] != count)
return false;

if (grid[r][c] == 0 && nextClue < clues.length)
if (clues[nextClue] == count)
return false;

int back = grid[r][c];
if (back == count)
nextClue++;

grid[r][c] = count;
for (int[] move : moves)
if (solve(r + move[1], c + move[0], count + 1, nextClue))
return true;

grid[r][c] = back;
return false;
}

static void printResult() {
for (int[] row : grid) {
for (int i : row) {
if (i == -1)
continue;
System.out.printf("%2d ", i);
}
System.out.println();
}
}
}
```
```49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5 ```

## Julia

See the Hidato module here.

```using .Hidato

const numbrixmoves = [[-1, 0], [0, -1], [0, 1], [1, 0]]

board, maxmoves, fixed, starts = hidatoconfigure(numbrix1)
printboard(board, " 0 ", "   ")
hidatosolve(board, maxmoves, numbrixmoves, fixed, starts[1][1], starts[1][2], 1)
printboard(board)

board, maxmoves, fixed, starts = hidatoconfigure(numbrix2)
printboard(board, " 0 ", "   ")
hidatosolve(board, maxmoves, numbrixmoves, fixed, starts[1][1], starts[1][2], 1)
printboard(board)
```
Output:
```
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79

```

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 numbrix1 = """
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0 """

const numbrix2 = """
0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0 """

const numbrixmoves = [[-1, 0], [0, -1], [0, 1], [1, 0]]

board, maxmoves, fixed, starts = hidatoconfigure(numbrix1)
printboard(board, " 0 ", "   ")
hidatosolve(board, maxmoves, numbrixmoves, fixed, starts[1][1], starts[1][2], 1)
printboard(board)

board, maxmoves, fixed, starts = hidatoconfigure(numbrix2)
printboard(board, " 0 ", "   ")
hidatosolve(board, maxmoves, numbrixmoves, fixed, starts[1][1], starts[1][2], 1)
printboard(board)
```
Output:
``` 0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

## Kotlin

Translation of: Java
```// version 1.2.0

val example1 = listOf(
"00,00,00,00,00,00,00,00,00",
"00,00,46,45,00,55,74,00,00",
"00,38,00,00,43,00,00,78,00",
"00,35,00,00,00,00,00,71,00",
"00,00,33,00,00,00,59,00,00",
"00,17,00,00,00,00,00,67,00",
"00,18,00,00,11,00,00,64,00",
"00,00,24,21,00,01,02,00,00",
"00,00,00,00,00,00,00,00,00"
)

val example2 = listOf(
"00,00,00,00,00,00,00,00,00",
"00,11,12,15,18,21,62,61,00",
"00,06,00,00,00,00,00,60,00",
"00,33,00,00,00,00,00,57,00",
"00,32,00,00,00,00,00,56,00",
"00,37,00,01,00,00,00,73,00",
"00,38,00,00,00,00,00,72,00",
"00,43,44,47,48,51,76,77,00",
"00,00,00,00,00,00,00,00,00"
)

val moves = listOf(1 to 0, 0 to 1, -1 to 0, 0 to -1)

lateinit var board: List<String>
lateinit var grid: List<IntArray>
lateinit var clues: IntArray
var totalToFill = 0

fun solve(r: Int, c: Int, count: Int, nextClue: Int): Boolean {
if (count > totalToFill) return true
val back = grid[r][c]
if (back != 0 && back != count) return false
if (back == 0 && nextClue < clues.size && clues[nextClue] == count) {
return false
}
var nextClue2 = nextClue
if (back == count) nextClue2++
grid[r][c] = count
for (m in moves) {
if (solve(r + m.second, c + m.first, count + 1, nextClue2)) return true
}
grid[r][c] = back
return false
}

fun printResult(n: Int) {
println("Solution for example \$n:")
for (row in grid) {
for (i in row) {
if (i == -1) continue
print("%2d ".format(i))
}
println()
}
}

fun main(args: Array<String>) {
for ((n, ex) in listOf(example1, example2).withIndex()) {
board = ex
val nRows = board.size + 2
val nCols = board[0].split(",").size + 2
var startRow = 0
var startCol = 0
grid = List(nRows) { IntArray(nCols) { -1 } }
totalToFill = (nRows - 2) * (nCols - 2)
val lst = mutableListOf<Int>()
for (r in 0 until nRows) {
if (r in 1 until nRows - 1) {
val row = board[r - 1].split(",")
for (c in 1 until nCols - 1) {
val value = row[c - 1].toInt()
if (value == 1) {
startRow = r
startCol = c
}
grid[r][c] = value
}
}
}
lst.sort()
clues = lst.toIntArray()
if (solve(startRow, startCol, 1, 0)) printResult(n + 1)
}
}
```
Output:
```Solution for example 1:

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

Solution for example 2:

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

## Mathematica/Wolfram Language

```ClearAll[NeighbourQ, CellDistance, VisualizeHidato, HiddenSingle, \
NakedN, HiddenN, ChainSearch, HidatoSolve, Cornering, ValidPuzzle, \
GapSearch, ReachDelete, GrowNeighbours]
NeighbourQ[cell1_, cell2_] := (CellDistance[cell1, cell2] === 1)
ValidPuzzle[cells_List, cands_List] :=
MemberQ[cands, {1}] \[And] MemberQ[cands, {Length[cells]}] \[And]
Length[cells] == Length[candidates] \[And]
MinMax[Flatten[cands]] === {1,
Length[cells]} \[And] (Union @@ cands === Range[Length[cells]])
CellDistance[cell1_, cell2_] := ManhattanDistance[cell1, cell2]
VisualizeHidato[cells_List, cands_List, path_ : {}] :=
Module[{grid, nums, cb, hx, pt},
grid = {EdgeForm[Thick],
If[Length[#2] > 1, {FaceForm[],
Rectangle[#1]}, {FaceForm[LightGray],
Rectangle[#1]}] &, {cells, cands}]};
nums =
If[Length[#1] == 1, Text[Style[First[#1], 16], #2 + 0.5 {1, 1}],
Text[
Tooltip[Style[Length[#1], Red, 10], #1], #2 +
0.5 {1, 1}]] &, {cands, cells}];
cb = CoordinateBounds[cells];
If[Length[path] > 0,
pt = Arrow[# + {0.5, 0.5} & /@ cells[[path]]];
,
pt = {};
];
Graphics[{grid, nums, pt},
PlotRange -> cb + {{-0.5, 1.5}, {-0.5, 1.5}},
ImageSize -> 60 (1 + cb[[1, 2]] - cb[[1, 1]])]
]
HiddenSingle[cands_List] := Module[{singles, newcands = cands},
singles = Cases[Tally[Flatten[cands]], {_, 1}];
If[Length[singles] > 0,
singles = Sort[singles[[All, 1]]];
newcands =
If[ContainsAny[#, singles], Intersection[#, singles], #] & /@
newcands;
newcands
,
cands
]
]
HiddenN[cands_List, n_Integer?(# > 1 &)] := Module[{tmp, out},
tmp = cands;
tmp = Join @@ MapIndexed[{#1, First[#2]} &, tmp, {2}];
tmp = Transpose /@ GatherBy[tmp, First];
tmp[[All, 1]] = tmp[[All, 1, 1]];
tmp = Select[tmp, 2 <= Length[Last[#]] <= n &];
If[Length[tmp] > 0,
tmp = Transpose /@ Subsets[tmp, {n}];
tmp[[All, 2]] = Union @@@ tmp[[All, 2]];
tmp = Select[tmp, Length[Last[#]] == n &];
If[Length[tmp] > 0,
(* for each tmp {cands,
cells} in each of the cells delete everything except the cands *)

out = cands;
Do[
Do[
out[[c]] = Select[out[[c]], MemberQ[t[[1]], #] &];
,
{c, t[[2]]}
]
,
{t, tmp}
];
out
,
cands
]
,
cands
]
]
NakedN[cands_List, n_Integer?(# > 1 &)] := Module[{tmp, newcands, ids},
tmp = {Range[Length[cands]], cands}\[Transpose];
tmp = Select[tmp, 2 <= Length[Last[#]] <= n &];
If[Length[tmp] > 0,
tmp = Transpose /@ Subsets[tmp, {n}];
tmp[[All, 2]] = Union @@@ tmp[[All, 2]];
tmp = Select[tmp, Length[Last[#]] == n &];
If[Length[tmp] > 0,
newcands = cands;
Do[
ids = Complement[Range[Length[newcands]], t[[1]]];
newcands[[ids]] =
DeleteCases[newcands[[ids]],
Alternatives @@ t[[2]], \[Infinity]];
,
{t, tmp}
];
newcands
,
cands
]
,
cands
]
]
Cornering[cells_List, cands_List] :=
Module[{newcands, neighbours, filled, neighboursfiltered, cellid,
filledneighours, begin, end, beginend},
filled = Flatten[MapIndexed[If[Length[#1] == 1, #2, {}] &, cands]];
begin = If[MemberQ[cands, {1}], {}, {1}];
end = If[MemberQ[cands, {Length[cells]}], {}, {Length[cells]}];
beginend = Join[begin, end];
neighbours = Outer[NeighbourQ, cells, cells, 1];
neighbours =
Association[
MapIndexed[
First[#2] -> {Complement[Flatten[Position[#1, True]], filled],
Intersection[Flatten[Position[#1, True]], filled]} &,
neighbours]];
KeyDropFrom[neighbours, filled];
neighbours = Select[neighbours, Length[First[#]] == 1 &];
If[Length[neighbours] > 0,
newcands = cands;
neighbours = KeyValueMap[List, neighbours];
Do[
cellid = n[[1]];
filledneighours = n[[2, 2]];
filledneighours = Join @@ cands[[filledneighours]];
filledneighours =
Union[filledneighours - 1, filledneighours + 1];
filledneighours = Union[filledneighours, beginend];
newcands[[cellid]] =
Intersection[newcands[[cellid]], filledneighours];
,
{n, neighbours}
];
newcands
,
cands
]
]
ChainSearch[cells_, cands_] := Module[{neighbours, sols, out},
neighbours = Outer[NeighbourQ, cells, cells, 1];
neighbours =
Association[
MapIndexed[First[#2] -> Flatten[Position[#1, True]] &,
neighbours]];
sols = Reap[ChainSearch[neighbours, cands, {}];][[2]];
If[Length[sols] > 0,
sols = sols[[1]];
If[Length[sols] > 1,
Print["multiple solutions found, showing first"];
];
sols = First[sols];
out = cands;
out[[sols]] = List /@ Range[Length[out]];
out
,
cands
]
]
ChainSearch[neighbours_, cands_List, solcellids_List] :=
Module[{largest, largestid, next, poss},
largest = Length[solcellids];
largestid = Last[solcellids, 0];
If[largest < Length[cands],
next = largest + 1;
poss =
Flatten[MapIndexed[If[MemberQ[#1, next], First[#2], {}] &, cands]];
If[Length[poss] > 0,
If[largest > 0,
poss = Intersection[poss, neighbours[largestid]];
];
poss = Complement[poss, solcellids]; (* can't be in previous path*)

If[Length[poss] > 0, (* there are 'next' ones iterate over,
calling this function *)
Do[
ChainSearch[neighbours, cands, Append[solcellids, p]]
,
{p, poss}
]
]
,
Print["There should be a next!"];
Abort[];
]
,
Sow[solcellids] (*
we found a solution with this ordering of cells *)
]
]
GrowNeighbours[neighbours_, set_List] :=
Module[{lastdone, ids, newneighbours, old},
old = Join @@ set[[All, All, 1]];
lastdone = Last[set];
ids = lastdone[[All, 1]];
newneighbours = Union @@ (neighbours /@ ids);
newneighbours = Complement[newneighbours, old]; (*only new ones*)

If[Length[newneighbours] > 0,
Append[set, Thread[{newneighbours, lastdone[[1, 2]] + 1}]]
,
set
]
]
ReachDelete[cells_List, cands_List, neighbours_, startid_] :=
Module[{seed, distances, val, newcands},
If[MatchQ[cands[[startid]], {_}],
val = cands[[startid, 1]];
seed = {{{startid, 0}}};
distances =
Join @@ FixedPoint[GrowNeighbours[neighbours, #] &, seed];
If[Length[distances] > 0,
distances = Select[distances, Last[#] > 0 &];
If[Length[distances] > 0,
newcands = cands;
distances[[All, 2]] =
Transpose[
val + Outer[Times, {-1, 1}, distances[[All, 2]] - 1]];
Do[newcands[[\[CurlyPhi][[1]]]] =
Complement[newcands[[\[CurlyPhi][[1]]]],
Range @@ \[CurlyPhi][[2]]];
, {\[CurlyPhi], distances}
];
newcands
,
cands
]
,
cands
]
,
Print["invalid starting point for neighbour search"];
Abort[];
]
]
GapSearch[cells_List, cands_List] :=
Module[{givensid, givens, neighbours},
givensid = Flatten[Position[cands, {_}]];
givens = {cells[[givensid]], givensid,
Flatten[cands[[givensid]]]}\[Transpose];
If[Length[givens] > 0,
givens = SortBy[givens, Last];
givens = Split[givens, Last[#2] == Last[#1] + 1 &];
givens = If[Length[#] <= 2, #, #[[{1, -1}]]] & /@ givens;
If[Length[givens] > 0,
givens = Join @@ givens;
If[Length[givens] > 0,
neighbours = Outer[NeighbourQ, cells, cells, 1];
neighbours =
Association[
MapIndexed[First[#2] -> Flatten[Position[#1, True]] &,
neighbours]];
givens = givens[[All, 2]];
Fold[ReachDelete[cells, #1, neighbours, #2] &, cands, givens]
,
cands
]
,
cands
]
,
cands
]
]
HidatoSolve[cells_List, cands_List] :=
Module[{newcands = cands, old},
Print@VisualizeHidato[cells, newcands];
If[ValidPuzzle[cells, cands] \[Or] 1 == 1,
old = -1;
newcands = GapSearch[cells, newcands];
While[old =!= newcands,
old = newcands;
newcands = GapSearch[cells, newcands];
If[old === newcands,
newcands = HiddenSingle[newcands];
If[old === newcands,
newcands = NakedN[newcands, 2];
newcands = HiddenN[newcands, 2];
If[old === newcands,
newcands = NakedN[newcands, 3];
newcands = HiddenN[newcands, 3];
If[old === newcands,
newcands = Cornering[cells, newcands];
If[old === newcands,
newcands = NakedN[newcands, 4];
newcands = HiddenN[newcands, 4];
If[old === newcands \[And] 2 == 3,
newcands = NakedN[newcands, 5];
newcands = HiddenN[newcands, 5];
If[old === newcands,
newcands = NakedN[newcands, 6];
newcands = HiddenN[newcands, 6];
If[old === newcands,
newcands = NakedN[newcands, 7];
newcands = HiddenN[newcands, 7];
If[old === newcands,
newcands = NakedN[newcands, 8];
newcands = HiddenN[newcands, 8];
]
]
]
]
]
]
]
]
]
];
If[Length[Flatten[newcands]] > Length[newcands], (*
if not solved do a depth-first brute force search*)

newcands = ChainSearch[cells, newcands];
];
Print@VisualizeHidato[cells, newcands];
newcands
,
Print[
"There seems to be something wrong with your Hidato puzzle. Check \
if the begin and endpoints are given, the cells and candidates have \
the same length, all the numbers are among the \
candidates\[Ellipsis]"]
]
]

puzz = "0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0";
puzz = StringSplit[#, " "] & /@
StringSplit[StringReplace[puzz, "  " -> " "], "\n"];
puzz = Map[StringTrim /* ToExpression, puzz, {2}];
puzz //= Transpose;
puzz //= Map[Reverse];
pos = Position[puzz, Except[0], {2}, Heads -> False];
clues = Thread[{pos, List /@ Extract[puzz, pos]}];
cells = Tuples[Range[9], 2];
candidates = ConstantArray[Range@Length[cells], Length[cells]];
indices = Flatten[Position[cells, #] & /@ clues[[All, 1]]];
candidates[[indices]] = clues[[All, 2]];
out = HidatoSolve[cells, candidates];

puzz = " 0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0";
puzz = StringSplit[#, " "] & /@
StringSplit[StringReplace[puzz, "  " -> " "], "\n"];
puzz = Map[StringTrim /* ToExpression, puzz, {2}];
puzz //= Transpose;
puzz //= Map[Reverse];
pos = Position[puzz, Except[0], {2}, Heads -> False];
clues = Thread[{pos, List /@ Extract[puzz, pos]}];
cells = Tuples[Range[9], 2];
candidates = ConstantArray[Range@Length[cells], Length[cells]];
indices = Flatten[Position[cells, #] & /@ clues[[All, 1]]];
candidates[[indices]] = clues[[All, 2]];
out = HidatoSolve[cells, candidates];
```
Output:

Outputs a graphical representation of the two numbrix puzzles and their solutions.

## Nim

Translation of: Go

With many changes, for instance using a “Numbrix” object as context, adding a procedure to create this object, etc.

```import algorithm, sequtils, strformat, strutils

const Moves = [(1, 0), (0, 1), (-1, 0), (0, -1)]

type Numbrix = object
grid: seq[seq[int]]
clues: seq[int]
totalToFill: Natural
startRow, startCol : Natural

proc initNumbrix(board: openArray[string]): Numbrix =

let nRows = board.len + 2
let nCols = board[0].split(',').len + 2
result.grid = newSeqWith(nRows, repeat(-1, nCols))
result.totalToFill = (nRows - 2) * (nCols - 2)

var list: seq[int]
for r in 0..board.high:
let row = board[r].split(',')
for c in 0..row.high:
let val = parseInt(row[c])
result.grid[r + 1][c + 1] = val
if val > 0:
if val == 1:
result.startRow = r + 1
result.startCol = c + 1

list.sort()
result.clues = list

proc solve(numbrix: var Numbrix; row, col, count: Natural; nextClue: int): bool =

if count > numbrix.totalToFill:
return true

let back = numbrix.grid[row][col]
if back notin [0, count]:
return false
if back == 0 and nextClue < numbrix.clues.len and numbrix.clues[nextClue] == count:
return false

var nextClue = nextClue
if back == count: inc nextClue

numbrix.grid[row][col] = count
for move in Moves:
if numbrix.solve(row + move[1], col + move[0], count + 1, nextClue):
return true
numbrix.grid[row][col] = back

proc print(numbrix: Numbrix) =
for row in numbrix.grid:
for val in row:
if val != -1:
stdout.write &"{val:2} "
echo()

when isMainModule:

const

Example1 = ["00,00,00,00,00,00,00,00,00",
"00,00,46,45,00,55,74,00,00",
"00,38,00,00,43,00,00,78,00",
"00,35,00,00,00,00,00,71,00",
"00,00,33,00,00,00,59,00,00",
"00,17,00,00,00,00,00,67,00",
"00,18,00,00,11,00,00,64,00",
"00,00,24,21,00,01,02,00,00",
"00,00,00,00,00,00,00,00,00"]

Example2 = ["00,00,00,00,00,00,00,00,00",
"00,11,12,15,18,21,62,61,00",
"00,06,00,00,00,00,00,60,00",
"00,33,00,00,00,00,00,57,00",
"00,32,00,00,00,00,00,56,00",
"00,37,00,01,00,00,00,73,00",
"00,38,00,00,00,00,00,72,00",
"00,43,44,47,48,51,76,77,00",
"00,00,00,00,00,00,00,00,00"]

for i, board in [1: Example1, 2: Example2]:
var numbrix = initNumbrix(board)
if numbrix.solve(numbrix.startRow, numbrix.startCol, 1, 0):
echo &"Solution for example {i}:"
numbrix.print()
else:
echo "No solution."
```
Output:
```Solution for example 1:

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

Solution for example 2:

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79```

## Perl

Tested on perl v5.26.1

```#!/usr/bin/perl

use strict;
use warnings;

\$_ = <<END;
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0
END

my \$gap = /.\n/ * \$-[0];
print;
s/ (?=\d\b)/0/g;
my \$max = sprintf "%02d", tr/0-9// / 2;

solve( '01', \$_ );

sub solve
{
my (\$have, \$in) = @_;
\$have eq \$max and exit !print "solution\n", \$in =~ s/\b0/ /gr;
if( \$in =~ ++(my \$want = \$have) )
{
\$in =~ /(\$have|\$want)( |.{\$gap})(\$have|\$want)/s and solve(\$want, \$in);
}
else
{
(\$_ = \$in) =~ s/\$have \K00/\$want/          and solve( \$want, \$_ ); # R
(\$_ = \$in) =~ s/\$have.{\$gap}\K00/\$want/s   and solve( \$want, \$_ ); # D
(\$_ = \$in) =~ s/00(?= \$have)/\$want/        and solve( \$want, \$_ ); # L
(\$_ = \$in) =~ s/00(?=.{\$gap}\$have)/\$want/s and solve( \$want, \$_ ); # U
}
}
```
Output  —  Example 1:

``` 0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0
solution
49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5
```

## Phix

```with javascript_semantics
include sets.e
sequence board, placed, px, py
integer w, h, limit, missing
bool solved

function get_moves(integer n)
sequence res = {}
integer x = px[n], y = py[n]
if x>1 and board[y,x-1]=0 then res &= {{x-1,y}} end if
if x<w and board[y,x+1]=0 then res &= {{x+1,y}} end if
if y>1 and board[y-1,x]=0 then res &= {{x,y-1}} end if
if y<h and board[y+1,x]=0 then res &= {{x,y+1}} end if
return res
end function

procedure solve()
if missing=0 then
solved = true
else
-- scan for next to place, which will be the lowest
-- of those with either n+1 or n-1 already placed,
-- checking that all needed can still be placed.
integer place
sequence moves
for n=limit to 1 by -1 do
if not placed[n] then
bool plus1 = false
if n<limit and placed[n+1] then
place = n
plus1 = true
moves = get_moves(n+1)
if length(moves)=0 then
return -- fail/backtrack
end if
end if
if n>1 and placed[n-1] then
place = n
if plus1 then
moves = intersection(moves,get_moves(n-1))
else
moves = get_moves(n-1)
end if
if length(moves)=0 then
return -- fail/backtrack
end if
end if
end if
end for
missing -= 1
for m in moves do
integer {x,y} = m
px[place] = x
py[place] = y
board[y,x] = place
placed[place] = true
solve()
if solved then return end if
placed[place] = false
board[y,x] = 0
end for
missing += 1
end if
end procedure

procedure Numbrix(string s)
atom t0 = time()
board = split(s,'\n')
for i,line in board do
board[i] = apply(split(substitute(line,'.','0')),to_number)
end for
w = length(board[1])
h = length(board)
limit = w*h
placed = repeat(false,limit)
px = repeat(0,limit)
py = repeat(0,limit)
missing = 0
for x=1 to w do
for y=1 to h do
integer byx = board[y][x]
if byx then
placed[byx] = true
px[byx] = x
py[byx] = y
else
missing += 1
end if
end for
end for
solved = false
solve()
printf(1,"%s\n\n",s)
if not solved then
puts(1,"No solutions\n\n")
else
integer nchars = length(sprintf("%d",limit))
string fmt = sprintf(" %%%dd",nchars)
printf(1,"solution found in %s:\n\n",elapsed(time()-t0))
board = apply(true,join_by,{board,1,w,{""},{""},{fmt}})
printf(1,"%s\n\n",{join(board,"\n")})
end if
end procedure

constant boards = {"""
.  .  .  .  .  .  .  .  .
.  . 46 45  . 55 74  .  .
. 38  .  . 43  .  . 78  .
. 35  .  .  .  .  . 71  .
.  . 33  .  .  . 59  .  .
. 17  .  .  .  .  . 67  .
. 18  .  . 11  .  . 64  .
.  . 24 21  .  1  2  .  .
.  .  .  .  .  .  .  .  .""","""
.  .  .  .  .  .  .  .  .
. 11 12 15 18 21 62 61  .
.  6  .  .  .  .  . 60  .
. 33  .  .  .  .  . 57  .
. 32  .  .  .  .  . 56  .
. 37  .  1  .  .  . 73  .
. 38  .  .  .  .  . 72  .
. 43 44 47 48 51 76 77  .
.  .  .  .  .  .  .  .  .""","""
17  .  .  . 11  .  .  . 59
. 15  .  .  6  .  . 61  .
.  .  3  .  .  . 63  .  .
.  .  .  . 66  .  .  .  .
23 24  . 68 67 78  . 54 55
.  .  .  . 72  .  .  .  .
.  . 35  .  .  . 49  .  .
. 29  .  . 40  .  . 47  .
31  .  .  . 39  .  .  . 45""","""
109   0   0   0   0   0   0   0   0   0   0   0   0   0  43
0   0   0   0   0   0   0  65   0   0   0   0   0   0   0
0   0 101 100   0  92   0  76   0  68   0  48   3   0   0
0   0 102  97   0   0  80   0  74   0   0  49   6   0   0
0   0   0   0   0   0  79   0  73   0   0   0   0   0   0
0   0 116   0   0   0   0   0   0   0   0   0  10   0   0
0   0   0 118 217   0   0   0   0   0  55  52   0   0   0
0 121 120   0   0   0   0 213   0   0   0   0  12  35   0
0   0   0 166 167   0   0   0   0   0 205 204   0   0   0
0   0 162   0   0   0   0   0   0   0   0   0  14   0   0
0   0   0   0   0   0 173   0 177   0   0   0   0   0   0
0   0 156 153   0   0 150   0 178   0   0 201  16   0   0
0   0 155 154   0 144   0 180   0 188   0 200  17   0   0
0   0   0   0   0   0   0 183   0   0   0   0   0   0   0
135   0   0   0   0   0   0   0   0   0   0   0   0   0  21"""}
papply(boards,Numbrix)
```
Output:
```  .  .  .  .  .  .  .  .  .
.  . 46 45  . 55 74  .  .
. 38  .  . 43  .  . 78  .
. 35  .  .  .  .  . 71  .
.  . 33  .  .  . 59  .  .
. 17  .  .  .  .  . 67  .
. 18  .  . 11  .  . 64  .
.  . 24 21  .  1  2  .  .
.  .  .  .  .  .  .  .  .

solution found in 0.1s:

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

.  .  .  .  .  .  .  .  .
. 11 12 15 18 21 62 61  .
.  6  .  .  .  .  . 60  .
. 33  .  .  .  .  . 57  .
. 32  .  .  .  .  . 56  .
. 37  .  1  .  .  . 73  .
. 38  .  .  .  .  . 72  .
. 43 44 47 48 51 76 77  .
.  .  .  .  .  .  .  .  .

solution found in 0.0s:

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79

17  .  .  . 11  .  .  . 59
. 15  .  .  6  .  . 61  .
.  .  3  .  .  . 63  .  .
.  .  .  . 66  .  .  .  .
23 24  . 68 67 78  . 54 55
.  .  .  . 72  .  .  .  .
.  . 35  .  .  . 49  .  .
. 29  .  . 40  .  . 47  .
31  .  .  . 39  .  .  . 45

solution found in 0.0s:

17 16 13 12 11 10  9 60 59
18 15 14  5  6  7  8 61 58
19 20  3  4 65 64 63 62 57
22 21  2  1 66 79 80 81 56
23 24 69 68 67 78 77 54 55
26 25 70 71 72 75 76 53 52
27 28 35 36 73 74 49 50 51
30 29 34 37 40 41 48 47 46
31 32 33 38 39 42 43 44 45

109   0   0   0   0   0   0   0   0   0   0   0   0   0  43
0   0   0   0   0   0   0  65   0   0   0   0   0   0   0
0   0 101 100   0  92   0  76   0  68   0  48   3   0   0
0   0 102  97   0   0  80   0  74   0   0  49   6   0   0
0   0   0   0   0   0  79   0  73   0   0   0   0   0   0
0   0 116   0   0   0   0   0   0   0   0   0  10   0   0
0   0   0 118 217   0   0   0   0   0  55  52   0   0   0
0 121 120   0   0   0   0 213   0   0   0   0  12  35   0
0   0   0 166 167   0   0   0   0   0 205 204   0   0   0
0   0 162   0   0   0   0   0   0   0   0   0  14   0   0
0   0   0   0   0   0 173   0 177   0   0   0   0   0   0
0   0 156 153   0   0 150   0 178   0   0 201  16   0   0
0   0 155 154   0 144   0 180   0 188   0 200  17   0   0
0   0   0   0   0   0   0 183   0   0   0   0   0   0   0
135   0   0   0   0   0   0   0   0   0   0   0   0   0  21

solution found in 0.5s:

109 108  87  86  85  84  83  64  63  62  61  46  45  44  43
110 107  88  89  90  91  82  65  66  67  60  47   2   1  42
111 106 101 100  99  92  81  76  75  68  59  48   3   4  41
112 105 102  97  98  93  80  77  74  69  58  49   6   5  40
113 104 103  96  95  94  79  78  73  70  57  50   7   8  39
114 115 116 225 224 223 222 221  72  71  56  51  10   9  38
123 122 117 118 217 218 219 220 209 208  55  52  11  36  37
124 121 120 119 216 215 214 213 210 207  54  53  12  35  34
125 164 165 166 167 168 169 212 211 206 205 204  13  32  33
126 163 162 161 160 171 170 175 176 191 192 203  14  31  30
127 128 157 158 159 172 173 174 177 190 193 202  15  28  29
130 129 156 153 152 151 150 179 178 189 194 201  16  27  26
131 132 155 154 143 144 149 180 181 188 195 200  17  24  25
134 133 138 139 142 145 148 183 182 187 196 199  18  23  22
135 136 137 140 141 146 147 184 185 186 197 198  19  20  21
```

## Picat

```import sat, util.

main([File]) =>
Dim = Lines.len(),
Board = new_array(Dim, Dim),
Max = Dim*Dim,
Board :: 1..Max,
Bvars = Board.vars(),
all_different(Bvars),

foreach ( R in 1..Dim )
Line = Lines[R].split(),
if( Line.len() != Dim ) then
printf("Line %d too short or too long, failing\n", R),
abort
end,
foreach ( C in 1..Dim )           % empty cell: _ or 0
if ( Line[C] != ['_'] ) then   % data as 49 _ _ 32 _ _...
Num = Line[C].to_int(),
if ( Num != 0 ) then        % data as 0 11 12 15 18...
Board[R,C] #= Num
end
end
end
end,

% each cell but that with value 1 must be +1 larger then one of its neighbours
% some numbrix puzzles do not have min and/or max values,
% but this method works for all cases
foreach ( R in 1..Dim, C in 1..Dim )
Nei = [(R1,C1) : (R1, C1) in [(R-1,C), (R,C+1), (R+1,C), (R,C-1)],
between(1, Dim, R1), between(1, Dim, C1)],
Consnei = [ Board[R,C] #= Board[R1,C1] + 1 : (R1,C1) in Nei ],
Board[R,C] #!= 1 #=> sum(Consnei) #= 1
end,

time2(solve(Bvars)),
printboard(Board).

printboard(A) =>
N = A.len,
nl,
foreach ( I in 1..N )
foreach ( J in 1..A[I].len )
if ( A[I,J] == 0 ) then
printf("    ")
else
printf("%4w", A[I,J])
end
end,
nl
end.```
Output:
```Solution 1:
49  50  51  52  53  54  75  76  81
48  47  46  45  44  55  74  77  80
37  38  39  40  43  56  73  78  79
36  35  34  41  42  57  72  71  70
31  32  33  14  13  58  59  68  69
30  17  16  15  12  61  60  67  66
29  18  19  20  11  62  63  64  65
28  25  24  21  10   1   2   3   4
27  26  23  22   9   8   7   6   5

Solution 2:
9  10  13  14  19  20  63  64  65
8  11  12  15  18  21  62  61  66
7   6   5  16  17  22  59  60  67
34  33   4   3  24  23  58  57  68
35  32  31   2  25  54  55  56  69
36  37  30   1  26  53  74  73  70
39  38  29  28  27  52  75  72  71
40  43  44  47  48  51  76  77  78
41  42  45  46  49  50  81  80  79

Problem, no starting (1) nor end (225) points (2.344 seconds):
109   0   0   0   0   0   0   0   0   0   0   0   0   0  43
0   0   0   0   0   0   0  65   0   0   0   0   0   0   0
0   0 101 100   0  92   0  76   0  68   0  48   3   0   0
0   0 102  97   0   0  80   0  74   0   0  49   6   0   0
0   0   0   0   0   0  79   0  73   0   0   0   0   0   0
0   0 116   0   0   0   0   0   0   0   0   0  10   0   0
0   0   0 118 217   0   0   0   0   0  55  52   0   0   0
0 121 120   0   0   0   0 213   0   0   0   0  12  35   0
0   0   0 166 167   0   0   0   0   0 205 204   0   0   0
0   0 162   0   0   0   0   0   0   0   0   0  14   0   0
0   0   0   0   0   0 173   0 177   0   0   0   0   0   0
0   0 156 153   0   0 150   0 178   0   0 201  16   0   0
0   0 155 154   0 144   0 180   0 188   0 200  17   0   0
0   0   0   0   0   0   0 183   0   0   0   0   0   0   0
135   0   0   0   0   0   0   0   0   0   0   0   0   0  21

Solution:
109 108  87  86  85  84  83  64  63  62  61  46  45  44  43
110 107  88  89  90  91  82  65  66  67  60  47   2   1  42
111 106 101 100  99  92  81  76  75  68  59  48   3   4  41
112 105 102  97  98  93  80  77  74  69  58  49   6   5  40
113 104 103  96  95  94  79  78  73  70  57  50   7   8  39
114 115 116 225 224 223 222 221  72  71  56  51  10   9  38
123 122 117 118 217 218 219 220 209 208  55  52  11  36  37
124 121 120 119 216 215 214 213 210 207  54  53  12  35  34
125 164 165 166 167 168 169 212 211 206 205 204  13  32  33
126 163 162 161 160 171 170 175 176 191 192 203  14  31  30
127 128 157 158 159 172 173 174 177 190 193 202  15  28  29
130 129 156 153 152 151 150 179 178 189 194 201  16  27  26
131 132 155 154 143 144 149 180 181 188 195 200  17  24  25
134 133 138 139 142 145 148 183 182 187 196 199  18  23  22
135 136 137 140 141 146 147 184 185 186 197 198  19  20  21
```

## Prolog

```/*
* Solver
*/
solve([A|T]) :-
numlist(1,81,S),
select(A,S,R),
solve_([A|T],R).

solve_([_],[]).
solve_([A,B|T],R) :-
move(A,B),
select(B,R,Rt),
solve_([B|T],Rt).

move(A,B) :- lr(A,B) ; lr(B,A) ; ud(A,B) ; ud(B,A).

% create the left-right mapping rules at compile time
term_expansion(lr(0,0),LrList) :-
findall(LR,
(between(1,81,N), M is N mod 9, dif(M,0), succ(N,N1), LR = lr(N,N1)),
LrList).
lr(0,0).

% create the up-down mapping rules at compile time
term_expansion(ud(0,0),UdList) :-
findall(UD,
(between(1,72,N), N9 is N + 9, UD = ud(N,N9)),
UdList).
ud(0,0).

/*
* Grid <-> Solvable List
*/
grid_solvable([],_,_).
grid_solvable([A|T],N,S) :-
(integer(A) -> nth1(A,S,N);true),
succ(N,N1),
grid_solvable(T,N1,S).

solvable_grid([],_,_).
solvable_grid([A|T],N,G) :-
nth1(A,G,N),
succ(N,N1),
solvable_grid(T,N1,G).

/*
* Print Grid
*/
print_cell(C) :-
C >= 10 -> format(' ~d', C)
; format('  ~d', C).
print_grid([],_).
print_grid([C|T],N) :-
print_cell(C),
(0 is N mod 9 -> nl ; true),
succ(N,N1),
print_grid(T,N1).

/*
* Numbrix!
*/
numbrix(L) :-
length(S, 81),
grid_solvable(L,1,S),
solve(S),
solvable_grid(S,1,P),
print_grid(P,1),
!.

test1 :- numbrix([
_,  _,  _,  _,  _,  _,  _,  _,  _,
_,  _, 46, 45,  _, 55, 74,  _,  _,
_, 38,  _,  _, 43,  _,  _, 78,  _,
_, 35,  _,  _,  _,  _,  _, 71,  _,
_,  _, 33,  _,  _,  _, 59,  _,  _,
_, 17,  _,  _,  _,  _,  _, 67,  _,
_, 18,  _,  _, 11,  _,  _, 64,  _,
_,  _, 24, 21,  _,  1,  2,  _,  _,
_,  _,  _,  _,  _,  _,  _,  _,  _
]).

test2 :- numbrix([
_,  _,  _,  _,  _,  _,  _,  _,  _,
_, 11, 12, 15, 18, 21, 62, 61,  _,
_,  6,  _,  _,  _,  _,  _, 60,  _,
_, 33,  _,  _,  _,  _,  _, 57,  _,
_, 32,  _,  _,  _,  _,  _, 56,  _,
_, 37,  _,  1,  _,  _,  _, 73,  _,
_, 38,  _,  _,  _,  _,  _, 72,  _,
_, 43, 44, 47, 48, 51, 76, 77,  _,
_,  _,  _,  _,  _,  _,  _,  _,  _
]).

test3 :- numbrix([
17,  _,  _,  _, 11,  _,  _,  _, 59,
_,  15,  _,  _,  6,  _,  _, 61,  _,
_,   _,  3,  _,  _,  _, 63,  _,  _,
_,   _,  _,  _, 66,  _,  _,  _,  _,
23, 24,  _, 68, 67, 78,  _, 54, 55,
_,   _,  _,  _, 72,  _,  _,  _,  _,
_,   _, 35,  _,  _,  _, 49,  _,  _,
_,  29,  _,  _, 40,  _,  _, 47,  _,
31,  _,  _,  _, 39,  _,  _,  _, 45
]).
```
Output:
```1 ?- test1.
49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5
true.

2 ?- test2.
9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
true.

3 ?- test3.
17 16 13 12 11 10  9 60 59
18 15 14  5  6  7  8 61 58
19 20  3  4 65 64 63 62 57
22 21  2  1 66 79 80 81 56
23 24 69 68 67 78 77 54 55
26 25 70 71 72 75 76 53 52
27 28 35 36 73 74 49 50 51
30 29 34 37 40 41 48 47 46
31 32 33 38 39 42 43 44 45
true.

4 ?-
```

## Python

 This example is incorrect. Please fix the code and remove this message.Details: 3rd solution has "00 00" in it where "02 01" shd be
```from sys import stdout
neighbours = [[-1, 0], [0, -1], [1, 0], [0, 1]]
exists = []
lastNumber = 0
wid = 0
hei = 0

def find_next(pa, x, y, z):
for i in range(4):
a = x + neighbours[i][0]
b = y + neighbours[i][1]
if wid > a > -1 and hei > b > -1:
if pa[a][b] == z:
return a, b

return -1, -1

def find_solution(pa, x, y, z):
if z > lastNumber:
return 1
if exists[z] == 1:
s = find_next(pa, x, y, z)
if s[0] < 0:
return 0
return find_solution(pa, s[0], s[1], z + 1)

for i in range(4):
a = x + neighbours[i][0]
b = y + neighbours[i][1]
if wid > a > -1 and hei > b > -1:
if pa[a][b] == 0:
pa[a][b] = z
r = find_solution(pa, a, b, z + 1)
if r == 1:
return 1
pa[a][b] = 0

return 0

def solve(pz, w, h):
global lastNumber, wid, hei, exists

lastNumber = w * h
wid = w
hei = h
exists = [0 for j in range(lastNumber + 1)]

pa = [[0 for j in range(h)] for i in range(w)]
st = pz.split()
idx = 0
for j in range(h):
for i in range(w):
if st[idx] == ".":
idx += 1
else:
pa[i][j] = int(st[idx])
exists[pa[i][j]] = 1
idx += 1

x = 0
y = 0
t = w * h + 1
for j in range(h):
for i in range(w):
if pa[i][j] != 0 and pa[i][j] < t:
t = pa[i][j]
x = i
y = j

return find_solution(pa, x, y, t + 1), pa

def show_result(r):
if r[0] == 1:
for j in range(hei):
for i in range(wid):
stdout.write(" {:0{}d}".format(r[1][i][j], 2))
print()
else:
stdout.write("No Solution!\n")

print()

r = solve(". . . . . . . . . . . 46 45 . 55 74 . . . 38 . . 43 . . 78 . . 35 . . . . . 71 . . . 33 . . . 59 . . . 17"
" . . . . . 67 . . 18 . . 11 . . 64 . . . 24 21 . 1  2 . . . . . . . . . . .", 9, 9)
show_result(r)

r = solve(". . . . . . . . . . 11 12 15 18 21 62 61 . .  6 . . . . . 60 . . 33 . . . . . 57 . . 32 . . . . . 56 . . 37"
" .  1 . . . 73 . . 38 . . . . . 72 . . 43 44 47 48 51 76 77 . . . . . . . . . .", 9, 9)
show_result(r)

r = solve("17 . . . 11 . . . 59 . 15 . . 6 . . 61 . . . 3 . . .  63 . . . . . . 66 . . . . 23 24 . 68 67 78 . 54 55"
" . . . . 72 . . . . . . 35 . . . 49 . . . 29 . . 40 . . 47 . 31 . . . 39 . . . 45", 9, 9)
show_result(r)
```
Output:
```
49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10 01 02 03 04
27 26 23 22 09 08 07 06 05
09 10 13 14 19 20 63 64 65
08 11 12 15 18 21 62 61 66
07 06 05 16 17 22 59 60 67
34 33 04 03 24 23 58 57 68
35 32 31 02 25 54 55 56 69
36 37 30 01 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
17 16 13 12 11 10 09 60 59
18 15 14 05 06 07 08 61 58
19 20 03 04 65 64 63 62 57
22 21 00 00 66 79 80 81 56
23 24 69 68 67 78 77 54 55
26 25 70 71 72 75 76 53 52
27 28 35 36 73 74 49 50 51
30 29 34 37 40 41 48 47 46
31 32 33 38 39 42 43 44 45

```

## Racket

This is a general "Hidato" style solver (which is why there is a search for a 0 start point (which supports Hopido). There is already a Racket implementation of Hidato, so to allow a variety of approaches to be demonstrated, the main library for this set of problems is here.

`hidato-family-solver.rkt`

```#lang racket
;;; Used in my solutions of:
;;; "Solve a Hidato Puzzle"
;;; "Solve a Holy Knights Tour"
;;; "Solve a Numbrix Puzzle"
;;; "Solve a Hopido Puzzle"

;;; As well as the solver being common, the solution renderer and input formats are common
(provide
;; Input:  list of neighbour offsets
;; Output: a solver function:
;;         Input:  a puzzle
;;         Output: either the solved puzzle or #f if impossible
solve-hidato-family
;; Input:  puzzle
;;         optional minimum cell width
;; Output: a pretty string that can be printed
puzzle->string)

;; Cell values are:
;; zero?     - unvisited
;; positive? - nth visitied
;; else      - unvisitable. In the puzzle layout, it's a _. In the hash it's a -1, so we can care less
;; A puzzle is a sequence of sequences of cell values
;; We work with a puzzle as a hash keyed on (cons row-num col-num)

;; Take a puzzle and get a working hash of it
(define (puzzle->hash p)
(for*/hash
(((r row-num) (in-parallel p (in-naturals)))
((v col-num) (in-parallel r (in-naturals)))
#:when (integer? v))
(values (cons row-num col-num) v)))

;; Takes a hash and recreates a vector of vectors puzzle
(define (hash->puzzle h# (blank '_))
(define keys (hash-keys h#))
(define n-rows (add1 (car (argmax car keys))))
(define n-cols (add1 (cdr (argmax cdr keys))))
(for/vector #:length n-rows ((r n-rows))
(for/vector #:length n-cols ((c n-cols))
(hash-ref h# (cons r c) blank))))

;; See "provide" section for description
(define (puzzle->string p (w #f))
(match p
[#f "unsolved"]
[(? sequence? s)
(define (max-n-digits p)
(and p (add1 (order-of-magnitude (* (vector-length p) (vector-length (vector-ref p 0)))))))
(define min-width (or w (max-n-digits p)))
(string-join
(for/list ((r s))
(string-join
(for/list ((c r)) (~a c #:align 'right #:min-width min-width))
" "))
"\n")]))

(define ((solve-hidato-family neighbour-offsets) board)
(define board# (puzzle->hash board))
;; reverse mapping, will only take note of positive values
(define targets# (for/hash ([(k v) (in-hash board#)] #:when (positive? v)) (values v k)))

(define (neighbours r.c)
(for/list ((r+.c+ neighbour-offsets))
(match-define (list r+ c+) r+.c+)
(match-define (cons r  c ) r.c)
(cons (+ r r+) (+ c c+))))

;; Count the moves, rather than check for "no more zeros" in puzzle
(define last-move (length (filter number? (hash-values board#))))

;; Depth first solution of the puzzle (we have to go deep, it's where the solutions are!
(define (inr-solve-pzl b# move r.c)
(cond
[(= move last-move) b#] ; no moves needed, so solved
[else
(for*/or ; check each neighbour as an option
((r.c+ (in-list (neighbours r.c)))
#:when (equal? (hash-ref targets# move r.c) r.c) ; we're where we should be!
#:when (match (hash-ref b# r.c+ -1) (0 #t) ((== m++) #t) (_ #f)))
(inr-solve-pzl (hash-set b# r.c+ m++) m++ r.c+))]))

(define (solution-starting-at n)
(define start-r.c (for/first (((k v) (in-hash board#)) #:when (= n v)) k))
(and start-r.c (inr-solve-pzl board# n start-r.c)))

(define sltn
(cond [(solution-starting-at 1) => values]
;; next clause starts from 0 for hopido
[(solution-starting-at 0) => values]))

(and sltn (hash->puzzle sltn)))
```
```#lang racket
(require "hidato-family-solver.rkt")

(define von-neumann-neighbour-offsets
'((+1 0) (-1 0) (0 +1) (0 -1)))

(define solve-numbrix (solve-hidato-family von-neumann-neighbour-offsets))

(displayln
(puzzle->string
(solve-numbrix
#(#(0  0  0  0  0  0  0  0  0)
#(0  0 46 45  0 55 74  0  0)
#(0 38  0  0 43  0  0 78  0)
#(0 35  0  0  0  0  0 71  0)
#(0  0 33  0  0  0 59  0  0)
#(0 17  0  0  0  0  0 67  0)
#(0 18  0  0 11  0  0 64  0)
#(0  0 24 21  0  1  2  0  0)
#(0  0  0  0  0  0  0  0  0)))))

(newline)

(displayln
(puzzle->string
(solve-numbrix
#(#(0  0  0  0  0  0  0  0  0)
#(0 11 12 15 18 21 62 61  0)
#(0  6  0  0  0  0  0 60  0)
#(0 33  0  0  0  0  0 57  0)
#(0 32  0  0  0  0  0 56  0)
#(0 37  0  1  0  0  0 73  0)
#(0 38  0  0  0  0  0 72  0)
#(0 43 44 47 48 51 76 77  0)
#(0  0  0  0  0  0  0  0  0)))))
```
Output:
```49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79```

## 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:

```my @adjacent =           [-1, 0],
[ 0, -1],          [ 0, 1],
[ 1, 0];
put "\n" xx 60;

solveboard q:to/END/;
__ __ __ __ __ __ __ __ __
__ __ 46 45 __ 55 74 __ __
__ 38 __ __ 43 __ __ 78 __
__ 35 __ __ __ __ __ 71 __
__ __ 33 __ __ __ 59 __ __
__ 17 __ __ __ __ __ 67 __
__ 18 __ __ 11 __ __ 64 __
__ __ 24 21 __  1  2 __ __
__ __ __ __ __ __ __ __ __
END

# And
put "\n" xx 60;

solveboard q:to/END/;
0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  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:
```49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5
1275 tries

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
4631 tries```

Oddly, reversing the tiebreaker rule that makes hidato run twice as fast causes this last example to run four times slower. Go figure...

## REXX

This solution is essentially same as the REXX Hidato puzzle solver.

Programming note: the coördinates for the cells used are the same as an X×Y grid, that is, the bottom left-most cell is (1,1) and the tenth cell on row 2 is (2,10).

Hidato   and   Numbrix   are registered trademarks.

```/*REXX program solves a  Numbrix (R) puzzle, it also displays the puzzle and solution.  */
maxR=   0;     maxC=   0;          maxX=   0;    /*define  maxR,  maxC,  and  maxX.     */
minR= 9e9;     minC= 9e9;          minX= 9e9;    /*   "    minR,  minC,   "   minX.     */
cells=  0                                        /*the number of cells  (so far).       */
parse arg xxx                                    /*get the cell definitions from the CL.*/
xxx=translate(xxx,  ',,,,,'  ,  "/\;:_")         /*also allow other characters as comma.*/
@.=
do  while xxx\='';       parse var  xxx    r c   marks  ','  xxx
do  while marks\='';               _=@.r.c
parse var marks  x  marks
if datatype(x, 'N')  then x= abs(x) / 1                /*normalize  │x│  */
minR= min(minR, r);       minC= min(minC, c)           /*find min R and C*/
maxR= max(maxR, r);       maxC= max(maxC, c)           /*  "  max "  "  "*/
if x==1    then do;   !r= r;   !c= c                   /*the START cell. */
end
if _\==''  then call err  "cell at"  r  c  'is already occupied with:'  _
@.r.c= x;       c=  c +1;           cells= cells + 1   /*assign a mark.  */
if x==.              then iterate                      /*is a hole?  Skip*/
if \datatype(x,'W')  then call err  'illegal marker specified:'    x
minX= min(minX, x);  maxX= max(maxX, x)                /*min  &  max   X.*/
end   /*while marks\='' */
end       /*while xxx  \='' */
call show                                        /* [↓]  is used for making fast moves. */
Nr = '0  1   0  -1    -1   1   1  -1'            /*possible  row     for the next move. */
Nc = '1  0  -1   0     1  -1   1  -1'            /*   "      column   "   "    "    "   */
pMoves= words(Nr)   -   4                        /*is this to be a Numbrix puzzle ?     */

do i=1  for pMoves;    Nr.i= word(Nr, i);    Nc.i= word(Nc, i)   /*for fast moves. */
end   /*i*/
say
if \next(2, !r, !c)  then call err   'No solution possible for this Numbrix puzzle.'
say 'A solution for the Numbrix puzzle exists.';          say;          call show
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
err:  say;   say '***error*** (from Numbrix puzzle): '    arg(1);       say;       exit 13
/*──────────────────────────────────────────────────────────────────────────────────────*/
next: procedure expose @. Nr. Nc. cells pMoves;  parse arg #,r,c;       ##= # + 1
do t=1  for pMoves                                      /* [↓]  try some moves. */
parse value   r+Nr.t  c+Nc.t    with   nr  nc           /*next move coördinates.*/
if @.nr.nc==.  then do;                  @.nr.nc= #     /*let's try this move.  */
if #==cells          then return 1  /*is this the last move?*/
if next(##, nr, nc)  then return 1
@.nr.nc=.                           /*undo the above move.  */
iterate                             /*go & try another move.*/
end
if @.nr.nc==#  then do                                  /*this a fill─in move ? */
if #==cells          then return 1  /*this is the last move.*/
if next(##, nr, nc)  then return 1  /*a fill─in move.       */
end
end   /*t*/
return 0                                                  /*this ain't working.   */
/*──────────────────────────────────────────────────────────────────────────────────────*/
show: if maxR<1 | maxC<1  then call err  'no legal cell was specified.'
if minX<1           then call err  'no  1  was specified for the puzzle start'
w= max(2, length(cells) );      do   r=maxR  to minR  by -1;   _=
do c=minC  to maxC;          _= _  right(@.r.c, w)
end   /*c*/
say _
end   /*r*/
say;      return
```
output   when using the input of:

1 1 . . . . . . . . ./2 1 . . 24 21 . 1 2 . ./3 1 . 18 . . 11 . . 64 ./4 1 . 17 . . . . . 67 ./5 1 . . 33 . . . 59 . ./6 1 . 35 . . . . . 71 ./7 1 . 38 . . 43 . . 78 ./8 1 . . 46 45 . 55 74 . ./9 1 . . . . . . . . .

```  .  .  .  .  .  .  .  .  .
.  . 46 45  . 55 74  .  .
. 38  .  . 43  .  . 78  .
. 35  .  .  .  .  . 71  .
.  . 33  .  .  . 59  .  .
. 17  .  .  .  .  . 67  .
. 18  .  . 11  .  . 64  .
.  . 24 21  .  1  2  .  .
.  .  .  .  .  .  .  .  .

A solution for the Numbrix puzzle exists.

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5
```
output   when using the input of:

1 1 . . . . . . . . .\2 1 . 43 44 47 48 51 76 77 .\3 1 . 38 . . . . . 72 .\4 1 . 37 . 1 . . . 73 .\5 1 . 32 . . . . . 56 .\6 1 . 33 . . . . . 57 .\7 1 . 6 . . . . . 60 .\8 1 . 11 12 15 18 21 62 61 .\9 1 . . . . . . . . .

```  .  .  .  .  .  .  .  .  .
. 11 12 15 18 21 62 61  .
.  6  .  .  .  .  . 60  .
. 33  .  .  .  .  . 57  .
. 32  .  .  .  .  . 56  .
. 37  .  1  .  .  . 73  .
. 38  .  .  .  .  . 72  .
. 43 44 47 48 51 76 77  .
.  .  .  .  .  .  .  .  .

A solution for the Numbrix puzzle exists.

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
```

## Ruby

This solution uses HLPsolver from here

```require 'HLPsolver'

ADJACENT = [[-1, 0], [0, -1], [0, 1], [1, 0]]

board1 = <<EOS
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0
EOS
HLPsolver.new(board1).solve

board2 = <<EOS
0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0
EOS
HLPsolver.new(board2).solve
```

Which produces:

```Problem:
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0

Solution:
49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

Problem:
0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0

Solution:
9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

## SystemVerilog

```//////////////////////////////////////////////////////////////////////////////
///  NumbrixSolver                                                         ///
///     Solve the puzzle, by using system verilog randomization engine     ///
//////////////////////////////////////////////////////////////////////////////
class NumbrixSolver;
rand int solvedBoard[][];
int fixedBoard[][];
int numCells;
////////////////////////////////////////////////////////////////////////////
/// Dynamically resize the board accordingly to the size of the reference///
/// board                                                                ///
////////////////////////////////////////////////////////////////////////////
constraint height {
solvedBoard.size == fixedBoard.size;
}
constraint width {
foreach(solvedBoard[i]) solvedBoard[i].size == fixedBoard[i].size;
}

////////////////////////////////////////////////////////////////////////////
///  Fix the positions defined in the input board                        ///
////////////////////////////////////////////////////////////////////////////
constraint fixed {
foreach(solvedBoard[i]) foreach(solvedBoard[i][j])
if(fixedBoard[i][j] != 0)solvedBoard[i][j] == fixedBoard[i][j];
}
////////////////////////////////////////////////////////////////////////////
///  Ensures that the whole board is filled from the number with numbers ///
///   1,2,3,...,numCells                                                 ///
////////////////////////////////////////////////////////////////////////////
constraint range {
foreach(solvedBoard[i])foreach(solvedBoard[i][j])
solvedBoard[i][j] inside {[1:numCells]};
}
////////////////////////////////////////////////////////////////////////////
///  Ensures that there is no repeated number, consequently every number ///
///  is present on the board                                             ///
////////////////////////////////////////////////////////////////////////////
constraint uniqueness {
foreach(solvedBoard[i1]) foreach(solvedBoard[i1][j1])
foreach(solvedBoard[i2]) foreach(solvedBoard[i2][j2])
if((i1 != i2) || (j1 != j2)) solvedBoard[i1][j1] != solvedBoard[i2][j2];
}

////////////////////////////////////////////////////////////////////////////
/// Ensures that exists one direction connecting the numbers in          ///
/// increasing order                                                     ///
////////////////////////////////////////////////////////////////////////////
constraint f_seq {
foreach(solvedBoard[i])foreach(solvedBoard[i][j])
(solvedBoard[i][j] == (numCells)) ||
(solvedBoard[(i < solvedBoard.size-1) ? (i+1): i][j]    ==
solvedBoard[i][j]+1) ||
(solvedBoard[i][(j < solvedBoard[i].size - 1) ? j+1: j] ==
solvedBoard[i][j]+1) ||
(solvedBoard[(i > 0) ? i-1: i][j]                       ==
solvedBoard[i][j]+1) ||
(solvedBoard[i][(j > 0)? j-1:j]                         ==
solvedBoard[i][j]+1);
}

function void pre_randomize();
// the multiplication is not supported in the constraints
numCells = fixedBoard.size * fixedBoard[0].size;
endfunction
function void printSolvedBoard();
foreach(solvedBoard[i]) begin
foreach(solvedBoard[j]) begin
\$write("%4d", solvedBoard[i][j]);
end
\$display("");
end
endfunction
endclass

//////////////////////////////////////////////////////////////////////////////
/// SolveNumBrix: A program demonstrating how to use NumbrixSolver class   ///
//////////////////////////////////////////////////////////////////////////////

program SolveNumbrix;
NumbrixSolver board;
initial begin
board = new;
board.fixedBoard = '{
'{0,  0,  0,  0,  0,  0,  0,  0,  0},
'{0,  0, 46, 45,  0, 55, 74,  0,  0},
'{0, 38,  0,  0, 43,  0,  0, 78,  0},
'{0, 35,  0,  0,  0,  0,  0, 71,  0},
'{0,  0, 33,  0,  0,  0, 59,  0,  0},
'{0, 17,  0,  0,  0,  0,  0, 67,  0},
'{0, 18,  0,  0, 11,  0,  0, 64,  0},
'{0,  0, 24, 21,  0,  1,  2,  0,  0},
'{0,  0,  0,  0,  0,  0,  0,  0,  0}};
if(board.randomize()) begin
\$display("Solution for the Example 1");
board.printSolvedBoard();
end
else begin
\$display("Failed to solve Example 1");
end

board.fixedBoard = '{
{0,  0,  0,  0,  0,  0,  0,  0,  0},
{0, 11, 12, 15, 18, 21, 62, 61,  0},
{0,  6,  0,  0,  0,  0,  0, 60,  0},
{0, 33,  0,  0,  0,  0,  0, 57,  0},
{0, 32,  0,  0,  0,  0,  0, 56,  0},
{0, 37,  0,  1,  0,  0,  0, 73,  0},
{0, 38,  0,  0,  0,  0,  0, 72,  0},
{0, 43, 44, 47, 48, 51, 76, 77,  0},
'{0,  0,  0,  0,  0,  0,  0,  0,  0}};

if(board.randomize()) begin
\$display("Solution for the Example 2");
board.printSolvedBoard();
end
else begin
\$display("Failed to solve Example 2");
end
\$finish;
end
endprogram
```

Running the above program in ncverilog

```  > ncverilog +sv numbrix.sv
Solution for the Example 1
49  50  51  52  53  54  75  76  81
48  47  46  45  44  55  74  77  80
37  38  39  40  43  56  73  78  79
36  35  34  41  42  57  72  71  70
31  32  33  14  13  58  59  68  69
30  17  16  15  12  61  60  67  66
29  18  19  20  11  62  63  64  65
28  25  24  21  10   1   2   3   4
27  26  23  22   9   8   7   6   5
Solution for the Example 2
9  10  13  14  19  20  63  64  65
8  11  12  15  18  21  62  61  66
7   6   5  16  17  22  59  60  67
34  33   4   3  24  23  58  57  68
35  32  31   2  25  54  55  56  69
36  37  30   1  26  53  74  73  70
39  38  29  28  27  52  75  72  71
40  43  44  47  48  51  76  77  78
41  42  45  46  49  50  81  80  79
```

## Tcl

Following loosely the structure of Solve_a_Hidato_puzzle#Tcl.

```# Loop over adjacent pairs in a list.
# Example:
#  % eachpair {a b} {1 2 3} {puts \$a \$b}
#  1 2
#  2 3
proc eachpair {varNames ls script} {
if {[lassign \$varNames _i _j] ne ""} {
return -code error "Must supply exactly two arguments"
}
tailcall foreach \$_i [lrange \$ls 0 end-1] \$_j [lrange \$ls 1 end] \$script
}

namespace eval numbrix {

namespace path {::tcl::mathop ::tcl::mathfunc}

proc parse {txt} {
set map [split [string trim \$txt] \n]
}

proc print {map} {
join [lmap row \$map {
join [lmap val \$row {
format %2d \$val
}] " "
}] \n
}

proc mark {map x y i} {
lset map \$x \$y \$i
}

proc moves {x y} {
foreach {dx dy} {
0  1
-1 0      1 0
0 -1
} {
lappend r [+ \$dx \$x] [+ \$dy \$y]
}
return \$r
}

proc rmap {map} {   ;# generate a reverse map in a dict {val {x y} ..}
set rmap {}
set h [llength \$map]
set w [llength [lindex \$map 0]]
set x \$w
while {[incr x -1]>=0} {
set y \$h
while {[incr y -1]>=0} {
set i [lindex \$map \$x \$y]
if {\$i} {
dict set rmap [lindex \$map \$x \$y] [list \$x \$y]
}
}
}
return \$rmap
}

proc gaps {rmap} {  ;# list all the gaps to be filled
set known [lsort -integer [dict keys \$rmap]]
set gaps {}
eachpair {i j} \$known {
if {\$j > \$i+1} {
lappend gaps \$i \$j
}
}
return \$gaps
}

proc fixgaps {map rmap gaps} {  ;# add a "tail" gap if needed
set w [llength \$map]
set h [llength [lindex \$map 0]]
set size [* \$h \$w]
set max [max {*}[dict keys \$rmap]]
if {\$max ne \$size} {
lappend gaps \$max Inf
}
return \$gaps
}

proc paths {map x0 y0 n} {  ;# generate all the maps with a single path filled legally
if {\$n == 0} {return [list \$map]}
set i [lindex \$map \$x0 \$y0]
set paths {}
foreach {x y} [moves \$x0 \$y0] {
set j [lindex \$map \$x \$y]
if {\$j eq ""} {
continue
} elseif {\$j == 0 && \$n == \$n+1} {
return [list [mark \$map \$x \$y [+ \$i 1]]]
} elseif {\$j == \$i+1} {
lappend paths \$map
continue
} elseif {\$j || (\$n == 1)} {
continue
} else {
lappend paths {*}[
paths [
mark \$map \$x \$y [+ \$i 1]
] \$x \$y [- \$n 1]
]
}
}
return \$paths
}

proc solve {map} {
# fixpoint map
while 1 {   ;# first we iteratively fill in paths with distinct solutions
set rmap [rmap \$map]
set gaps [gaps \$rmap]
set gaps [fixgaps \$map \$rmap \$gaps]
if {\$gaps eq ""} {
return \$map
}
set oldmap \$map
foreach {i j} \$gaps {
lassign [dict get \$rmap \$i] x0 y0
set n [- \$j \$i]
set paths [paths \$map \$x0 \$y0 \$n]
if {\$paths eq ""} {
return ""
} elseif {[llength \$paths] == 1} {
#puts "solved \$i..\$j"
#puts [print \$map]
set map [lindex \$paths 0]
}
;# we could intersect the paths to maybe get some tiles
}
if {\$map eq \$oldmap} {
break
}
}
#puts "unique paths exhausted - going DFS"
try {   ;# for any left over paths, go DFS
;# we might want to sort the gaps first
foreach {i j} \$gaps {
lassign [dict get \$rmap \$i] x0 y0
set n [- \$j \$i]
set paths [paths \$map \$x0 \$y0 \$n]
foreach path \$paths {
#puts "recursing on \$i..\$j"
set sol [solve \$path]
if {\$sol ne ""} {
return \$sol
}
}
}
}
}

namespace export {[a-z]*}
namespace ensemble create
}

set puzzles {
{
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0
}

{
0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0
}
}

foreach puzzle \$puzzles {
set map [numbrix parse \$puzzle]
puts "\n== Puzzle [incr i] =="
puts [numbrix print \$map]
set sol [numbrix solve \$map]
if {\$sol ne ""} {
puts "\n== Solution \$i =="
puts [numbrix print \$sol]
} else {
puts "\n== No Solution for Puzzle \$i =="
}
}
```
Output:
```== Puzzle 1 ==
0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0

== Solution 1 ==
49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

== Puzzle 2 ==
0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0

== Solution 2 ==
9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

## Wren

Library: Wren-sort
Library: Wren-fmt
```import "./sort" for Sort
import "./fmt" for Fmt

var example1 = [
"00,00,00,00,00,00,00,00,00",
"00,00,46,45,00,55,74,00,00",
"00,38,00,00,43,00,00,78,00",
"00,35,00,00,00,00,00,71,00",
"00,00,33,00,00,00,59,00,00",
"00,17,00,00,00,00,00,67,00",
"00,18,00,00,11,00,00,64,00",
"00,00,24,21,00,01,02,00,00",
"00,00,00,00,00,00,00,00,00"
]

var example2 = [
"00,00,00,00,00,00,00,00,00",
"00,11,12,15,18,21,62,61,00",
"00,06,00,00,00,00,00,60,00",
"00,33,00,00,00,00,00,57,00",
"00,32,00,00,00,00,00,56,00",
"00,37,00,01,00,00,00,73,00",
"00,38,00,00,00,00,00,72,00",
"00,43,44,47,48,51,76,77,00",
"00,00,00,00,00,00,00,00,00"
]

var moves = [ [1, 0], [0, 1], [-1, 0], [0, -1] ]

var board = []
var grid  = []
var clues = []
var totalToFill = 0

var solve // recursive
solve = Fn.new { |r, c, count, nextClue|
if (count > totalToFill) return true
var back = grid[r][c]
if (back != 0 && back != count) return false
if (back == 0 && nextClue < clues.count && clues[nextClue] == count) {
return false
}
if (back == count) nextClue = nextClue + 1
grid[r][c] = count
for (m in moves) {
if (solve.call(r + m[1], c + m[0], count + 1, nextClue)) return true
}
grid[r][c] = back
return false
}

var printResult = Fn.new { |n|
System.print("Solution for example %(n):")
for (row in grid) {
for (i in row) if (i != -1) Fmt.write("\$2d ", i)
System.print()
}
}

var n = 0
for (ex in [example1, example2]) {
board = ex
var nRows = board.count + 2
var nCols = board[0].split(",").count + 2
var startRow = 0
var startCol = 0
grid = List.filled(nRows, null)
for (i in 0...nRows) grid[i] = List.filled(nCols, -1)
totalToFill = (nRows - 2) * (nCols - 2)
var lst = []
for (r in 0...nRows) {
if (r >= 1 && r < nRows - 1) {
var row = board[r - 1].split(",")
for (c in 1...nCols - 1) {
var value = Num.fromString(row[c - 1])
if (value == 1) {
startRow = r
startCol = c
}
grid[r][c] = value
}
}
}
Sort.quick(lst)
clues = lst
if (solve.call(startRow, startCol, 1, 0)) printResult.call(n + 1)
n = n + 1
}
```
Output:
```Solution for example 1:

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

Solution for example 2:

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```

## zkl

 This example is incorrect. Please fix the code and remove this message.Details: [Maybe], see python issue, which is not evident in the output here...
Translation of: Python

This code solves Hidato, Hopido and Numbrix puzzles.

```     // Solve Hidato/Hopido/Numbrix puzzles
class Puzzle{  // hold info concerning this puzzle
var board, nrows,ncols, cells,
start,      // (r,c) where 1 is located, Void if no 1
terminated, // if board holds highest numbered cell
given,	   // all the pre-loaded cells
adj,        // a list of (r,c) that are valid next cells
;

fcn print_board{
d:=Dictionary(-1,"  ", 0,"__");
foreach r in (board){
r.pump(String,'wrap(c){ "%2s ".fmt(d.find(c,c)) }).println();
}
}
lines:=s.split("\n");
ncols,nrows=lines[0].split().len(),lines.len();
board=nrows.pump(List(), ncols.pump(List(),-1).copy);
given,start=List(),Void;
cells,terminated=0,True;
foreach r,row in (lines.enumerate()){
foreach c,cell in (row.split().enumerate()){
if(cell=="X") continue;   // X == not in play, leave at -1
cells+=1;
val:=cell.toInt();
board[r][c]=val;
given.append(val);
if(val==1) start=T(r,c);
}
}
println("Number of cells = ",cells);
if(not given.holds(cells)){ given.append(cells); terminated=False; }
given=given.filter().sort();
}
fcn solve{   //-->Bool
if(start) return(_solve(start.xplode()));
foreach r,c in (nrows,ncols){
if(board[r][c]==0 and _solve(r,c)) return(True);
}
False
}
fcn [private] _solve(r,c,n=1, next=0){
if(n>given[-1])                       		   return(True);
if(not ( (0<=r<nrows) and (0<=c<ncols) ))		   return(False);
if(board[r][c] and board[r][c]!=n)                   return(False);
if(terminated and board[r][c]==0 and given[next]==n) return(False);

back:=0;
if(board[r][c]==n){ next+=1; back=n; }

board[r][c]=n;
foreach i,j in (adj){ if(self.fcn(r+i,c+j,n+1, next)) return(True) }
board[r][c]=back;
False
}
} // Puzzle```
```hi1:=  // 0==empty cell, X==not a cell
#<<<
"0  0  0  0  0  0  0  0  0
0  0 46 45  0 55 74  0  0
0 38  0  0 43  0  0 78  0
0 35  0  0  0  0  0 71  0
0  0 33  0  0  0 59  0  0
0 17  0  0  0  0  0 67  0
0 18  0  0 11  0  0 64  0
0  0 24 21  0  1  2  0  0
0  0  0  0  0  0  0  0  0";
#<<<

hi2:=  // 0==empty cell, X==not a cell
#<<<
"0  0  0  0  0  0  0  0  0
0 11 12 15 18 21 62 61  0
0  6  0  0  0  0  0 60  0
0 33  0  0  0  0  0 57  0
0 32  0  0  0  0  0 56  0
0 37  0  1  0  0  0 73  0
0 38  0  0  0  0  0 72  0
0 43 44 47 48 51 76 77  0
0  0  0  0  0  0  0  0  0";
#<<<
T( 0,-1),         T( 0,1),
T( 1,0) );

foreach hi in (T(hi1,hi2)){
puzzle.print_board();
puzzle.solve();
println();
puzzle.print_board();
println();
}```
Output:
```Number of cells = 81
__ __ __ __ __ __ __ __ __
__ __ 46 45 __ 55 74 __ __
__ 38 __ __ 43 __ __ 78 __
__ 35 __ __ __ __ __ 71 __
__ __ 33 __ __ __ 59 __ __
__ 17 __ __ __ __ __ 67 __
__ 18 __ __ 11 __ __ 64 __
__ __ 24 21 __  1  2 __ __
__ __ __ __ __ __ __ __ __

49 50 51 52 53 54 75 76 81
48 47 46 45 44 55 74 77 80
37 38 39 40 43 56 73 78 79
36 35 34 41 42 57 72 71 70
31 32 33 14 13 58 59 68 69
30 17 16 15 12 61 60 67 66
29 18 19 20 11 62 63 64 65
28 25 24 21 10  1  2  3  4
27 26 23 22  9  8  7  6  5

Number of cells = 81
__ __ __ __ __ __ __ __ __
__ 11 12 15 18 21 62 61 __
__  6 __ __ __ __ __ 60 __
__ 33 __ __ __ __ __ 57 __
__ 32 __ __ __ __ __ 56 __
__ 37 __  1 __ __ __ 73 __
__ 38 __ __ __ __ __ 72 __
__ 43 44 47 48 51 76 77 __
__ __ __ __ __ __ __ __ __

9 10 13 14 19 20 63 64 65
8 11 12 15 18 21 62 61 66
7  6  5 16 17 22 59 60 67
34 33  4  3 24 23 58 57 68
35 32 31  2 25 54 55 56 69
36 37 30  1 26 53 74 73 70
39 38 29 28 27 52 75 72 71
40 43 44 47 48 51 76 77 78
41 42 45 46 49 50 81 80 79
```