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
- Task
Write a program to solve puzzles of this ilk, demonstrating your program by solving the above examples. Extra credit for other interesting examples.
- Related tasks
- A* search algorithm
- Solve a Holy Knight's tour
- Knight's tour
- N-queens problem
- Solve a Hidato puzzle
- Solve a Holy Knight's tour
- Solve a Hopido puzzle
- Solve the no connection puzzle
11l
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.
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
Dart
Pretty much a straight transcription of the Kotlin solution.
/// Based on https://rosettacode.org/wiki/Solve_a_Numbrix_puzzle#Kotlin
import 'package:more/more.dart';
import 'package:sprintf/sprintf.dart';
const examples = [
[
"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"
],
[
"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"
]
];
const moves = [
[1, 0],
[0, 1],
[-1, 0],
[0, -1]
];
late List<String> board;
late List<List<int>> grid;
late List<int> clues;
var totalToFill = 0;
bool solve(int r, int c, int count, int nextClue) {
if (count > totalToFill) return true;
var back = grid[r][c];
if (back != 0 && back != count) return false;
if (back == 0 && nextClue < clues.length && clues[nextClue] == count) {
return false;
}
var nextClue2 = nextClue;
if (back == count) nextClue2++;
grid[r][c] = count;
var res = moves.any((m) => solve(r + m[1], c + m[0], count + 1, nextClue2));
if (!res) grid[r][c] = back;
return res;
}
void printResult(int n) {
print("Solution for example $n:");
for (var row in grid) {
print([
for (var e in row)
if (e != -1) sprintf("%2d ", [e])
].join());
}
}
main() {
for (var each in examples.indexed()) {
board = each.value;
var nRows = board.length + 2, nCols = board[0].split(",").length + 2;
var startRow = 0, startCol = 0;
grid = List.generate(nRows, (i) => List.filled(nCols, -1));
totalToFill = (nRows - 2) * (nCols - 2);
var lst = <int>[];
for (var r in 1.to(nRows - 1)) {
var row = board[r - 1].split(",");
for (var c in 1.to(nCols - 1)) {
var value = int.parse(row[c - 1]);
if (value > 0) lst.add(value);
if (value == 1) {
startRow = r;
startCol = c;
}
grid[r][c] = value;
}
}
clues = (lst..sort()).toList();
if (solve(startRow, startCol, 1, 0)) printResult(each.index + 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
Elixir
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
"""
HLPsolver.solve(board1, adjacent)
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
"""
HLPsolver.solve(board2, adjacent)
- 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
FreeBASIC
Type Move
dx As Integer
dy As Integer
End Type
Dim Shared As Move moves(3) = {(0, 1), (1, 0), (0, -1), (-1, 0)}
Dim Shared As Integer grid(10, 10)
Dim Shared As Integer clues(99)
Dim Shared As Integer clueCount, totalToFill
Dim Shared As String example1(8)
example1(0) = "00,00,00,00,00,00,00,00,00"
example1(1) = "00,00,46,45,00,55,74,00,00"
example1(2) = "00,38,00,00,43,00,00,78,00"
example1(3) = "00,35,00,00,00,00,00,71,00"
example1(4) = "00,00,33,00,00,00,59,00,00"
example1(5) = "00,17,00,00,00,00,00,67,00"
example1(6) = "00,18,00,00,11,00,00,64,00"
example1(7) = "00,00,24,21,00,01,02,00,00"
example1(8) = "00,00,00,00,00,00,00,00,00"
Dim Shared As String example2(8)
example2(0) = "00,00,00,00,00,00,00,00,00"
example2(1) = "00,11,12,15,18,21,62,61,00"
example2(2) = "00,06,00,00,00,00,00,60,00"
example2(3) = "00,33,00,00,00,00,00,57,00"
example2(4) = "00,32,00,00,00,00,00,56,00"
example2(5) = "00,37,00,01,00,00,00,73,00"
example2(6) = "00,38,00,00,00,00,00,72,00"
example2(7) = "00,43,44,47,48,51,76,77,00"
example2(8) = "00,00,00,00,00,00,00,00,00"
Sub SortArray(arr() As Integer, n As Integer)
Dim As Integer i, j
For i = 0 To n - 2
For j = 0 To n - 2 - i
If arr(j) > arr(j + 1) Then Swap arr(j), arr(j + 1)
Next j
Next i
End Sub
Function solve(r As Integer, c As Integer, count As Integer, nextClue As Integer) As Boolean
If count > totalToFill Then Return True
Dim As Integer back = grid(r, c)
If back <> 0 Andalso back <> count Then Return False
If back = 0 Andalso nextClue < clueCount Andalso clues(nextClue) = count Then Return False
If back = count Then nextClue += 1
grid(r, c) = count
For i As Integer = 0 To 3
If solve(r + moves(i).dy, c + moves(i).dx, count + 1, nextClue) Then Return True
Next i
grid(r, c) = back
Return False
End Function
Sub printSolution(n As Integer)
Print "Solution for example " & n & !":\n"
For r As Integer = 1 To 9
For c As Integer = 1 To 9
If grid(r, c) = -1 Then Continue For
Print Using "## "; grid(r, c);
Next c
Print
Next r
Print
End Sub
Sub main()
Dim As Integer n, i, r, c
Dim As String board(8)
For n = 0 To 1
If n = 0 Then
For i = 0 To 8: board(i) = example1(i): Next
Else
For i = 0 To 8: board(i) = example2(i): Next
End If
Dim As Integer nRows = Ubound(board) + 3
Dim As Integer nCols = 11
Dim As Integer startRow = 0, startCol = 0
totalToFill = (nRows - 2) * (nCols - 2)
clueCount = 0
For r = 0 To nRows - 1
For c = 0 To nCols - 1
grid(r, c) = -1
Next c
If r >= 1 Andalso r < nRows - 1 Then
Dim As String row = board(r - 1)
Dim As Integer posic = 1
For c = 1 To nCols - 2
Dim As String cell = Mid(row, posic, 2)
posic += 3
Dim As Integer valor = Cint(cell)
If valor > 0 Then
clues(clueCount) = valor
clueCount += 1
End If
If valor = 1 Then
startRow = r
startCol = c
End If
grid(r, c) = valor
Next c
End If
Next r
SortArray(clues(), clueCount)
If solve(startRow, startCol, 1, 0) Then printSolution(n + 1)
Next n
End Sub
main()
Sleep
- Output:
Same as Go entry.
Go
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)
puzzle := showPuzzle("Input",readPuzzle())
QMouse(puzzle,findStart(puzzle),&null,0)
showPuzzle("Output", solvePuzzle(puzzle)) | write("No solution!")
end
procedure readPuzzle()
# Start with a reduced puzzle space
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
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)
lst.add(val);
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
jq
Adapted from Wren
Works with jq, the C implementation of jq
Works with gojq, the Go implementation of jq
Works with jaq, the Rust implementation of jq
The following solution takes advantage of jq's support for backtracking. There is thus no need to undo a trial placement manually.
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
def moves: [ [1, 0], [0, 1], [-1, 0], [0, -1] ];
# "All solutions"
# Can the puzzle be solved by placing the token $count at row $r, column $c?
# Input: { grid, clues, totalToFill }
def solve($r; $c; $count; $nextClue):
if $count > .totalToFill then .
else .grid[$r][$c] as $back
| if ($back != 0 and $back != $count)
or ($back == 0 and $nextClue < (.clues|length) and .clues[$nextClue] == $count)
then empty
else if $back == $count then .nextClue += 1 end
| .grid[$r][$c] = $count
| moves[] as $m
| solve($r + $m[1]; $c + $m[0]; $count + 1; $nextClue)
end
end ;
def printResult:
.grid[1:-1][] as $row
| reduce range(1; $row|length -1) as $i ("";
. + (if $i != -1 then $row[$i] else "" end | lpad(3)) );
def task($board):
{ clues: [] }
| (($board|length) + 2) as $nRows
| ($board[0]|split(",")|length + 2) as $nCols
| .startRow = 0
| .startCol = 0
| [range(0; $nCols) | -1] as $m1
| .grid = [range(0; $nRows) | $m1]
| .totalToFill = ($nRows - 2) * ($nCols - 2)
| reduce range(0; $nRows) as $r (.;
if $r >= 1 and $r < $nRows - 1
then ($board[$r - 1]|split(",")) as $row
| reduce range(1; $nCols - 1) as $c (.;
($row[$c - 1] | sub("^0";"") | tonumber) as $value
| if $value > 0 then .clues += [$value] end
| if $value == 1
then .startRow = $r
| .startCol = $c
end
| .grid[$r][$c] = $value )
end )
| .clues |= sort
| .startRow as $startRow
| .startCol as $startCol
| first(solve($startRow; $startCol; 1; 0))
| printResult ;
### Examples
def 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"
];
def 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"
];
"Example 1", task(example1),
"\nExample 2", task(example2)- Output:
As per the requirements.
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 0 049 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 50 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 09 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
// 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 > 0) lst.add(value)
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],
MapThread[
If[Length[#2] > 1, {FaceForm[],
Rectangle[#1]}, {FaceForm[LightGray],
Rectangle[#1]}] &, {cells, cands}]};
nums =
MapThread[
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
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:
list.add val
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]) =>
Lines = read_file_lines(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
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
This algorithm only works with s specified starting point.
# numbrix.py by Xing216
MOVES = [(1, 0), (0, 1), (-1, 0), (0, -1)]
class Numbrix:
def __init__(self) -> None:
self.grid: list[list[int]]
self.clues: list[int]
self.totalToFill: int
self.startRow: int
self.StartCol: int
def solve(self, row, col, count:int, nextClue: int) -> bool:
if count > self.totalToFill:
return True
back = self.grid[row][col]
if back not in [0, count]:
return False
if back == 0 and nextClue < len(self.clues) and self.clues[nextClue] == count:
return False
nextClue = nextClue
if back == count: nextClue += 1
self.grid[row][col] = count
for move in MOVES:
if self.solve(row + move[1], col + move[0], count + 1, nextClue):
return True
self.grid[row][col] = back
def xprint(self):
for row in self.grid:
for val in row:
if val != -1:
print(f"{val:02d} ",end='')
print()
def initNumbrix(board: list[str]) -> Numbrix:
result = Numbrix()
nRows = len(board)
nCols = len(board[0].split(','))
result.grid = [[-1]*(nCols + 2) for _ in range(nRows + 2)]
result.totalToFill = nRows * nCols
xlist = []
for r, row in enumerate(board):
row = row.split(',')
for c, tile in enumerate(row):
val = int(tile)
result.grid[r + 1][c + 1] = val
if val > 0:
xlist.append(val)
if val == 1:
result.startRow = r + 1
result.startCol = c + 1
xlist.sort()
result.clues = xlist
return result
if __name__ == "__main__":
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"]
EXAMPLE3 = ["17,00,00,00,11,00,00,00,59",
"00,15,00,00,06,00,00,61,00",
"00,00,03,00,00,00,63,00,00",
"00,00,00,01,66,00,00,00,00",
"23,24,00,68,67,78,00,54,55",
"00,00,00,00,72,00,00,00,00",
"00,00,35,00,00,00,49,00,00",
"00,29,00,00,40,00,00,47,00",
"31,00,00,00,39,00,00,00,45"]
for i, board in enumerate([EXAMPLE1,EXAMPLE2, EXAMPLE3]):
numbrix = initNumbrix(board)
if numbrix.solve(numbrix.startRow, numbrix.startCol, 1, 0):
print(f"Solution for example {i+1}:", end="")
numbrix.xprint()
else:
"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 01 02 03 04 27 26 23 22 09 08 07 06 05 Solution for example 2: 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 Solution for example 3: 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
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
;; about number type checking.
;; 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
(define m++ (add1 move))
(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:
- Solve a Hidato puzzle
- Solve a Hopido puzzle
- Solve a Holy Knight's tour
- Solve a Numbrix puzzle
- Solve the no connection puzzle
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) {
eager gather for @adjacent {
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
Uiua
Uses experimental astar operator. Rewritten to avoid using now deprecated λ (swizzle) operator :-(
# Experimental!
G ← [[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]]
S ← /×△G # Total size.
Width ← ⧻⊢G
Dirs ← [∩(¯.)1Width] # D4 directions.
ManD ← /+⌵-∩(⊟⊃(◿|⌊÷)Width) # Manhattan dist.
Ns ← ▽:⟜≡(=0⊡)⊙¤▽⊸≡(↧⊃(≥0|<S))+Dirs ¤ # Valid empty Ns.
Next ← +1⊢⊚=S⊗+1⇡S # Next unplaced number.
Nodes ← ⍣(≡(⍜⊡⋅∘)⊃(⋅∘|¤⋅⋅∘|¤∘)⊙Ns:⊗-1,,Next..|[]) # Valid next boards from here.
Placed ← ⊏(⍏⊸≡(⊡1))▽⊙(⍉⊟)±,⊗.. # [pos num] sorted by nums.
# Ensure each pair of placed numbers have ManD <= number differemce.
Heur ← ⨬(-1S|999)=0/↧≡(≥⊙(⌵/-)ManD°⊟°⊟⍉)◫2Placed
astar(Nodes|Heur|¬∊0)♭G
↯△G⊡¯1°□⊢⊙◌- Output:
╭─
╷ 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
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 > 0) lst.add(value)
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 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();
}
}
fcn init(s,adjacent){
adj=adjacent;
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
}
} // Puzzlehi1:= // 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";
#<<<
adjacent:=T( T(-1,0),
T( 0,-1), T( 0,1),
T( 1,0) );
foreach hi in (T(hi1,hi2)){
puzzle:=Puzzle(hi); puzzle.adjacent=adjacent;
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
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