Solve a Hidato puzzle
You are encouraged to solve this task according to the task description, using any language you may know.
The task is to write a program which solves Hidato puzzles.
The rules are:
- You are given a grid with some numbers placed in it. The other squares in the grid will be blank.
- The grid is not necessarily rectangular.
- The grid may have holes in it.
- The grid is always connected.
- The number “1” is always present, as is another number that is equal to the number of squares in the grid. Other numbers are present so as to force the solution to be unique.
- It may be assumed that the difference between numbers present on the grid is not greater than lucky 13.
- The aim is to place a natural number in each blank square so that in the sequence of numbered squares from “1” upwards, each square is in the wp:Moore neighborhood of the squares immediately before and after it in the sequence (except for the first and last squares, of course, which only have one-sided constraints).
- Thus, if the grid was overlaid on a chessboard, a king would be able to make legal moves along the path from first to last square in numerical order.
- A square may only contain one number.
- In a proper Hidato puzzle, the solution is unique.
For example the following problem
has the following solution, with path marked on it:
Bracmat
<lang bracmat>(
( hidato
= Line solve lowest Ncells row column rpad
, Board colWidth maxDigits start curCol curRow
, range head line cellN solution output tail
. out$!arg
& @(!arg:? ((%@:>" ") ?:?arg))
& 0:?row:?column
& :?Board
& ( Line
= token
. whl
' ( @(!arg:?token [3 ?arg)
& ( ( @(!token:? "_" ?)
& :?token
| @(!token:? #?token (|" " ?))
)
& (!token.!row.!column) !Board:?Board
|
)
& 1+!column:?column
)
)
& whl
' ( @(!arg:?line \n ?arg)
& Line$!line
& 1+!row:?row
& 0:?column
)
& Line$!arg
& ( range
= hi lo
. (!arg+1:?hi)+-2:?lo
& '($lo|$arg|$hi)
)
& ( solve
= ToDo cellN row column head tail remainder
, candCell Solved rowCand colCand pattern recurse
. !arg:(?ToDo.?cellN.?row.?column)
& range$!row:(=?row)
& range$!column:(=?column)
&
' ( ?head ($cellN.?rowCand.?colCand) ?tail
& (!rowCand.!colCand):($row.$column)
& !recurse
| ?head
(.($row.$column):(?rowCand.?colCand))
(?tail&!recurse)
. ((!rowCand.!colCand).$cellN)
: ?candCell
& ( !head !tail:
& out$found!
& !candCell
| solve
$ ( !head !tail
. $cellN+1
. !rowCand
. !colCand
)
: ?remainder
& !candCell+!remainder
)
: ?Solved
)
: (=?pattern.?recurse)
& !ToDo:!pattern
& !Solved
)
& infinity:?lowest
& ( !Board
: ? (<!lowest:#%?lowest.?start) (?&~)
| solve$(!Board.!lowest.!start):?solution
)
& :?output
& 0:?curCol
& !solution:((?curRow.?).?)+?+[?Ncells
& @(!Ncells:? [?maxDigits)
& 1+!maxDigits:?colWidth
& ( rpad
= len
. !arg:(?arg.?len)
& @(str$(!arg " "):?arg [!len ?)
& !arg
)
& whl
' ( !solution:((?row.?column).?cellN)+?solution
& ( !row:>!curRow:?curRow
& !output \n:?output
& 0:?curCol
|
)
& whl
' ( !curCol+1:~>!column:?curCol
& !output rpad$(.!colWidth):?output
)
& !output rev$(rpad$(rev$(str$(!cellN " ")).!colWidth))
: ?output
& !curCol+1:?curCol
)
& str$!output
)
& "
__ 33 35 __ __
__ __ 24 22 __
__ __ __ 21 __ __
__ 26 __ 13 40 11
27 __ __ __ 9 __ 1
__ __ 18 __ __
__ 7 __ __
5 __"
: ?board
& out$(hidato$!board) );</lang> Output:
__ 33 35 __ __
__ __ 24 22 __
__ __ __ 21 __ __
__ 26 __ 13 40 11
27 __ __ __ 9 __ 1
__ __ 18 __ __
__ 7 __ __
5 __
found!
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
C
Depth-first graph, with simple connectivity check to reject some impossible situations early. The checks slow down simpler puzzles significantly, but can make some deep recursions backtrack much earilier. <lang c>#include <stdio.h>
- include <stdlib.h>
- include <string.h>
- include <ctype.h>
int *board, *flood, *known, top = 0, w, h;
static inline int idx(int y, int x) { return y * w + x; }
int neighbors(int c, int *p) /* @c cell @p list of neighbours @return amount of neighbours
- /
{ int i, j, n = 0; int y = c / w, x = c % w;
for (i = y - 1; i <= y + 1; i++) { if (i < 0 || i >= h) continue; for (j = x - 1; j <= x + 1; j++) if (!(j < 0 || j >= w || (j == x && i == y) || board[ p[n] = idx(i,j) ] == -1)) n++; }
return n; }
void flood_fill(int c) /* fill all free cells around @c with “1” and write output to variable “flood” @c cell
- /
{ int i, n[8], nei;
nei = neighbors(c, n); for (i = 0; i < nei; i++) { // for all neighbours if (board[n[i]] || flood[n[i]]) continue; // if cell is not free, choose another neighbour
flood[n[i]] = 1; flood_fill(n[i]); } }
/* Check all empty cells are reachable from higher known cells.
Should really do more checks to make sure cell_x and cell_x+1 share enough reachable empty cells; I'm lazy. Will implement if a good counter example is presented. */
int check_connectity(int lowerbound) { int c; memset(flood, 0, sizeof(flood[0]) * w * h); for (c = lowerbound + 1; c <= top; c++) if (known[c]) flood_fill(known[c]); // mark all free cells around known cells
for (c = 0; c < w * h; c++) if (!board[c] && !flood[c]) // if there are free cells which could not be reached from flood_fill return 0;
return 1; }
void make_board(int x, int y, const char *s) { int i;
w = x, h = y;
top = 0;
x = w * h;
known = calloc(x + 1, sizeof(int));
board = calloc(x, sizeof(int));
flood = calloc(x, sizeof(int));
while (x--) board[x] = -1;
for (y = 0; y < h; y++) for (x = 0; x < w; x++) { i = idx(y, x);
while (isspace(*s)) s++;
switch (*s) { case '_': board[i] = 0; case '.': break; default: known[ board[i] = strtol(s, 0, 10) ] = i; if (board[i] > top) top = board[i]; }
while (*s && !isspace(*s)) s++; } }
void show_board(const char *s) { int i, j, c;
printf("\n%s:\n", s);
for (i = 0; i < h; i++, putchar('\n')) for (j = 0; j < w; j++) { c = board[ idx(i, j) ]; printf(!c ? " __" : c == -1 ? " " : " %2d", c); } }
int fill(int c, int n) { int i, nei, p[8], ko, bo;
if ((board[c] && board[c] != n) || (known[n] && known[n] != c)) return 0;
if (n == top) return 1;
ko = known[n]; bo = board[c]; board[c] = n;
if (check_connectity(n)) { nei = neighbors(c, p); for (i = 0; i < nei; i++) if (fill(p[i], n + 1)) return 1; }
board[c] = bo; known[n] = ko; return 0; }
int main() { make_board(
- define USE_E 0
- if (USE_E == 0)
8,8, " __ 33 35 __ __ .. .. .." " __ __ 24 22 __ .. .. .." " __ __ __ 21 __ __ .. .." " __ 26 __ 13 40 11 .. .." " 27 __ __ __ 9 __ 1 .." " . . __ __ 18 __ __ .." " . .. . . __ 7 __ __" " . .. .. .. . . 5 __"
- elif (USE_E == 1)
3, 3, " . 4 ." " _ 7 _" " 1 _ _"
- else
50, 3, " 1 _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . 74" " . . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ ." " . . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ ."
- endif
);
show_board("Before"); fill(known[1], 1); show_board("After"); /* "40 lbs in two weeks!" */
return 0; }</lang>
- Output:
Before:
__ 33 35 __ __
__ __ 24 22 __
__ __ __ 21 __ __
__ 26 __ 13 40 11
27 __ __ __ 9 __ 1
__ __ 18 __ __
__ 7 __ __
5 __
After:
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
C++
<lang cpp>
- include <iostream>
- include <sstream>
- include <iterator>
- include <vector>
//------------------------------------------------------------------------------ using namespace std;
//------------------------------------------------------------------------------ struct node {
int val; unsigned char neighbors;
}; //------------------------------------------------------------------------------ class hSolver { public:
hSolver()
{
dx[0] = -1; dx[1] = 0; dx[2] = 1; dx[3] = -1; dx[4] = 1; dx[5] = -1; dx[6] = 0; dx[7] = 1; dy[0] = -1; dy[1] = -1; dy[2] = -1; dy[3] = 0; dy[4] = 0; dy[5] = 1; dy[6] = 1; dy[7] = 1;
}
void solve( vector<string>& puzz, int max_wid )
{
if( puzz.size() < 1 ) return; wid = max_wid; hei = static_cast<int>( puzz.size() ) / wid; int len = wid * hei, c = 0; max = 0; arr = new node[len]; memset( arr, 0, len * sizeof( node ) ); weHave = new bool[len + 1]; memset( weHave, 0, len + 1 );
for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ ) { if( ( *i ) == "*" ) { arr[c++].val = -1; continue; } arr[c].val = atoi( ( *i ).c_str() ); if( arr[c].val > 0 ) weHave[arr[c].val] = true; if( max < arr[c].val ) max = arr[c].val; c++; }
solveIt(); c = 0; for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ ) { if( ( *i ) == "." ) { ostringstream o; o << arr[c].val; ( *i ) = o.str(); } c++; } delete [] arr; delete [] weHave;
}
private:
bool search( int x, int y, int w )
{
if( w == max ) return true;
node* n = &arr[x + y * wid]; n->neighbors = getNeighbors( x, y ); if( weHave[w] ) { for( int d = 0; d < 8; d++ ) { if( n->neighbors & ( 1 << d ) ) { int a = x + dx[d], b = y + dy[d]; if( arr[a + b * wid].val == w ) if( search( a, b, w + 1 ) ) return true; } } return false; }
for( int d = 0; d < 8; d++ ) { if( n->neighbors & ( 1 << d ) ) { int a = x + dx[d], b = y + dy[d]; if( arr[a + b * wid].val == 0 ) { arr[a + b * wid].val = w; if( search( a, b, w + 1 ) ) return true; arr[a + b * wid].val = 0; } } } return false;
}
unsigned char getNeighbors( int x, int y )
{
unsigned char c = 0; int m = -1, a, b; for( int yy = -1; yy < 2; yy++ ) for( int xx = -1; xx < 2; xx++ ) { if( !yy && !xx ) continue; m++; a = x + xx, b = y + yy; if( a < 0 || b < 0 || a >= wid || b >= hei ) continue; if( arr[a + b * wid].val > -1 ) c |= ( 1 << m ); } return c;
}
void solveIt()
{
int x, y; findStart( x, y ); if( x < 0 ) { cout << "\nCan't find start point!\n"; return; } search( x, y, 2 );
}
void findStart( int& x, int& y )
{
for( int b = 0; b < hei; b++ ) for( int a = 0; a < wid; a++ ) if( arr[a + wid * b].val == 1 ) { x = a; y = b; return; } x = y = -1;
}
int wid, hei, max, dx[8], dy[8]; node* arr; bool* weHave;
}; //------------------------------------------------------------------------------ int main( int argc, char* argv[] ) {
int wid; string p = ". 33 35 . . * * * . . 24 22 . * * * . . . 21 . . * * . 26 . 13 40 11 * * 27 . . . 9 . 1 * * * . . 18 . . * * * * * . 7 . . * * * * * * 5 ."; wid = 8; //string p = "54 . 60 59 . 67 . 69 . . 55 . . 63 65 . 72 71 51 50 56 62 . * * * * . . . 14 * * 17 . * 48 10 11 * 15 . 18 . 22 . 46 . * 3 . 19 23 . . 44 . 5 . 1 33 32 . . 43 7 . 36 . 27 . 31 42 . . 38 . 35 28 . 30"; wid = 9; //string p = ". 58 . 60 . . 63 66 . 57 55 59 53 49 . 65 . 68 . 8 . . 50 . 46 45 . 10 6 . * * * . 43 70 . 11 12 * * * 72 71 . . 14 . * * * 30 39 . 15 3 17 . 28 29 . . 40 . . 19 22 . . 37 36 . 1 20 . 24 . 26 . 34 33"; wid = 9;
istringstream iss( p ); vector<string> puzz; copy( istream_iterator<string>( iss ), istream_iterator<string>(), back_inserter<vector<string> >( puzz ) ); hSolver s; s.solve( puzz, wid );
int c = 0;
for( vector<string>::iterator i = puzz.begin(); i != puzz.end(); i++ )
{
if( ( *i ) != "*" && ( *i ) != "." ) { if( atoi( ( *i ).c_str() ) < 10 ) cout << "0"; cout << ( *i ) << " "; } else cout << " "; if( ++c >= wid ) { cout << endl; c = 0; }
} cout << endl << endl; return system( "pause" );
} //-------------------------------------------------------------------------------------------------- </lang> Output:
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 09 10 01
15 16 18 08 02
17 07 06 03
05 04
56 58 54 60 61 62 63 66 67
57 55 59 53 49 47 65 64 68
09 08 52 51 50 48 46 45 69
10 06 07 44 43 70
05 11 12 72 71 42
04 14 13 30 39 41
15 03 17 18 28 29 38 31 40
02 16 19 22 23 27 37 36 32
01 20 21 24 25 26 35 34 33
D
More C-Style Version
This version retains some of the characteristics of the original C version. It uses global variables, it doesn't enforce immutability and purity. This style is faster to write for prototypes, short programs or less important code, but in larger programs you usually want more strictness to avoid some bugs and increase long-term maintainability.
<lang d>import std.stdio, std.array, std.conv, std.algorithm, std.string;
int[][] board; int[] given, start;
void setup(string s) {
auto lines = s.splitLines; auto cols = lines[0].split.length; auto rows = lines.length; given.length = 0;
board = new int[][](rows + 2, cols + 2);
foreach (row; board)
row[] = -1;
foreach (r, row; lines) {
foreach (c, cell; row.split) {
switch (cell) {
case "__":
board[r + 1][c + 1] = 0;
break;
case ".":
break;
default:
int val = cell.to!int;
board[r + 1][c + 1] = val;
given ~= val;
if (val == 1)
start = [r + 1, c + 1];
}
}
}
given.sort();
}
bool solve(int r, int c, int n, int next = 0) {
if (n > given.back)
return true;
if (board[r][c] && board[r][c] != n)
return false;
if (board[r][c] == 0 && given[next] == n)
return false;
int back = board[r][c];
board[r][c] = n;
foreach (i; -1 .. 2)
foreach (j; -1 .. 2)
if (solve(r + i, c + j, n + 1, next + (back == n)))
return true;
board[r][c] = back; return false;
}
void printBoard() {
foreach (row; board) {
foreach (c; row)
writef(c == -1 ? " . " : c ? "%2d " : "__ ", c);
writeln;
}
}
void main() {
auto hi = "__ 33 35 __ __ . . .
__ __ 24 22 __ . . .
__ __ __ 21 __ __ . .
__ 26 __ 13 40 11 . .
27 __ __ __ 9 __ 1 .
. . __ __ 18 __ __ .
. . . . __ 7 __ __
. . . . . . 5 __";
hi.setup; printBoard; "\nFound:".writeln; solve(start[0], start[1], 1); printBoard;
}</lang>
- Output:
. . . . . . . . . . . __ 33 35 __ __ . . . . . __ __ 24 22 __ . . . . . __ __ __ 21 __ __ . . . . __ 26 __ 13 40 11 . . . . 27 __ __ __ 9 __ 1 . . . . . __ __ 18 __ __ . . . . . . . __ 7 __ __ . . . . . . . . 5 __ . . . . . . . . . . . Found: . . . . . . . . . . . 32 33 35 36 37 . . . . . 31 34 24 22 38 . . . . . 30 25 23 21 12 39 . . . . 29 26 20 13 40 11 . . . . 27 28 14 19 9 10 1 . . . . . 15 16 18 8 2 . . . . . . . 17 7 6 3 . . . . . . . . 5 4 . . . . . . . . . . .
Stronger Version
This version uses a little stronger typing, performs tests a run-time with contracts, it doesn't use global variables, it enforces immutability and purity where possible, and produces a correct text output for both larger ad small boards. This style is more fit for larger programs, or when you want the code to be less bug-prone or a little more efficient.
With this coding style the changes in the code become less bug-prone, but also more laborious. This version is also faster, its total runtime is about 0.02 seconds or less. <lang d>import std.stdio, std.conv, std.ascii, std.array, std.string,
std.algorithm, std.exception, std.range, std.typetuple;
struct Hidato {
// alias Cell = RangedValue!(int, -1, int.max);
alias Cell = int;
alias Pos = size_t;
enum : Cell { emptyCell = -1, unknownCell = 0 }
immutable Cell boardMax; immutable size_t nCols, nRows; Cell[] board; Pos[] known; bool[] flood;
this(in string input) @safe pure
in {
assert(!input.strip.empty);
} out {
assert(nCols > 0 && nRows > 0);
immutable size = nCols * nRows;
assert(board.length == size);
assert(known.length == size + 1);
assert(flood.length == size);
assert(boardMax > 0 && boardMax <= size);
assert(board.reduce!max == boardMax);
assert(board.canFind(1) && board.canFind(boardMax));
assert(flood.all!(f => f == 0));
assert(known.all!(rc => rc >= 0 && rc < size));
foreach (immutable i, immutable cell; board) {
assert(cell == Hidato.emptyCell ||
cell == Hidato.unknownCell ||
(cell >= 1 && cell <= size));
if (cell > 0)
assert(i == known[cast(size_t)cell]);
}
} body {
bool[Cell] pathSeen; // A set.
/*immutable*/ const lines = input.splitLines;
this.nRows = lines.length;
this.nCols = lines[0].split.length;
immutable size = nCols * nRows;
this.board = new typeof(this.board[0])[size];
this.board[] = emptyCell;
this.known = new typeof(this.known[0])[size + 1];
this.flood = new typeof(this.flood[0])[size];
auto boardMaxMutable = Cell.min;
Pos i = 0;
foreach (immutable row; lines) {
assert(row.split.length == nCols,
text("Wrong cols n.: ", row.split.length));
foreach (immutable cell; row.split) {
switch (cell) {
case "_":
this.board[i] = Hidato.unknownCell;
break;
case ".":
this.board[i] = Hidato.emptyCell;
break;
default: // Known.
immutable val = cell.to!Cell;
enforce(val > 0, "Path numbers must be > 0.");
enforce(val !in pathSeen,
text("Duplicated path number: ", val));
pathSeen[val] = true;
this.board[i] = val;
this.known[val] = i;
boardMaxMutable = max(boardMaxMutable, val);
}
i++;
}
}
this.boardMax = boardMaxMutable; }
private Pos idx(in size_t r, in size_t c) const pure nothrow {
return r * nCols + c;
}
private uint nNeighbors(in Pos pos, ref Pos[8] neighbours)
const pure nothrow {
immutable r = pos / nCols;
immutable c = pos % nCols;
typeof(return) n = 0;
foreach (immutable sr; TypeTuple!(-1, 0, 1)) {
immutable size_t i = r + sr; // Can wrap-around.
if (i >= nRows)
continue;
foreach (immutable sc; TypeTuple!(-1, 0, 1)) {
immutable size_t j = c + sc; // Can wrap-around.
if ((sc != 0 || sr != 0) && j < nCols) {
immutable pos2 = idx(i, j);
neighbours[n] = pos2;
if (board[pos2] != Hidato.emptyCell)
n++;
}
}
}
return n; }
/// Fill all free cells around 'cell' with true and write
/// output to variable "flood".
private void floodFill(in Pos pos) pure nothrow {
Pos[8] n = void;
// For all neighbours.
foreach (immutable i; 0 .. nNeighbors(pos, n)) {
// If pos is not free, choose another neighbour.
if (board[n[i]] || flood[n[i]])
continue;
flood[n[i]] = true;
floodFill(n[i]);
}
}
/// Check all empty cells are reachable from higher known cells.
private bool checkConnectity(in uint lowerBound) pure nothrow {
flood[] = false;
foreach (immutable i; lowerBound + 1 .. boardMax + 1)
if (known[i])
floodFill(known[i]);
foreach (immutable i; 0 .. nCols * nRows)
// If there are free cells which could not be
// reached from floodFill.
if (!board[i] && !flood[i])
return false;
return true;
}
private bool fill(in Pos pos, in uint n) pure nothrow {
if ((board[pos] && board[pos] != n) ||
(known[n] && known[n] != pos))
return false;
if (n == boardMax)
return true;
immutable ko = known[n];
immutable bo = board[pos];
board[pos] = n;
Pos[8] p = void;
if (checkConnectity(n))
foreach (immutable i; 0 .. nNeighbors(pos, p))
if (fill(p[i], n + 1))
return true;
board[pos] = bo;
known[n] = ko;
return false;
}
void solve() pure nothrow
in {
assert(!known.empty);
} body {
fill(known[1], 1);
}
string toString() const pure {
immutable d = [Hidato.emptyCell: ".",
Hidato.unknownCell: "_"];
immutable form = "%" ~ text(boardMax.text.length + 1) ~ "s";
string result;
foreach (immutable r; 0 .. nRows) {
foreach (immutable c; 0 .. nCols) {
immutable cell = board[idx(r, c)];
result ~= format(form, d.get(cell, cell.text));
}
result ~= "\n";
}
return result;
}
}
void solveHidato(in string problem) {
auto game = problem.Hidato;
writeln("Problem:\n", game);
game.solve;
writeln("Solution:\n", game);
}
void main() {
solveHidato(" _ 33 35 _ _ . . .
_ _ 24 22 _ . . .
_ _ _ 21 _ _ . .
_ 26 _ 13 40 11 . .
27 _ _ _ 9 _ 1 .
. . _ _ 18 _ _ .
. . . . _ 7 _ _
. . . . . . 5 _");
solveHidato(". 4 .
_ 7 _
1 _ _");
solveHidato(
"1 _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . 74
. . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . . . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ ." );
}</lang>
- Output:
Problem: _ 33 35 _ _ . . . _ _ 24 22 _ . . . _ _ _ 21 _ _ . . _ 26 _ 13 40 11 . . 27 _ _ _ 9 _ 1 . . . _ _ 18 _ _ . . . . . _ 7 _ _ . . . . . . 5 _ Solution: 32 33 35 36 37 . . . 31 34 24 22 38 . . . 30 25 23 21 12 39 . . 29 26 20 13 40 11 . . 27 28 14 19 9 10 1 . . . 15 16 18 8 2 . . . . . 17 7 6 3 . . . . . . 5 4 Problem: . 4 . _ 7 _ 1 _ _ Solution: . 4 . 3 7 5 1 2 6 Problem: 1 _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . 74 . . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . . . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . Solution: 1 2 3 . . 8 9 . . 14 15 . . 20 21 . . 26 27 . . 32 33 . . 38 39 . . 44 45 . . 50 51 . . 56 57 . . 62 63 . . 68 69 . . 74 . . 4 . 7 . 10 . 13 . 16 . 19 . 22 . 25 . 28 . 31 . 34 . 37 . 40 . 43 . 46 . 49 . 52 . 55 . 58 . 61 . 64 . 67 . 70 . 73 . . . . 5 6 . . 11 12 . . 17 18 . . 23 24 . . 29 30 . . 35 36 . . 41 42 . . 47 48 . . 53 54 . . 59 60 . . 65 66 . . 71 72 .
Erlang
To simplify the code I start a new process for searching each potential path through the grid. This means that the default maximum number of processes had to be raised ("erl +P 50000" works for me). The task takes about 1-2 seconds on a low level Mac mini. If faster times are needed, or even less performing hardware is used, some optimisation should be done. <lang Erlang> -module( solve_hidato_puzzle ).
-export( [create/2, solve/1, task/0] ).
-compile({no_auto_import,[max/2]}).
create( Grid_list, Number_list ) ->
Squares = lists:flatten( [create_column(X, Y) || {X, Y} <- Grid_list] ),
lists:foldl( fun store/2, dict:from_list(Squares), Number_list ).
print( Grid_list ) when is_list(Grid_list) -> print( create(Grid_list, []) ); print( Grid_dict ) ->
Max_x = max_x( Grid_dict ), Max_y = max_y( Grid_dict ), Print_row = fun (Y) -> [print(X, Y, Grid_dict) || X <- lists:seq(1, Max_x)], io:nl() end, [Print_row(Y) || Y <- lists:seq(1, Max_y)].
solve( Dict ) ->
{find_start, [Start]} = {find_start, dict:fold( fun start/3, [], Dict )},
Max = dict:size( Dict ),
{stop_ok, {Max, Max, [Stop]}} = {stop_ok, dict:fold( fun stop/3, {Max, 0, []}, Dict )},
My_pid = erlang:self(),
erlang:spawn( fun() -> path(Start, Stop, Dict, My_pid, []) end ),
receive
{grid, Grid, path, Path} -> {Grid, Path}
end.
task() ->
%% Square is {X, Y}, N}. N = 0 for empty square. These are created if not present.
%% Leftmost column is X=1. Top row is Y=1.
%% Optimised for the example, grid is a list of {X, {Y_min, Y_max}}.
%% When there are holes, X is repeated as many times as needed with two new Y values each time.
Start = {{7,5}, 1},
Stop = {{5,4}, 40},
Grid_list = [{1, {1,5}}, {2, {1,5}}, {3, {1,6}}, {4, {1,6}}, {5, {1,7}}, {6, {3,7}}, {7, {5,8}}, {8, {7,8}}],
Number_list = [Start, Stop, {{1,5}, 27}, {{2,1}, 33}, {{2,4}, 26}, {{3,1}, 35}, {{3,2}, 24},
{{4,2}, 22}, {{4,3}, 21}, {{4,4}, 13}, {{5,5}, 9}, {{5,6}, 18}, {{6,4}, 11}, {{6,7}, 7}, {{7,8}, 5}],
Grid = create( Grid_list, Number_list ),
io:fwrite( "Start grid~n" ),
print( Grid ),
{New_grid, Path} = solve( create(Grid_list, Number_list) ),
io:fwrite( "Start square ~p, Stop square ~p.~nPath ~p~n", [Start, Stop, Path] ),
print( New_grid ).
create_column( X, {Y_min, Y_max} ) -> [{{X, Y}, 0} || Y <- lists:seq(Y_min, Y_max)].
is_filled( Dict ) -> [] =:= dict:fold( fun keep_0_square/3, [], Dict ).
keep_0_square( Key, 0, Acc ) -> [Key | Acc]; keep_0_square( _Key, _Value, Acc ) -> Acc.
max( Position, Keys ) ->
[Square | _T] = lists:reverse( lists:keysort(Position, Keys) ), Square.
max_x( Dict ) ->
{X, _Y} = max( 1, dict:fetch_keys(Dict) ),
X.
max_y( Dict ) ->
{_X, Y} = max( 2, dict:fetch_keys(Dict) ),
Y.
neighbourhood( Square, Dict ) ->
Potentials = neighbourhood_potential_squares( Square ),
neighbourhood_squares( dict:find(Square, Dict), Potentials, Dict ).
neighbourhood_potential_squares( {X, Y} ) -> [{Sx, Sy} || Sx <- [X-1, X, X+1], Sy <- [Y-1, Y, Y+1], {X, Y} =/= {Sx, Sy}].
neighbourhood_squares( {ok, Value}, Potentials, Dict ) ->
Square_values = lists:flatten( [neighbourhood_square_value(X, dict:find(X, Dict)) || X <- Potentials] ),
Next_value = Value + 1,
neighbourhood_squares_next_value( lists:keyfind(Next_value, 2, Square_values), Square_values, Next_value ).
neighbourhood_squares_next_value( {Square, Value}, _Square_values, Value ) -> [{Square, Value}]; neighbourhood_squares_next_value( false, Square_values, Value ) -> [{Square, Value} || {Square, Y} <- Square_values, Y =:= 0].
neighbourhood_square_value( Square, {ok, Value} ) -> [{Square, Value}]; neighbourhood_square_value( _Square, error ) -> [].
path( Square, Square, Dict, Pid, Path ) -> path_correct( is_filled(Dict), Pid, [Square | Path], Dict ); path( Square, Stop, Dict, Pid, Path ) ->
Reversed_path = [Square | Path],
Neighbours = neighbourhood( Square, Dict ),
[erlang:spawn( fun() -> path(Next_square, Stop, dict:store(Next_square, Value, Dict), Pid, Reversed_path) end ) || {Next_square, Value} <- Neighbours].
path_correct( true, Pid, Path, Dict ) -> Pid ! {grid, Dict, path, lists:reverse( Path )}; path_correct( false, _Pid, _Path, _Dict ) -> dead_end.
print( X, Y, Dict ) -> print_number( dict:find({X, Y}, Dict) ).
print_number( {ok, 0} ) -> io:fwrite( "~3s", ["."] ); % . is less distracting than 0 print_number( {ok, Value} ) -> io:fwrite( "~3b", [Value] ); print_number( error ) -> io:fwrite( "~3s", [" "] ).
start( Key, 1, Acc ) -> [Key | Acc]; % Allow check that we only have one key with value 1. start( _Key, _Value, Acc ) -> Acc.
stop( Key, Max, {Max, Max_found, Stops} ) -> {Max, erlang:max(Max, Max_found), [Key | Stops]}; % Allow check that we only have one key with value Max. stop( _Key, Value, {Max, Max_found, Stops} ) -> {Max, erlang:max(Value, Max_found), Stops}. % Allow check that Max is Max.
store( {Key, Value}, Dict ) -> dict:store( Key, Value, Dict ). </lang>
- Output:
2> solve_hidato_puzzle:task().
Start grid
. 33 35 . .
. . 24 22 .
. . . 21 . .
. 26 . 13 40 11
27 . . . 9 . 1
. . 18 . .
. 7 . .
5 .
Start square {{7,5},1}, Stop square {{5,4},40}.
Path [{7,5}, {7,6}, {8,7}, {8,8}, {7,8}, {7,7}, {6,7}, {6,6}, {5,5}, {6,5}, {6,4}, {5,3}, {4,4}, {3,5}, {3,6}, {4,6}, {5,7}, {5,6}, {4,5}, {3,4},
{4,3}, {4,2}, {3,3}, {3,2}, {2,3}, {2,4}, {1,5},{2,5}, {1,4}, {1,3}, {1,2}, {1,1}, {2,1}, {2,2}, {3,1}, {4,1}, {5,1}, {5,2}, {6,3}, {5,4}]
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
Icon and Unicon
This is an Unicon-specific solution but could easily be adjusted to work in Icon. <lang unicon>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 := -1 nCells := maxCols := 0 every line := !&input do { put(p,[: -1 | gencells(line) | -1 :]) maxCols <:= *p[-1] } put(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["#"] := -1; cMap["_"] := 0
cMap[-1] := " "; cMap[0] := "_"
}
s ? while not pos(0) do {
w := (tab(many(WS))|"", tab(many(NWS))) | break
w := numeric(\cMap[w]|w)
if -1 ~= w then nCells +:= 1
suspend w
}
end
procedure showPuzzle(label, p)
write(label," with ",nCells," cells:")
every r := !p do {
every c := !r do writes(right((\cMap[c]|c),*nCells+1))
write()
}
return p
end
procedure findStart(p)
if \p[r := !*p][c := !*p[r]] = 1 then return Pos(r,c)
end
procedure solvePuzzle(puzzle)
if path := \best then {
repeat {
loc := path.getLoc()
puzzle[loc.r][loc.c] := path.getVal()
path := \path.getParent() | break
}
return puzzle
}
end
class QMouse(puzzle, loc, parent, val)
method getVal(); return val; end method getLoc(); return loc; end method getParent(); return parent; end method atEnd(); return (nCells = val) = puzzle[loc.r][loc.c]; end method goNorth(); return visit(loc.r-1,loc.c); end method goNE(); return visit(loc.r-1,loc.c+1); end method goEast(); return visit(loc.r, loc.c+1); end method goSE(); return visit(loc.r+1,loc.c+1); end method goSouth(); return visit(loc.r+1,loc.c); end method goSW(); return visit(loc.r+1,loc.c-1); end method goWest(); return visit(loc.r, loc.c-1); end method goNW(); return visit(loc.r-1,loc.c-1); end
method visit(r,c)
if /best & validPos(r,c) then return Pos(r,c)
end
method validPos(r,c)
xv := puzzle[r][c]
if xv = (val+1) then return
if xv = 0 then { # make sure this path hasn't already gone there
ancestor := self
while xl := (ancestor := \ancestor.getParent()).getLoc() do
if (xl.r = r) & (xl.c = c) then fail
return
}
end
initially (p, l, a, v)
puzzle := p loc := l parent := a val := v+1 if atEnd() then return best := self if \best then return QMouse(puzzle, goNorth(), self, val) QMouse(puzzle, goNE(), self, val) QMouse(puzzle, goEast(), self, val) QMouse(puzzle, goSE(), self, val) QMouse(puzzle, goSouth(), self, val) QMouse(puzzle, goSW(), self, val) QMouse(puzzle, goWest(), self, val) QMouse(puzzle, goNW(), self, val)
end</lang>
Sample run:
->hd <hd.in
Input with 40 cells:
_ 33 35 _ _
_ _ 24 22 _
_ _ _ 21 _ _
_ 26 _ 13 40 11
27 _ _ _ 9 _ 1
_ _ 18 _ _
_ 7 _ _
5 _
Output with 40 cells:
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
->
Mathprog
<lang mathprog>/*Hidato.mathprog, part of KuKu by Nigel Galloway
Find a solution to a Hidato problem
Nigel_Galloway@operamail.com April 1st., 2011
- /
param ZBLS; param ROWS; param COLS; param D := 1; set ROWSR := 1..ROWS; set COLSR := 1..COLS; set ROWSV := (1-D)..(ROWS+D); set COLSV := (1-D)..(COLS+D); param Iz{ROWSR,COLSR}, integer, default 0; set ZBLSV := 1..(ZBLS+1); set ZBLSR := 1..ZBLS;
var BR{ROWSV,COLSV,ZBLSV}, binary;
void0{r in ROWSV, z in ZBLSR,c in (1-D)..0}: BR[r,c,z] = 0; void1{r in ROWSV, z in ZBLSR,c in (COLS+1)..(COLS+D)}: BR[r,c,z] = 0; void2{c in COLSV, z in ZBLSR,r in (1-D)..0}: BR[r,c,z] = 0; void3{c in COLSV, z in ZBLSR,r in (ROWS+1)..(ROWS+D)}: BR[r,c,z] = 0; void4{r in ROWSV,c in (1-D)..0}: BR[r,c,ZBLS+1] = 1; void5{r in ROWSV,c in (COLS+1)..(COLS+D)}: BR[r,c,ZBLS+1] = 1; void6{c in COLSV,r in (1-D)..0}: BR[r,c,ZBLS+1] = 1; void7{c in COLSV,r in (ROWS+1)..(ROWS+D)}: BR[r,c,ZBLS+1] = 1;
Izfree{r in ROWSR, c in COLSR, z in ZBLSR : Iz[r,c] = -1}: BR[r,c,z] = 0; Iz1{Izr in ROWSR, Izc in COLSR, r in ROWSR, c in COLSR, z in ZBLSR : Izr=r and Izc=c and Iz[Izr,Izc]=z}: BR[r,c,z] = 1;
rule1{z in ZBLSR}: sum{r in ROWSR, c in COLSR} BR[r,c,z] = 1; rule2{r in ROWSR, c in COLSR}: sum{z in ZBLSV} BR[r,c,z] = 1; rule3{r in ROWSR, c in COLSR, z in ZBLSR}: BR[0,0,z+1] + BR[r-1,c-1,z+1] + BR[r-1,c,z+1] + BR[r-1,c+1,z+1] + BR[r,c-1,z+1] + BR[r,c+1,z+1] + BR[r+1,c-1,z+1] + BR[r+1,c,z+1] + BR[r+1,c+1,z+1] - BR[r,c,z] >= 0;
solve;
for {r in ROWSR} {
for {c in COLSR} {
printf " %2d", sum{z in ZBLSR} BR[r,c,z]*z;
}
printf "\n";
} data;
param ROWS := 8; param COLS := 8; param ZBLS := 40; param Iz: 1 2 3 4 5 6 7 8 :=
1 . 33 35 . . -1 -1 -1 2 . . 24 22 . -1 -1 -1 3 . . . 21 . . -1 -1 4 . 26 . 13 40 11 -1 -1 5 27 . . . 9 . 1 -1 6 -1 -1 . . 18 . . -1 7 -1 -1 -1 -1 . 7 . . 8 -1 -1 -1 -1 -1 -1 5 . ; end;</lang>
Using the data in the model produces the following:
- Output:
>glpsol --minisat --math Hidato.mathprog GLPSOL: GLPK LP/MIP Solver, v4.47 Parameter(s) specified in the command line: --minisat --math Hidato.mathprog Reading model section from Hidato.mathprog... Reading data section from Hidato.mathprog... 64 lines were read Generating void0... Generating void1... Generating void2... Generating void3... Generating void4... Generating void5... Generating void6... Generating void7... Generating Izfree... Generating Iz1... Generating rule1... Generating rule2... Generating rule3... Model has been successfully generated Will search for ANY feasible solution Translating to CNF-SAT... Original problem has 5279 rows, 4100 columns, and 33359 non-zeros 2520 covering inequalities 2719 partitioning equalities Solving CNF-SAT problem... Instance has 7076 variables, 24047 clauses, and 77735 literals ==================================[MINISAT]=================================== | Conflicts | ORIGINAL | LEARNT | Progress | | | Clauses Literals | Limit Clauses Literals Lit/Cl | | ============================================================================== | 0 | 21432 75120 | 7144 0 0 0.0 | 0.000 % | ============================================================================== SATISFIABLE Objective value = 0.000000000e+000 Time used: 0.0 secs Memory used: 14.5 Mb (15192264 bytes) 32 33 35 36 37 0 0 0 31 34 24 22 38 0 0 0 30 25 23 21 12 39 0 0 29 26 20 13 40 11 0 0 27 28 14 19 9 10 1 0 0 0 15 16 18 8 2 0 0 0 0 0 17 7 6 3 0 0 0 0 0 0 5 4 Model has been successfully processed
Modelling Evil Case 1:
data; param ROWS := 3; param COLS := 3; param ZBLS := 7; param Iz: 1 2 3 := 1 -1 4 -1 2 . 7 . 3 1 . . ; end;
Produces:
>glpsol --minisat --math Hidato.mathprog --data Evil1.data GLPSOL: GLPK LP/MIP Solver, v4.47 Parameter(s) specified in the command line: --minisat --math Hidato.mathprog --data Evil1.data Reading model section from Hidato.mathprog... Hidato.mathprog:47: warning: data section ignored 47 lines were read Reading data section from Evil1.data... 11 lines were read Generating void0... Generating void1... Generating void2... Generating void3... Generating void4... Generating void5... Generating void6... Generating void7... Generating Izfree... Generating Iz1... Generating rule1... Generating rule2... Generating rule3... Model has been successfully generated Will search for ANY feasible solution Translating to CNF-SAT... Original problem has 256 rows, 200 columns, and 935 non-zeros 56 covering inequalities 193 partitioning equalities Solving CNF-SAT problem... Instance has 337 variables, 1237 clauses, and 4094 literals ==================================[MINISAT]=================================== | Conflicts | ORIGINAL | LEARNT | Progress | | | Clauses Literals | Limit Clauses Literals Lit/Cl | | ============================================================================== | 0 | 1060 3917 | 353 0 0 0.0 | 0.000 % | ============================================================================== SATISFIABLE Objective value = 0.000000000e+000 Time used: 0.0 secs Memory used: 0.8 Mb (861188 bytes) 0 4 0 3 7 5 1 2 6 Model has been successfully processed
Modelling Evil Case 2 - The Snake in the Grass:
data; param ROWS := 3; param COLS := 50; param ZBLS := 74; param Iz: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 := 1 1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 74 2 -1 -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 . -1 3 -1 -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 -1 . . -1 ; end;
Produces:
G:\IAJAAR4.47>glpsol --minisat --math Hidato.mathprog --data Evil2.data GLPSOL: GLPK LP/MIP Solver, v4.47 Parameter(s) specified in the command line: --minisat --math Hidato.mathprog --data Evil2.data Reading model section from Hidato.mathprog... Hidato.mathprog:47: warning: data section ignored 47 lines were read Reading data section from Evil2.data... Evil2.data:11: warning: final NL missing before end of file 11 lines were read Generating void0... Generating void1... Generating void2... Generating void3... Generating void4... Generating void5... Generating void6... Generating void7... Generating Izfree... Generating Iz1... Generating rule1... Generating rule2... Generating rule3... Model has been successfully generated Will search for ANY feasible solution Translating to CNF-SAT... Original problem has 25500 rows, 19500 columns, and 147452 non-zeros 11026 covering inequalities 14400 partitioning equalities Solving CNF-SAT problem... Instance has 31338 variables, 98310 clauses, and 305726 literals ==================================[MINISAT]=================================== | Conflicts | ORIGINAL | LEARNT | Progress | | | Clauses Literals | Limit Clauses Literals Lit/Cl | | ============================================================================== | 0 | 84134 291550 | 28044 0 0 0.0 | 0.000 % | | 101 | 31135 126809 | 30848 98 5496 56.1 | 65.521 % | | 251 | 31135 126809 | 33933 244 12470 51.1 | 66.552 % | | 476 | 27353 115512 | 37327 446 23819 53.4 | 68.160 % | | 814 | 26574 113330 | 41059 770 42161 54.8 | 69.586 % | | 1321 | 25432 110534 | 45165 1262 83658 66.3 | 70.056 % | ============================================================================== SATISFIABLE Objective value = 0.000000000e+000 Time used: 1.0 secs Memory used: 60.9 Mb (63862624 bytes) 1 2 3 0 0 8 9 0 0 14 15 0 0 20 21 0 0 26 27 0 0 32 33 0 0 38 39 0 0 44 45 0 0 50 51 0 0 56 57 0 0 62 63 0 0 68 69 0 0 74 0 0 4 0 7 0 10 0 13 0 16 0 19 0 22 0 25 0 28 0 31 0 34 0 37 0 40 0 43 0 46 0 49 0 52 0 55 0 58 0 61 0 64 0 67 0 70 0 73 0 0 0 0 5 6 0 0 11 12 0 0 17 18 0 0 23 24 0 0 29 30 0 0 35 36 0 0 41 42 0 0 47 48 0 0 53 54 0 0 59 60 0 0 65 66 0 0 71 72 0 Model has been successfully processed
Modelling Evil Case 3 - A fatter snake in the Grass:
data; param ROWS := 4; param COLS := 46; param ZBLS := 82; param Iz: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 := 1 1 0 -1 -1 -1 0 0 -1 -1 -1 0 0 -1 -1 -1 0 0 -1 -1 -1 0 0 -1 -1 -1 0 0 -1 -1 -1 0 0 -1 -1 -1 0 0 -1 -1 -1 0 0 -1 -1 -1 82 2 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 3 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 0 -1 0 -1 -1 4 0 0 0 -1 -1 0 0 0 -1 -1 0 0 0 -1 -1 0 0 0 -1 -1 0 0 0 -1 -1 0 0 0 -1 -1 0 0 0 -1 -1 0 0 0 -1 -1 0 0 0 -1 -1 -1 ; end;
Produces:
>glpsol --minisat --math Hidato.mathprog --data Evil3.data GLPSOL: GLPK LP/MIP Solver, v4.47 Parameter(s) specified in the command line: --minisat --math Hidato.mathprog --data Evil3.data Reading model section from Hidato.mathprog... Hidato.mathprog:47: warning: data section ignored 47 lines were read Reading data section from Evil3.data... 12 lines were read Generating void0... Generating void1... Generating void2... Generating void3... Generating void4... Generating void5... Generating void6... Generating void7... Generating Izfree... Generating Iz1... Generating rule1... Generating rule2... Generating rule3... Model has been successfully generated Will search for ANY feasible solution Translating to CNF-SAT... Original problem has 32684 rows, 23904 columns, and 198488 non-zeros 15006 covering inequalities 17596 partitioning equalities Solving CNF-SAT problem... Instance has 39792 variables, 130040 clauses, and 407222 literals ==================================[MINISAT]=================================== | Conflicts | ORIGINAL | LEARNT | Progress | | | Clauses Literals | Limit Clauses Literals Lit/Cl | | ============================================================================== | 0 | 112710 389892 | 37570 0 0 0.0 | 0.000 % | ============================================================================== SATISFIABLE Objective value = 0.000000000e+000 Time used: 0.0 secs Memory used: 80.2 Mb (84067912 bytes) 1 2 0 0 0 10 11 0 0 0 19 20 0 0 0 28 29 0 0 0 37 38 0 0 0 46 47 0 0 0 55 56 0 0 0 64 65 0 0 0 73 74 0 0 0 82 0 0 3 0 9 0 0 12 0 18 0 0 21 0 27 0 0 30 0 36 0 0 39 0 45 0 0 48 0 54 0 0 57 0 63 0 0 66 0 72 0 0 75 0 81 0 0 4 0 8 0 0 13 0 17 0 0 22 0 26 0 0 31 0 35 0 0 40 0 44 0 0 49 0 53 0 0 58 0 62 0 0 67 0 71 0 0 76 0 80 0 0 5 6 7 0 0 14 15 16 0 0 23 24 25 0 0 32 33 34 0 0 41 42 43 0 0 50 51 52 0 0 59 60 61 0 0 68 69 70 0 0 77 78 79 0 0 0 Model has been successfully processed
Nimrod
<lang nimrod>import strutils, algorithm
var board: array[0..19, array[0..19, int]] var given, start: seq[int] = @[] var rows, cols: int = 0
proc setup(s: string) =
var lines = s.splitLines() cols = lines[0].split().len() rows = lines.len()
for i in 0 .. rows + 1:
for j in 0 .. cols + 1:
board[i][j] = -1
for r, row in pairs(lines):
for c, cell in pairs(row.split()):
case cell
of "__" :
board[r + 1][c + 1] = 0
continue
of "." : continue
else :
var val = parseInt(cell)
board[r + 1][c + 1] = val
given.add(val)
if (val == 1):
start.add(r + 1)
start.add(c + 1)
given.sort(cmp[int], Ascending)
proc solve(r, c, n: int, next: int = 0): Bool =
if n > given[high(given)]:
return True
if board[r][c] < 0:
return False
if (board[r][c] > 0 and board[r][c] != n):
return False
if (board[r][c] == 0 and given[next] == n):
return False
var back = board[r][c]
board[r][c] = n
for i in -1 .. 1:
for j in -1 .. 1:
if back == n:
if (solve(r + i, c + j, n + 1, next + 1)): return True
else:
if (solve(r + i, c + j, n + 1, next)): return True
board[r][c] = back
result = False
proc printBoard() =
for r in 0 .. rows + 1:
for cellid,c in pairs(board[r]):
if cellid > cols + 1: break
if c == -1:
write(stdout, " . ")
elif c == 0:
write(stdout, "__ ")
else:
write(stdout, "$# " % align($c,2))
writeln(stdout, "")
var hi: string = """__ 33 35 __ __ . . . __ __ 24 22 __ . . . __ __ __ 21 __ __ . . __ 26 __ 13 40 11 . . 27 __ __ __ 9 __ 1 . . . __ __ 18 __ __ . . . . . __ 7 __ __ . . . . . . 5 __"""
setup(hi) printBoard() echo("") echo("Found:") discard solve(start[0], start[1], 1) printBoard()</lang>
- Output:
. . . . . . . . . . . __ 33 35 __ __ . . . . . __ __ 24 22 __ . . . . . __ __ __ 21 __ __ . . . . __ 26 __ 13 40 11 . . . . 27 __ __ __ 9 __ 1 . . . . . __ __ 18 __ __ . . . . . . . __ 7 __ __ . . . . . . . . 5 __ . . . . . . . . . . . Found: . . . . . . . . . . . 32 33 35 36 37 . . . . . 31 34 24 22 38 . . . . . 30 25 23 21 12 39 . . . . 29 26 20 13 40 11 . . . . 27 28 14 19 9 10 1 . . . . . 15 16 18 8 2 . . . . . . . 17 7 6 3 . . . . . . . . 5 4 . . . . . . . . . . .
Perl
<lang perl>use strict; use List::Util 'max';
our (@grid, @known, $n);
sub show_board { for my $r (@grid) { print map(!defined($_) ? ' ' : $_ ? sprintf("%3d", $_) : ' __' , @$r), "\n" } }
sub parse_board { @grid = map{[map(/^_/ ? 0 : /^\./ ? undef: $_, split ' ')]} split "\n", shift(); for my $y (0 .. $#grid) { for my $x (0 .. $#{$grid[$y]}) { $grid[$y][$x] > 0 and $known[$grid[$y][$x]] = "$y,$x"; } } $n = max(map { max @$_ } @grid); }
sub neighbors { my ($y, $x) = @_; my @out; for ( [-1, -1], [-1, 0], [-1, 1], [ 0, -1], [ 0, 1], [ 1, -1], [ 1, 0], [ 1, 1]) { my $y1 = $y + $_->[0]; my $x1 = $x + $_->[1]; next if $x1 < 0 || $y1 < 0; next unless defined $grid[$y1][$x1]; push @out, "$y1,$x1"; } @out }
sub try_fill { my ($v, $coord) = @_; return 1 if $v > $n;
my ($y, $x) = split ',', $coord; my $old = $grid[$y][$x];
return if $old && $old != $v; return if exists $known[$v] and $known[$v] ne $coord;
$grid[$y][$x] = $v; print "\033[0H"; show_board();
try_fill($v + 1, $_) && return 1 for neighbors($y, $x);
$grid[$y][$x] = $old; return }
parse_board
- ". 4 .
- _ 7 _
- 1 _ _";
- " 1 _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . 74
- . . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _ . _
- . . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _ . . _ _
- ";
"__ 33 35 __ __ .. .. .. . __ __ 24 22 __ .. .. .. . __ __ __ 21 __ __ .. .. . __ 26 __ 13 40 11 .. .. . 27 __ __ __ 9 __ 1 .. . . . __ __ 18 __ __ .. . . .. . . __ 7 __ __ . . .. .. .. . . 5 __ .";
print "\033[2J";
try_fill(1, $known[1]);</lang>
- Output:
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
Perl 6
<lang perl6>my @grid; my @known;
sub show_board() {
my $width = $*max.chars;
for @grid -> $r {
say do for @$r { when Rat { ' ' x $width } when 0 { '_' x $width } default { .fmt("%{$width}d") } }
}
}
sub neighbors($y,$x --> List) {
gather for [-1, -1], [-1, 0], [-1, 1],
[ 0, -1], [ 0, 1], [ 1, -1], [ 1, 0], [ 1, 1]
{
my $y1 = $y + .[0]; my $x1 = $x + .[1]; take [$y1,$x1] if defined @grid[$y1][$x1];
}
}
sub try_fill($v, $coord [$y,$x] --> Bool) {
return True if $v > $*max;
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] = $v; print "\e[0H"; show_board();
for neighbors($y, $x) {
return True if try_fill $v + 1, $_;
}
@grid[$y][$x] = $old; return False;
}
sub hidato($board) {
my $*max = [max] +«$board.comb(/\d+/);
@grid = $board.lines.map: -> $line {
[ $line.words.map: { /^_/ ?? 0 !! /^\./ ?? Rat !! $_ } ]
}
for ^@grid -> $y {
for ^@grid[$y] -> $x { if @grid[$y][$x] -> $v { @known[$v] = [$y,$x]; } }
} print "\e[0H\e[0J"; try_fill 1, @known[1];
}
hidato q:to/END/;
__ 33 35 __ __ .. .. ..
__ __ 24 22 __ .. .. .. __ __ __ 21 __ __ .. .. __ 26 __ 13 40 11 .. .. 27 __ __ __ 9 __ 1 .. .. .. __ __ 18 __ __ .. .. .. .. .. __ 7 __ __ .. .. .. .. .. .. 5 __ END</lang>
PicoLisp
<lang PicoLisp>(load "@lib/simul.l")
(de hidato (Lst)
(let Grid (grid (length (maxi length Lst)) (length Lst))
(mapc
'((G L)
(mapc
'((This Val)
(nond
(Val
(with (: 0 1 1) (con (: 0 1))) # Cut off west
(with (: 0 1 -1) (set (: 0 1))) # east
(with (: 0 -1 1) (con (: 0 -1))) # south
(with (: 0 -1 -1) (set (: 0 -1))) # north
(set This) )
((=T Val) (=: val Val)) ) )
G L ) )
Grid
(apply mapcar (reverse Lst) list) )
(let Todo
(by '((This) (: val)) sort
(mapcan '((Col) (filter '((This) (: val)) Col))
Grid ) )
(let N 1
(with (pop 'Todo)
(recur (N Todo)
(unless (> (inc 'N) (; Todo 1 val))
(find
'((Dir)
(with (Dir This)
(cond
((= N (: val))
(if (cdr Todo) (recurse N @) T) )
((not (: val))
(=: val N)
(or (recurse N Todo) (=: val NIL)) ) ) ) )
(quote
west east south north
((X) (or (south (west X)) (west (south X))))
((X) (or (north (west X)) (west (north X))))
((X) (or (south (east X)) (east (south X))))
((X) (or (north (east X)) (east (north X)))) ) ) ) ) ) ) )
(disp Grid 0
'((This)
(if (: val) (align 3 @) " ") ) ) ) )</lang>
Test: <lang PicoLisp>(hidato
(quote
(T 33 35 T T)
(T T 24 22 T)
(T T T 21 T T)
(T 26 T 13 40 11)
(27 T T T 9 T 1)
(NIL NIL T T 18 T T)
(NIL NIL NIL NIL T 7 T T)
(NIL NIL NIL NIL NIL NIL 5 T) ) )</lang>
Output:
+---+---+---+---+---+---+---+---+
8 | 32 33 35 36 37| | | |
+ + + + + +---+---+---+
7 | 31 34 24 22 38| | | |
+ + + + + +---+---+---+
6 | 30 25 23 21 12 39| | |
+ + + + + + +---+---+
5 | 29 26 20 13 40 11| | |
+ + + + + + +---+---+
4 | 27 28 14 19 9 10 1| |
+---+---+ + + + + +---+
3 | | | 15 16 18 8 2| |
+---+---+---+---+ + + +---+
2 | | | | | 17 7 6 3|
+---+---+---+---+---+---+ + +
1 | | | | | | | 5 4|
+---+---+---+---+---+---+---+---+
a b c d e f g h
Prolog
Works with SWI-Prolog and library(clpfd) written by Markus Triska.
Puzzle solved is from the Wilkipedia page : http://en.wikipedia.org/wiki/Hidato
<lang Prolog>:- use_module(library(clpfd)).
hidato :- init1(Li), % skip first blank line init2(1, 1, 10, Li), my_write(Li).
init1(Li) :-
Li = [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, A, 33, 35, B, C, 0, 0, 0, 0,
0, D, E, 24, 22, F, 0, 0, 0, 0,
0, G, H, I, 21, J, K, 0, 0, 0,
0, L, 26, M, 13, 40, 11, 0, 0, 0,
0, 27, N, O, P, 9, Q, 1, 0, 0,
0, 0, 0, R, S, 18, T, U, 0, 0,
0, 0, 0, 0, 0, V, 7, W, X, 0,
0, 0, 0, 0, 0, 0, 0, 5, Y, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
LV = [ A, 33, 35, B, C, D, E, 24, 22, F, G, H, I, 21, J, K, L, 26, M, 13, 40, 11, 27, N, O, P, 9, Q, 1, R, S, 18, T, U, V, 7, W, X, 5, Y],
LV ins 1..40,
all_distinct(LV).
% give the constraints % Stop before the last line init2(_N, Col, Max_Col, _L) :- Col is Max_Col - 1.
% skip zeros init2(N, Lig, Col, L) :- I is N + Lig * Col, element(I, L, 0), !, V is N+1, ( V > Col -> N1 = 1, Lig1 is Lig + 1; N1 = V, Lig1 = Lig), init2(N1, Lig1, Col, L).
% skip first column
init2(1, Lig, Col, L) :-
!,
init2(2, Lig, Col, L) .
% skip last column init2(Col, Lig, Col, L) :- !, Lig1 is Lig+1, init2(1, Lig1, Col, L).
% V5 V3 V6 % V1 V V2 % V7 V4 V8 % general case init2(N, Lig, Col, L) :- I is N + Lig * Col, element(I, L, V),
I1 is I - 1, I2 is I + 1, I3 is I - Col, I4 is I + Col, I5 is I3 - 1, I6 is I3 + 1, I7 is I4 - 1, I8 is I4 + 1,
maplist(compute_BI(L, V), [I1,I2,I3,I4,I5,I6,I7,I8], VI, BI),
sum(BI, #=, SBI),
( ((V #= 1 #\/ V #= 40) #/\ SBI #= 1) #\/ (V #\= 1 #/\ V #\= 40 #/\ SBI #= 2)) #<==> 1,
labeling([ffc, enum], [V | VI]),
N1 is N+1, init2(N1, Lig, Col, L).
compute_BI(L, V, I, VI, BI) :- element(I, L, VI), VI #= 0 #==> BI #= 0, ( VI #\= 0 #/\ (V - VI #= 1 #\/ VI - V #= 1)) #<==> BI.
% display the result my_write([0, A, B, C, D, E, F, G, H, 0 | T]) :- maplist(my_write_1, [A, B, C, D, E, F, G, H]), nl, my_write(T).
my_write([]).
my_write_1(0) :- write(' ').
my_write_1(X) :- writef('%3r', [X]).</lang>
- Output:
?- hidato.
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
true
Python
<lang python>board = [] given = [] start = None
def setup(s):
global board, given, start lines = s.splitlines() ncols = len(lines[0].split()) nrows = len(lines) board = [[-1] * (ncols + 2) for _ in xrange(nrows + 2)]
for r, row in enumerate(lines):
for c, cell in enumerate(row.split()):
if cell == "__" :
board[r + 1][c + 1] = 0
continue
elif cell == ".":
continue # -1
else:
val = int(cell)
board[r + 1][c + 1] = val
given.append(val)
if val == 1:
start = (r + 1, c + 1)
given.sort()
def solve(r, c, n, next=0):
if n > given[-1]:
return True
if board[r][c] and board[r][c] != n:
return False
if 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
for i in xrange(-1, 2):
for j in xrange(-1, 2):
if solve(r + i, c + j, n + 1, next):
return True
board[r][c] = back
return False
def print_board():
d = {-1: " ", 0: "__"}
bmax = max(max(r) for r in board)
form = "%" + str(len(str(bmax)) + 1) + "s"
for r in board[1:-1]:
print "".join(form % d.get(c, str(c)) for c in r[1:-1])
hi = """\ __ 33 35 __ __ . . . __ __ 24 22 __ . . . __ __ __ 21 __ __ . . __ 26 __ 13 40 11 . . 27 __ __ __ 9 __ 1 .
. . __ __ 18 __ __ . . . . . __ 7 __ __ . . . . . . 5 __"""
setup(hi) print_board() solve(start[0], start[1], 1) print print_board()</lang>
- Output:
__ 33 35 __ __
__ __ 24 22 __
__ __ __ 21 __ __
__ 26 __ 13 40 11
27 __ __ __ 9 __ 1
__ __ 18 __ __
__ 7 __ __
5 __
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
Racket
Algorithm is depth first search for each number, repeating for all numbers in ascending order. It currently runs slowish due to temporary shortcomings in untyped Racket's array indexing, but finished immediately when tested with custom 2d vector library.
<lang Racket>
- lang racket
(require math/array)
- f = not a legal position, #t = blank position
(define board
(array
#[#[#t 33 35 #t #t #f #f #f]
#[#t #t 24 22 #t #f #f #f]
#[#t #t #t 21 #t #t #f #f]
#[#t 26 #t 13 40 11 #f #f]
#[27 #t #t #t 9 #t 1 #f]
#[#f #f #t #t 18 #t #t #f]
#[#f #f #f #f #t 7 #t #t]
#[#f #f #f #f #f #f 5 #t]]))
- filters elements with the predicate, returning the element and its indices
(define (array-indices-of a f)
(for*/list ([i (range 0 (vector-ref (array-shape a) 0))]
[j (range 0 (vector-ref (array-shape a) 1))]
#:when (f (array-ref a (vector i j))))
(list (array-ref a (vector i j)) i j)))
- returns a list, each element is a list of the number followed by i and j indices
- sorted ascending by number
(define (goal-list v) (sort (array-indices-of v number?) (λ (a b) (< (car a) (car b)))))
- every direction + start position that's on the board
(define (legal-moves a i0 j0)
(for*/list ([i (range (sub1 i0) (+ i0 2))]
[j (range (sub1 j0) (+ j0 2))]
;cartesian product -1..1 and -1..1, except 0 0
#:when (and (not (and (= i i0) (= j j0)))
;make sure it's on the board
(<= 0 i (sub1 (vector-ref (array-shape a) 0)))
(<= 0 j (sub1 (vector-ref (array-shape a) 1)))
;make sure it's an actual position too (the real board isn't square)
(array-ref a (vector i j))))
(cons i j)))
- find path through array, returning list of coords from start to finish
(define (hidato-path a)
;get starting position as first goal
(match-let ([(cons (list n i j) goals) (goal-list a)])
(let hidato ([goals goals] [n n] [i i] [j j] [path '()])
(match goals
;no more goals, return path
['() (reverse (cons (cons i j) path))]
;get next goal
[(cons (list n-goal i-goal j-goal) _)
(let ([move (cons i j)])
;already visiting a spot or taking too many moves to reach the next goal is no good
(cond [(or (member move path) (> n n-goal)) #f]
;taking the right number of moves to be at the goal square is good
;so go to the next goal
[(and (= n n-goal) (= i i-goal) (= j j-goal))
(hidato (cdr goals) n i j path)]
;depth first search using every legal move to find next goal
[else (ormap (λ (m) (hidato goals (add1 n) (car m) (cdr m) (cons move path)))
(legal-moves a i j))]))]))))
- take a path and insert it into the array
(define (put-path a path)
(let ([a (array->mutable-array a)])
(for ([n (range 1 (add1 (length path)))] [move path])
(array-set! a (vector (car move) (cdr move)) n))
a))
- main function
(define (hidato board) (put-path board (hidato-path board))) </lang>
- Output:
> (hidato board) (mutable-array #[#[32 33 35 36 37 #f #f #f] #[31 34 24 22 38 #f #f #f] #[30 25 23 21 12 39 #f #f] #[29 26 20 13 40 11 #f #f] #[27 28 14 19 9 10 1 #f] #[#f #f 15 16 18 8 2 #f] #[#f #f #f #f 17 7 6 3] #[#f #f #f #f #f #f 5 4]])
Ruby
Without Warnsdorff
The following class provides functionality for solving a hidato problem: <lang ruby>
- Solve a Hidato Puzzle
- Nigel_Galloway
- May 5th., 2012.
class Cell
def initialize(row=0, col=0, value=nil)
@adj = [[row-1,col-1],[row,col-1],[row+1,col-1],[row-1,col],[row+1,col],[row-1,col+1],[row,col+1],[row+1,col+1]]
@t = false
$zbl[value] = false unless value == nil
@value = value
end
def try(value)
return false if @value == nil
return true if value > E
return false if @t
return false if @value > 0 and @value != value
return false if @value == 0 and not $zbl[value]
@t = true
@adj.each{|x|
if (Board[x[0]][x[1]].try(value+1)) then
@value = value
return true
end
}
@t = false
return false
end
def value
return @value
end
end </lang> Which may be used as follows to solve Evil Case 1: <lang ruby> Rows = 3 Cols = 3 E = 7 $zbl = Array.new(E+1,true) Board = [[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new(1,2,4) ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new(2,1,0) ,Cell.new(2,2,7) ,Cell.new(2,3,0) ,Cell.new()],
[Cell.new(),Cell.new(3,1,1) ,Cell.new(3,2,0) ,Cell.new(3,3,0) ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()]]
require 'benchmark' puts Benchmark.measure {Board[3][1].try(1)} (1..Rows).each{|r|
(1..Cols).each{|c| printf("%2s",Board[r][c].value)}
puts ""}
</lang> Which produces:
>ruby Program.rb 0.000000 0.000000 0.000000 ( 0.000000) 4 3 7 5 1 2 6
Which may be used as follows to solve this tasks example: <lang ruby> Rows = 8 Cols = 8 E = 40 $zbl = Array.new(E+1,true) Board = [[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new(1,1,0) ,Cell.new(1,2,33) ,Cell.new(1,3,35) ,Cell.new(1,4,0) ,Cell.new(1,5,0) ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new(2,1,0) ,Cell.new(2,2,0) ,Cell.new(2,3,24) ,Cell.new(2,4,22) ,Cell.new(2,5,0) ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new(3,1,0) ,Cell.new(3,2,0) ,Cell.new(3,3,0) ,Cell.new(3,4,21) ,Cell.new(3,5,0) ,Cell.new(3,6,0) ,Cell.new() ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new(4,1,0) ,Cell.new(4,2,26) ,Cell.new(4,3,0) ,Cell.new(4,4,13) ,Cell.new(4,5,40) ,Cell.new(4,6,11) ,Cell.new() ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new(5,1,27) ,Cell.new(5,2,0) ,Cell.new(5,3,0) ,Cell.new(5,4,0) ,Cell.new(5,5,9) ,Cell.new(5,6,0) ,Cell.new(5,7,1) ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new(6,3,0) ,Cell.new(6,4,0) ,Cell.new(6,5,18) ,Cell.new(6,6,0) ,Cell.new(6,7,0) ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new(7,5,0) ,Cell.new(7,6,7) ,Cell.new(7,7,0) ,Cell.new(7,8,0) ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new(8,7,5) ,Cell.new(8,8,0) ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()]]
require 'benchmark' puts Benchmark.measure {Board[5][7].try(1)} (1..Rows).each{|r|
(1..Cols).each{|c| printf("%3s",Board[r][c].value)}
puts ""}
</lang>
>ruby Program.rb
0.000000 0.000000 0.000000 ( 0.000000)
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
Which may be used as follows to solve The Snake in the Grass: <lang ruby> Rows = 3 Cols = 50 E = 74 $zbl = Array.new(E+1,true) Board = [[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new(1,1,1) ,Cell.new(1,2,0) ,Cell.new(1,3,0) ,Cell.new() ,Cell.new() ,Cell.new(1,6,0) ,Cell.new(1,7,0) ,Cell.new() ,Cell.new() ,Cell.new(1,10,0) ,Cell.new(1,11,0) ,Cell.new() ,Cell.new() ,Cell.new(1,14,0) ,Cell.new(1,15,0) ,Cell.new() ,Cell.new() ,Cell.new(1,18,0) ,Cell.new(1,19,0) ,Cell.new() ,Cell.new() ,Cell.new(1,22,0) ,Cell.new(1,23,0) ,Cell.new() ,Cell.new() ,Cell.new(1,26,0) ,Cell.new(1,27,0) ,Cell.new() ,Cell.new() ,Cell.new(1,30,0) ,Cell.new(1,31,0) ,Cell.new() ,Cell.new() ,Cell.new(1,34,0) ,Cell.new(1,35,0) ,Cell.new() ,Cell.new() ,Cell.new(1,38,0) ,Cell.new(1,39,0) ,Cell.new() ,Cell.new() ,Cell.new(1,42,0) ,Cell.new(1,43,0) ,Cell.new() ,Cell.new() ,Cell.new(1,46,0) ,Cell.new(1,47,0) ,Cell.new() ,Cell.new() ,Cell.new(1,50,74),Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new(2,3,0) ,Cell.new() ,Cell.new(2,5,0) ,Cell.new() ,Cell.new(2,7,0) ,Cell.new() ,Cell.new(2,9,0) ,Cell.new() ,Cell.new(2,11,0) ,Cell.new() ,Cell.new(2,13,0) ,Cell.new() ,Cell.new(2,15,0) ,Cell.new() ,Cell.new(2,17,0) ,Cell.new() ,Cell.new(2,19,0) ,Cell.new() ,Cell.new(2,21,0) ,Cell.new() ,Cell.new(2,23,0) ,Cell.new() ,Cell.new(2,25,0) ,Cell.new() ,Cell.new(2,27,0) ,Cell.new() ,Cell.new(2,29,0) ,Cell.new() ,Cell.new(2,31,0) ,Cell.new() ,Cell.new(2,33,0) ,Cell.new() ,Cell.new(2,35,0) ,Cell.new() ,Cell.new(2,37,0) ,Cell.new() ,Cell.new(2,39,0) ,Cell.new() ,Cell.new(2,41,0) ,Cell.new() ,Cell.new(2,43,0) ,Cell.new() ,Cell.new(2,45,0) ,Cell.new() ,Cell.new(2,47,0) ,Cell.new() ,Cell.new(2,49,0) ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new(3,4,0) ,Cell.new(3,5,0) ,Cell.new() ,Cell.new() ,Cell.new(3,8,0) ,Cell.new(3,9,0) ,Cell.new() ,Cell.new() ,Cell.new(3,12,0) ,Cell.new(3,13,0) ,Cell.new() ,Cell.new() ,Cell.new(3,16,0) ,Cell.new(3,17,0) ,Cell.new() ,Cell.new() ,Cell.new(3,20,0) ,Cell.new(3,21,0) ,Cell.new() ,Cell.new() ,Cell.new(3,24,0) ,Cell.new(3,25,0) ,Cell.new() ,Cell.new() ,Cell.new(3,28,0) ,Cell.new(3,29,0) ,Cell.new() ,Cell.new() ,Cell.new(3,32,0) ,Cell.new(3,33,0) ,Cell.new() ,Cell.new() ,Cell.new(3,36,0) ,Cell.new(3,37,0) ,Cell.new() ,Cell.new() ,Cell.new(3,40,0) ,Cell.new(3,41,0) ,Cell.new() ,Cell.new() ,Cell.new(3,44,0) ,Cell.new(3,45,0) ,Cell.new() ,Cell.new() ,Cell.new(3,48,0) ,Cell.new(3,49,0) ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()]]
require 'benchmark' puts Benchmark.measure {Board[1][1].try(1)} (1..Rows).each{|r|
(1..Cols).each{|c| printf("%3s",Board[r][c].value)}
puts ""}
</lang> Which produces:
>ruby Program.rb
87.626000 0.000000 87.626000 ( 87.640800)
1 2 3 8 9 14 15 20 21 26 27 32 33 38 39 44 45 50 51 56 57 62 63 68 69 74
4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73
5 6 11 12 17 18 23 24 29 30 35 36 41 42 47 48 53 54 59 60 65 66 71 72
Note that if The Snake in the Grass is modified to look more like a genuine Hidato by shortening the path between hints: <lang ruby> Rows = 3 Cols = 50 E = 74 $zbl = Array.new(E+1,true) Board = [[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new(1,1,1) ,Cell.new(1,2,0) ,Cell.new(1,3,0) ,Cell.new() ,Cell.new() ,Cell.new(1,6,0) ,Cell.new(1,7,0) ,Cell.new() ,Cell.new() ,Cell.new(1,10,0) ,Cell.new(1,11,0) ,Cell.new() ,Cell.new() ,Cell.new(1,14,0) ,Cell.new(1,15,0) ,Cell.new() ,Cell.new() ,Cell.new(1,18,0) ,Cell.new(1,19,0) ,Cell.new() ,Cell.new() ,Cell.new(1,22,0) ,Cell.new(1,23,0) ,Cell.new() ,Cell.new() ,Cell.new(1,26,0) ,Cell.new(1,27,0) ,Cell.new() ,Cell.new() ,Cell.new(1,30,0) ,Cell.new(1,31,0) ,Cell.new() ,Cell.new() ,Cell.new(1,34,0) ,Cell.new(1,35,0) ,Cell.new() ,Cell.new() ,Cell.new(1,38,0) ,Cell.new(1,39,0) ,Cell.new() ,Cell.new() ,Cell.new(1,42,0) ,Cell.new(1,43,0) ,Cell.new() ,Cell.new() ,Cell.new(1,46,0) ,Cell.new(1,47,0) ,Cell.new() ,Cell.new() ,Cell.new(1,50,74),Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new(2,3,0) ,Cell.new() ,Cell.new(2,5,0) ,Cell.new() ,Cell.new(2,7,0) ,Cell.new() ,Cell.new(2,9,0) ,Cell.new() ,Cell.new(2,11,0) ,Cell.new() ,Cell.new(2,13,0) ,Cell.new() ,Cell.new(2,15,0) ,Cell.new() ,Cell.new(2,17,0) ,Cell.new() ,Cell.new(2,19,0) ,Cell.new() ,Cell.new(2,21,0) ,Cell.new() ,Cell.new(2,23,0) ,Cell.new() ,Cell.new(2,25,0) ,Cell.new() ,Cell.new(2,27,0) ,Cell.new() ,Cell.new(2,29,0) ,Cell.new() ,Cell.new(2,31,0) ,Cell.new() ,Cell.new(2,33,0) ,Cell.new() ,Cell.new(2,35,0) ,Cell.new() ,Cell.new(2,37,0) ,Cell.new() ,Cell.new(2,39,0) ,Cell.new() ,Cell.new(2,41,0) ,Cell.new() ,Cell.new(2,43,0) ,Cell.new() ,Cell.new(2,45,0) ,Cell.new() ,Cell.new(2,47,0) ,Cell.new() ,Cell.new(2,49,0) ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new(3,4,0) ,Cell.new(3,5,0) ,Cell.new() ,Cell.new() ,Cell.new(3,8,11) ,Cell.new(3,9,0) ,Cell.new() ,Cell.new() ,Cell.new(3,12,0) ,Cell.new(3,13,0) ,Cell.new() ,Cell.new() ,Cell.new(3,16,23) ,Cell.new(3,17,0) ,Cell.new() ,Cell.new() ,Cell.new(3,20,0) ,Cell.new(3,21,0) ,Cell.new() ,Cell.new() ,Cell.new(3,24,35) ,Cell.new(3,25,0) ,Cell.new() ,Cell.new() ,Cell.new(3,28,0) ,Cell.new(3,29,0) ,Cell.new() ,Cell.new() ,Cell.new(3,32,47) ,Cell.new(3,33,0) ,Cell.new() ,Cell.new() ,Cell.new(3,36,0) ,Cell.new(3,37,0) ,Cell.new() ,Cell.new() ,Cell.new(3,40,0) ,Cell.new(3,41,0) ,Cell.new() ,Cell.new() ,Cell.new(3,44,65) ,Cell.new(3,45,0) ,Cell.new() ,Cell.new() ,Cell.new(3,48,0) ,Cell.new(3,49,0) ,Cell.new() ,Cell.new()],
[Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new()]]
</lang> Produces the following in 0 secs instead of 87.62
>ruby Program.rb
0.000000 0.000000 0.000000 ( 0.000000)
1 2 3 8 9 14 15 20 21 26 27 32 33 38 39 44 45 50 51 56 57 62 63 68 69 74
4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73
5 6 11 12 17 18 23 24 29 30 35 36 41 42 47 48 53 54 59 60 65 66 71 72
header|With Warnsdorff
I modify Cell as follows to implement Warnsdorff <lang ruby>
- Solve a Hidato Puzzle with Warnsdorff like logic applied
- Nigel_Galloway
- May 6th., 2012.
class Cell
def initialize(row=0, col=0, value=nil)
@adj = [[row-1,col-1],[row,col-1],[row+1,col-1],[row-1,col],[row+1,col],[row-1,col+1],[row,col+1],[row+1,col+1]]
@t = false
$zbl[value] = false unless value == nil
@value = value
end
def try(value)
return false if @value == nil
return true if value > E
return false if @t
return false if @value > 0 and @value != value
return false if @value == 0 and not $zbl[value]
@t = true
h = Hash.new
n = 0
@adj.each{|x|
h[Board[x[0]][x[1]].wdof*10+n] = Board[x[0]][x[1]]
n += 1
}
h.sort.each{|key,cell|
if (cell.try(value+1)) then
@value = value
return true
end
}
@t = false
return false
end
def wdon
return 0 if @value == nil
return 0 if @value > 0
return 0 if @t
return 1
end
def wdof
res = 0
@adj.each{|x| res += Board[x[0]][x[1]].wdon}
return res
end
def value
return @value
end
end </lang> The usage remains unchanged from above and produces:
>ruby Program.rb
0.000000 0.000000 0.000000 ( 0.000000)
4
3 7 5
1 2 6
>ruby Program.rb
0.046000 0.000000 0.046000 ( 0.046800)
32 33 35 36 37
31 34 24 22 38
30 25 23 21 12 39
29 26 20 13 40 11
27 28 14 19 9 10 1
15 16 18 8 2
17 7 6 3
5 4
>ruby Program.rb
0.016000 0.000000 0.016000 ( 0.015600)
1 2 3 8 9 14 15 20 21 26 27 32 33 38 39 44 45 50 51 56 57 62 63 68 69 74
4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73
5 6 11 12 17 18 23 24 29 30 35 36 41 42 47 48 53 54 59 60 65 66 71 72
This cell may be used to solve Knight's Tour: <lang ruby> class Cell
def initialize(row=0, col=0, value=nil)
@adj = [[row-1,col-2],[row-2,col-1],[row-2,col+1],[row-1,col+2],[row+1,col+2],[row+2,col+1],[row+2,col-1],[row+1,col-2]]
@t = false
$zbl[value] = false unless value == nil
@value = value
end
def try(value)
return false if @value == nil
return true if value > E
return false if @t
return false if @value > 0 and @value != value
return false if @value == 0 and not $zbl[value]
@t = true
h = Hash.new
n = 0
@adj.each{|x|
h[Board[x[0]][x[1]].wdof*10+n] = Board[x[0]][x[1]]
n += 1
}
h.sort.each{|key,cell|
if (cell.try(value+1)) then
@value = value
return true
end
}
@t = false
return false
end
def wdon
return 0 if @value == nil
return 0 if @value > 0
return 0 if @t
return 1
end
def wdof
res = 0
@adj.each{|x| res += Board[x[0]][x[1]].wdon}
return res
end
def value
return @value
end
end </lang> Where only line 3 @adj= <the list> is changed. A possible test for this is: <lang ruby> Rows = 9 Cols = 9 E = 64 $zbl = Array.new(E+1,true) Board = [[Cell.new(),Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new(2,2,0),Cell.new(2,3,0),Cell.new(2,4,0),Cell.new(2,5,0),Cell.new(2,6,0),Cell.new(2,7,0),Cell.new(2,8,0),Cell.new(2,9,0) ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new(3,2,0),Cell.new(3,3,0),Cell.new(3,4,0),Cell.new(3,5,0),Cell.new(3,6,0),Cell.new(3,7,0),Cell.new(3,8,0),Cell.new(3,9,0) ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new(4,2,0),Cell.new(4,3,0),Cell.new(4,4,0),Cell.new(4,5,0),Cell.new(4,6,0),Cell.new(4,7,0),Cell.new(4,8,0),Cell.new(4,9,0) ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new(5,2,0),Cell.new(5,3,1),Cell.new(5,4,0),Cell.new(5,5,0),Cell.new(5,6,0),Cell.new(5,7,0),Cell.new(5,8,0),Cell.new(5,9,0) ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new(6,2,0),Cell.new(6,3,0),Cell.new(6,4,0),Cell.new(6,5,0),Cell.new(6,6,0),Cell.new(6,7,0),Cell.new(6,8,0),Cell.new(6,9,0) ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new(7,2,0),Cell.new(7,3,0),Cell.new(7,4,0),Cell.new(7,5,0),Cell.new(7,6,0),Cell.new(7,7,0),Cell.new(7,8,0),Cell.new(7,9,0) ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new(8,2,0),Cell.new(8,3,0),Cell.new(8,4,0),Cell.new(8,5,0),Cell.new(8,6,0),Cell.new(8,7,0),Cell.new(8,8,0),Cell.new(8,9,0) ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new(9,2,0),Cell.new(9,3,0),Cell.new(9,4,0),Cell.new(9,5,0),Cell.new(9,6,0),Cell.new(9,7,0),Cell.new(9,8,0),Cell.new(9,9,0),Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new(),Cell.new()],
[Cell.new(),Cell.new(),Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new() ,Cell.new(),Cell.new()]]
require 'benchmark' puts Benchmark.measure {Board[5][3].try(1)} (1..Rows).each{|r|
(1..Cols).each{|c| printf("%3s",Board[r][c].value)}
puts ""}
</lang> Which produces:
>ruby Program.rb
0.000000 0.000000 0.000000 ( 0.000000)
23 20 3 32 25 10 5 8
2 35 24 21 4 7 26 11
19 22 33 36 31 28 9 6
34 1 50 29 48 37 12 27
51 18 53 44 61 30 47 38
54 43 56 49 58 45 62 13
17 52 41 60 15 64 39 46
42 55 16 57 40 59 14 63
Tcl
<lang tcl>proc init {initialConfiguration} {
global grid max filled
set max 1
set y 0
foreach row [split [string trim $initialConfiguration "\n"] "\n"] {
set x 0 set rowcontents {} foreach cell $row { if {![string is integer -strict $cell]} {set cell -1} lappend rowcontents $cell set max [expr {max($max, $cell)}] if {$cell > 0} { dict set filled $cell [list $y $x] } incr x } lappend grid $rowcontents incr y
}
}
proc findseps {} {
global max filled
set result {}
for {set i 1} {$i < $max-1} {incr i} {
if {[dict exists $filled $i]} { for {set j [expr {$i+1}]} {$j <= $max} {incr j} { if {[dict exists $filled $j]} { if {$j-$i > 1} { lappend result [list $i $j [expr {$j-$i}]] } break } } }
} return [lsort -integer -index 2 $result]
}
proc makepaths {sep} {
global grid filled
lassign $sep from to len
lassign [dict get $filled $from] y x
set result {}
foreach {dx dy} {-1 -1 -1 0 -1 1 0 -1 0 1 1 -1 1 0 1 1} {
discover [expr {$x+$dx}] [expr {$y+$dy}] [expr {$from+1}] $to \ [list [list $from $x $y]] $grid
} return $result
} proc discover {x y n limit path model} {
global filled
# Check for illegal
if {[lindex $model $y $x] != 0} return
upvar 1 result result
lassign [dict get $filled $limit] ly lx
# Special case
if {$n == $limit-1} {
if {abs($x-$lx)<=1 && abs($y-$ly)<=1 && !($lx==$x && $ly==$y)} { lappend result [lappend path [list $n $x $y] [list $limit $lx $ly]] } return
}
# Check for impossible
if {abs($x-$lx) > $limit-$n || abs($y-$ly) > $limit-$n} return
# Recursive search
lappend path [list $n $x $y]
lset model $y $x $n
incr n
foreach {dx dy} {-1 -1 -1 0 -1 1 0 -1 0 1 1 -1 1 0 1 1} {
discover [expr {$x+$dx}] [expr {$y+$dy}] $n $limit $path $model
}
}
proc applypath {path} {
global grid filled
puts "Found unique path for [lindex $path 0 0] -> [lindex $path end 0]"
foreach cell [lrange $path 1 end-1] {
lassign $cell n x y lset grid $y $x $n dict set filled $n [list $y $x]
}
}
proc printgrid {} {
global grid max
foreach row $grid {
foreach cell $row { puts -nonewline [format " %*s" [string length $max] [expr { $cell==-1 ? "." : $cell }]] } puts ""
}
}
proc solveHidato {initialConfiguration} {
init $initialConfiguration
set limit [llength [findseps]]
while {[llength [set seps [findseps]]] && [incr limit -1]>=0} {
foreach sep $seps { if {[llength [set paths [makepaths $sep]]] == 1} { applypath [lindex $paths 0] break } }
} puts "" printgrid
}</lang> Demonstrating (dots are “outside” the grid, and zeroes are the cells to be filled in): <lang tcl>solveHidato "
0 33 35 0 0 . . .
0 0 24 22 0 . . .
0 0 0 21 0 0 . .
0 26 0 13 40 11 . .
27 0 0 0 9 0 1 .
. . 0 0 18 0 0 .
. . . . 0 7 0 0
. . . . . . 5 0
"</lang>
- Output:
Found unique path for 5 -> 7 Found unique path for 7 -> 9 Found unique path for 9 -> 11 Found unique path for 11 -> 13 Found unique path for 33 -> 35 Found unique path for 18 -> 21 Found unique path for 1 -> 5 Found unique path for 35 -> 40 Found unique path for 22 -> 24 Found unique path for 24 -> 26 Found unique path for 27 -> 33 Found unique path for 13 -> 18 32 33 35 36 37 . . . 31 34 24 22 38 . . . 30 25 23 21 12 39 . . 29 26 20 13 40 11 . . 27 28 14 19 9 10 1 . . . 15 16 18 8 2 . . . . . 17 7 6 3 . . . . . . 5 4
More complex cases are solvable with an extended version of this code, though that has more onerous version requirements.