Knight's tour: Difference between revisions
m (→{{header|Ada}}) |
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Ada.Text_IO.Put_Line("No Solution"); |
Ada.Text_IO.Put_Line("No Solution"); |
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end if; |
end if; |
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end Tour_IO; |
end Tour_IO; |
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end Knights_Tour;</lang> |
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Here is the main program: |
Here is the main program: |
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Revision as of 00:05, 1 March 2012
You are encouraged to solve this task according to the task description, using any language you may know.
Problem: you have a standard 8x8 chessboard, empty but for a single knight on some square. Your task is to emit a series of legal knight moves that result in the knight visiting every square on the chessboard exactly once. Note that it is not a requirement that the knight should end within a knight's move of its start position, (this would have then been a closed tour).
Input and output may be textual or graphical, according to the conventions of the programming environment. If textual, squares should be indicated in Algebraic Notation. The output should indicate the order in which the knight visits the squares, starting with the initial position. The form of the output may be a diagram of the board with the squares numbered ordering to visitation sequence, or a textual list of algebraic coordinates in order, or even an actual animation of the knight moving around the chessboard.
Input: starting square
Output: move sequence
- Cf.
Ada
First, we specify a naive implementation with naive backtracking:
<lang Ada>generic
Size: Integer;
package Knights_Tour is
subtype Index is Integer range 1 .. Size; type Tour is array (Index, Index) of Natural;
function Get_Tour(Start_X, Start_Y: Index) return Tour; -- finds tour via backtracking -- either no tour has been found, (Get_Tour(X, Y)=0 for all X, Y in Index) -- or the Result(X,Y)=K if and only if I,J is visited at the K-th move
procedure Tour_IO(The_Tour: Tour); -- writes The_Tour to the output using Ada.Text_IO;
end Knights_Tour;</lang>
Here is the implementation:
<lang Ada>with Ada.Text_IO, Ada.Integer_Text_IO;
package body Knights_Tour is
Visited: Tour; -- Visited(I, J)=0: Not yet visited -- Visited(I, J)=K: Visited during the k-th move
type Pair is array(1..2) of Integer; type Pair_Array is array (Positive range <>) of Pair;
Pairs: constant Pair_Array (1..8) := ((-2,1),(-1,2),(1,2),(2,1),(2,-1),(1,-2),(-1,-2),(-2,-1)); -- possible places to visit if Done
function Get_Tour(Start_X, Start_Y: Index) return Tour is
procedure Visit(X, Y: Index; Move_Number: Positive; Found: out Boolean) is
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move_Number;
if Move_Number = Integer(Index'Last * Index'Last) then
Found := True;
else
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Visit(XX, YY, Move_Number+1, Found); -- recursion
if Found then
return; -- no need to search further
end if;
end if;
end loop;
Visited(X, Y) := 0; -- undo previous mark
end if;
end Visit;
Done: Boolean;
begin
Visited := (others => (others => 0));
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := (others => (others => 0));
end if;
return Visited;
end Get_Tour;
procedure Tour_IO(The_Tour: Tour) is
begin
if The_Tour(1, 1) /= 0 then
for I in Index loop
for J in Index loop
Ada.Integer_Text_IO.Put(The_Tour(I, J), 4);
end loop;
Ada.Text_IO.New_Line;
end loop;
else
Ada.Text_IO.Put_Line("No Solution");
end if;
end Tour_IO;
end Knights_Tour;</lang>
Here is the main program:
<lang Ada>with Knights_Tour, Ada.Command_Line;
procedure Test_Knight is
Size: Positive := Positive'Value(Ada.Command_Line.Argument(1));
package KT is new Knights_Tour(Size => Size);
begin
KT.Tour_IO(KT.Get_Tour(1, 1));
end Test_Knight;</lang>
For small sizes, this already works well (< 1 sec for size 8). Sample output:
>./test_knight 8 1 38 55 34 3 36 19 22 54 47 2 37 20 23 4 17 39 56 33 46 35 18 21 10 48 53 40 57 24 11 16 5 59 32 45 52 41 26 9 12 44 49 58 25 62 15 6 27 31 60 51 42 29 8 13 64 50 43 30 61 14 63 28 7
For larger sizes we'll use Warnsdorff's heuristic (without any thoughtful tie breaking). We extend the specification by
<lang Ada> function Warnsdorff_Get_Tour(Start_X, Start_Y: Index) return Tour;
-- uses Warnsdorff heurisitic to find a tour faster -- same interface as Get_Tour</lang>
and the implementation by <lang Ada> function Warnsdorff_Get_Tour(Start_X, Start_Y: Index) return Tour is
Done: Boolean;
function Neighbors(X, Y: Index) return Natural is
Result: Natural := 0;
begin
for P in Pairs'Range loop
if X+Pairs(P)(1) in Index and then Y+Pairs(P)(2) in Index and then
Visited(X+Pairs(P)(1), Y+Pairs(P)(2)) = 0 then
Result := Result + 1;
end if;
end loop;
return Result;
end Neighbors;
procedure Sort(Options: in out Pair_Array) is
N_Bors: array(Options'Range) of Natural;
K: Positive range Options'Range;
N: Natural;
P: Pair;
begin
for Opt in Options'Range loop
N_Bors(Opt) := Neighbors(Options(Opt)(1), Options(Opt)(2));
end loop;
for Opt in Options'Range loop
K := Opt;
for Alternative in Opt+1 .. Options'Last loop
if N_Bors(Alternative) < N_Bors(Opt) then
K := Alternative;
end if;
end loop;
N := N_Bors(Opt);
N_Bors(Opt) := N_Bors(K);
N_Bors(K) := N;
P := Options(Opt);
Options(Opt) := Options(K);
Options(K) := P;
end loop;
end Sort;
procedure Visit(X, Y: Index; Move: Positive; Found: out Boolean) is
Next_Count: Natural range 0 .. 8 := 0;
Next_Steps: Pair_Array(1 .. 8);
XX, YY: Integer;
begin
Found := False;
Visited(X, Y) := Move;
if Move = Integer(Index'Last * Index'Last) then
Found := True;
else
-- consider all possible places to go
for P in Pairs'Range loop
XX := X + Pairs(P)(1);
YY := Y + Pairs(P)(2);
if (XX in Index) and then (YY in Index)
and then Visited(XX, YY) = 0 then
Next_Count := Next_Count+1;
Next_Steps(Next_Count) := (XX, YY);
end if;
end loop;
Sort(Next_Steps(1 .. Next_Count));
for N in 1 .. Next_Count loop
Visit(Next_Steps(N)(1), Next_Steps(N)(2), Move+1, Found);
if Found then
return; -- no need to search further
end if;
end loop;
-- if we didn't return above, we have to undo our move
Visited(X, Y) := 0;
end if;
end Visit;
begin
Visited := (others => (others => 0));
Visit(Start_X, Start_Y, 1, Done);
if not Done then
Visited := (others => (others => 0));
end if;
return Visited;
end Warnsdorff_Get_Tour;</lang>
The modification for the main program is trivial: <lang Ada>with Knights_Tour, Ada.Command_Line;
procedure Test_Fast is
Size: Positive := Positive'Value(Ada.Command_Line.Argument(1));
package KT is new Knights_Tour(Size => Size);
begin
KT.Tour_IO(KT.Warnsdorff_Get_Tour(1, 1));
end Test_Fast;</lang>
This works still well for somewhat larger sizes:
>./test_fast 24 1 108 45 52 3 112 57 60 5 62 131 144 7 64 147 170 9 66 187 192 11 68 71 190 46 51 2 111 56 53 4 113 130 59 6 63 146 169 8 65 186 215 10 67 188 191 12 69 107 44 109 54 123 114 129 58 61 132 145 168 143 148 185 214 171 198 225 216 193 70 189 72 50 47 122 115 110 55 140 133 128 167 142 149 184 213 172 199 226 255 246 197 224 217 194 13 43 106 49 124 139 134 127 166 141 150 183 212 173 200 227 254 247 242 223 256 245 196 73 218 48 121 116 135 126 165 138 151 182 211 174 201 228 253 248 241 290 263 304 243 222 257 14 195 105 42 125 164 137 152 181 210 175 202 229 252 249 240 289 264 329 308 291 262 303 244 219 74 120 117 136 153 180 163 176 203 230 267 250 239 288 265 328 309 334 345 330 305 292 221 258 15 41 104 119 160 177 204 231 268 209 238 287 266 251 310 335 344 357 332 307 346 261 302 75 220 118 159 154 205 162 179 208 237 286 269 324 311 336 327 438 333 418 347 356 331 306 293 16 259 103 40 161 178 207 232 285 270 323 312 337 326 483 416 343 422 437 358 419 298 349 260 301 76 158 155 206 233 284 271 236 313 338 325 482 415 342 439 484 417 420 423 348 355 360 299 294 17 39 102 157 272 235 314 339 322 481 414 341 492 497 514 421 440 485 436 359 424 297 350 77 300 156 273 234 315 276 283 478 413 340 493 480 513 530 491 498 515 452 441 454 435 354 361 18 295 101 38 275 282 397 412 321 494 479 512 557 496 543 534 529 490 499 486 451 442 425 296 351 78 274 279 316 277 320 477 410 511 570 495 554 535 556 531 542 533 516 453 444 455 434 353 362 19 37 100 281 398 411 396 575 476 567 558 561 544 553 536 521 528 489 500 487 450 443 426 79 352 280 317 278 319 402 409 510 569 560 571 566 555 550 541 532 537 522 517 460 445 456 433 20 363 99 36 389 378 399 576 395 574 475 568 559 562 545 552 525 520 527 488 501 462 449 364 427 80 94 379 318 401 388 403 408 509 572 565 474 551 540 549 538 523 518 461 446 459 432 457 366 21 35 98 93 390 377 400 573 394 375 508 563 546 373 524 519 526 371 502 463 466 365 448 81 428 380 95 382 385 404 387 376 407 564 473 374 507 548 539 372 503 464 467 370 447 458 431 22 367 383 34 97 92 391 32 405 90 393 30 547 88 471 28 505 86 469 26 465 84 369 24 429 82 96 381 384 33 386 91 392 31 406 89 472 29 506 87 470 27 504 85 468 25 430 83 368 23
Bracmat
<lang> ( knightsTour
= validmoves WarnsdorffSort algebraicNotation init solve
, x y fieldsToVisit
. ~
| ( validmoves
= x y jumps moves
. !arg:(?x.?y)
& :?moves
& ( jumps
= dx dy Fs fxs fys fx fy
. !arg:(?dx.?dy)
& 1 -1:?Fs
& !Fs:?fxs
& whl
' ( !fxs:%?fx ?fxs
& !Fs:?fys
& whl
' ( !fys:%?fy ?fys
& ( (!x+!fx*!dx.!y+!fy*!dy)
: (>0:<9.>0:<9)
|
)
!moves
: ?moves
)
)
)
& jumps$(1.2)
& jumps$(2.1)
& !moves
)
& ( init
= fields x y
. :?fields
& 0:?x
& whl
' ( 1+!x:<9:?x
& 0:?y
& whl
' ( 1+!y:<9:?y
& (!x.!y) !fields:?fields
)
)
& !fields
)
& init$:?fieldsToVisit
& ( WarnsdorffSort
= sum moves elm weightedTerms
. ( weightedTerms
= pos alts fieldsToVisit moves move weight
. !arg:(%?pos ?alts.?fieldsToVisit)
& ( !fieldsToVisit:!pos
& (0.!pos)
| !fieldsToVisit:? !pos ?
& validmoves$!pos:?moves
& 0:?weight
& whl
' ( !moves:%?move ?moves
& ( !fieldsToVisit:? !move ?
& !weight+1:?weight
|
)
)
& (!weight.!pos)
| 0
)
+ weightedTerms$(!alts.!fieldsToVisit)
| 0
)
& weightedTerms$!arg:?sum
& :?moves
& whl
' ( !sum:(#.?elm)+?sum
& !moves !elm:?moves
)
& !moves
)
& ( solve
= pos alts fieldsToVisit A Z tailOfSolution
. !arg:(%?pos ?alts.?fieldsToVisit)
& ( !fieldsToVisit:?A !pos ?Z
& ( !A !Z:&
| solve
$ ( WarnsdorffSort$(validmoves$!pos.!A !Z)
. !A !Z
)
)
| solve$(!alts.!fieldsToVisit)
)
: ?tailOfSolution
& !pos !tailOfSolution
)
& ( algebraicNotation
= x y
. !arg:(?x.?y) ?arg
& str$(chr$(asc$a+!x+-1) !y " ")
algebraicNotation$!arg
|
)
& @(!arg:?x #?y)
& asc$!x+-1*asc$a+1:?x
& str
$ (algebraicNotation$(solve$((!x.!y).!fieldsToVisit)))
)
& out$(knightsTour$a1);</lang>
<lang>a1 b3 a5 b7 d8 f7 h8 g6 f8 h7 g5 h3 g1 e2 c1 a2 b4 a6 b8 c6 a7 c8 e7 g8 h6 g4 h2 f1 d2 b1 a3 c2 e1 f3 h4 g2 e3 d1 b2 a4 c3 b5 d4 f5 d6 c4 e5 d3 f2 h1 g3 e4 c5 d7 b6 a8 c7 d5 f4 e6 g7 e8 f6 h5 </lang>
C
For an animated version using OpenGL, see Knight's tour/C.
The following draws on console the progress of the horsie. Specify board size on commandline, or use default 8. <lang c>#include <stdio.h>
- include <stdlib.h>
- include <string.h>
- include <unistd.h>
typedef unsigned char cell; int dx[] = { -2, -2, -1, 1, 2, 2, 1, -1 }; int dy[] = { -1, 1, 2, 2, 1, -1, -2, -2 };
void init_board(int w, int h, cell **a, cell **b) { int i, j, k, x, y, p = w + 4, q = h + 4; /* b is board; a is board with 2 rows padded at each side */ a[0] = (cell*)(a + q); b[0] = a[0] + 2;
for (i = 1; i < q; i++) { a[i] = a[i-1] + p; b[i] = a[i] + 2; }
memset(a[0], 255, p * q); for (i = 0; i < h; i++) { for (j = 0; j < w; j++) { for (k = 0; k < 8; k++) { x = j + dx[k], y = i + dy[k]; if (b[i+2][j] == 255) b[i+2][j] = 0; b[i+2][j] += x >= 0 && x < w && y >= 0 && y < h; } } } }
- define E "\033["
int walk_board(int w, int h, int x, int y, cell **b) { int i, nx, ny, least; int steps = 0; printf(E"H"E"J"E"%d;%dH"E"32m[]"E"m", y + 1, 1 + 2 * x);
while (1) { /* occupy cell */ b[y][x] = 255;
/* reduce all neighbors' neighbor count */ for (i = 0; i < 8; i++) b[ y + dy[i] ][ x + dx[i] ]--;
/* find neighbor with lowest neighbor count */ least = 255; for (i = 0; i < 8; i++) { if (b[ y + dy[i] ][ x + dx[i] ] < least) { nx = x + dx[i]; ny = y + dy[i]; least = b[ny][nx]; } }
if (least > 7) { printf(E"%dH", h + 2); return steps == w * h - 1; }
if (steps++) printf(E"%d;%dH[]", y + 1, 1 + 2 * x); x = nx, y = ny; printf(E"%d;%dH"E"31m[]"E"m", y + 1, 1 + 2 * x); fflush(stdout); usleep(120000); } }
int solve(int w, int h) { int x = 0, y = 0; cell **a, **b; a = malloc((w + 4) * (h + 4) + sizeof(cell*) * (h + 4)); b = malloc((h + 4) * sizeof(cell*));
while (1) { init_board(w, h, a, b); if (walk_board(w, h, x, y, b + 2)) { printf("Success!\n"); return 1; } if (++x >= w) x = 0, y++; if (y >= h) { printf("Failed to find a solution\n"); return 0; } printf("Any key to try next start position"); getchar(); } }
int main(int c, char **v) { int w, h; if (c < 2 || (w = atoi(v[1])) <= 0) w = 8; if (c < 3 || (h = atoi(v[2])) <= 0) h = w; solve(w, h);
return 0; }</lang>
C++
Uses Warnsdorff's rule and (iterative) backtracking if that fails.
<lang cpp>#include <iostream>
- include <iomanip>
- include <array>
- include <string>
- include <tuple>
- include <algorithm>
using namespace std;
template<int N = 8> class Board { public:
array<pair<int, int>, 8> moves; array<array<int, N>, N> data;
Board()
{
moves[0] = make_pair(2, 1);
moves[1] = make_pair(1, 2);
moves[2] = make_pair(-1, 2);
moves[3] = make_pair(-2, 1);
moves[4] = make_pair(-2, -1);
moves[5] = make_pair(-1, -2);
moves[6] = make_pair(1, -2);
moves[7] = make_pair(2, -1);
}
array<int, 8> sortMoves(int x, int y) const
{
array<tuple<int, int>, 8> counts;
for(int i = 0; i < 8; ++i)
{
int dx = get<0>(moves[i]);
int dy = get<1>(moves[i]);
int c = 0;
for(int j = 0; j < 8; ++j)
{
int x2 = x + dx + get<0>(moves[j]);
int y2 = y + dy + get<1>(moves[j]);
if (x2 < 0 || x2 >= N || y2 < 0 || y2 >= N)
continue;
if(data[y2][x2] != 0)
continue;
c++;
}
counts[i] = make_tuple(c, i);
}
// Shuffle to randomly break ties
random_shuffle(counts.begin(), counts.end());
// Lexicographic sort
sort(counts.begin(), counts.end());
array<int, 8> out;
for(int i = 0; i < 8; ++i)
out[i] = get<1>(counts[i]);
return out;
}
void solve(string start)
{
for(int v = 0; v < N; ++v)
for(int u = 0; u < N; ++u)
data[v][u] = 0;
int x0 = start[0] - 'a';
int y0 = N - (start[1] - '0');
data[y0][x0] = 1;
array<tuple<int, int, int, array<int, 8>>, N*N> order;
order[0] = make_tuple(x0, y0, 0, sortMoves(x0, y0));
int n = 0;
while(n < N*N-1)
{
int x = get<0>(order[n]);
int y = get<1>(order[n]);
bool ok = false;
for(int i = get<2>(order[n]); i < 8; ++i)
{
int dx = moves[get<3>(order[n])[i]].first;
int dy = moves[get<3>(order[n])[i]].second;
if(x+dx < 0 || x+dx >= N || y+dy < 0 || y+dy >= N)
continue;
if(data[y + dy][x + dx] != 0)
continue;
++n;
get<2>(order[n]) = i + 1;
data[y+dy][x+dx] = n + 1;
order[n] = make_tuple(x+dx, y+dy, 0, sortMoves(x+dx, y+dy));
ok = true;
break;
}
if(!ok) // Failed. Backtrack.
{
data[y][x] = 0;
--n;
}
}
}
template<int N> friend ostream& operator<<(ostream &out, const Board<N> &b);
};
template<int N> ostream& operator<<(ostream &out, const Board<N> &b) {
for (int v = 0; v < N; ++v)
{
for (int u = 0; u < N; ++u)
{
if (u != 0) out << ",";
out << setw(3) << b.data[v][u];
}
out << endl;
}
return out;
}
int main() {
Board<5> b1;
b1.solve("c3");
cout << b1 << endl;
Board<8> b2;
b2.solve("b5");
cout << b2 << endl;
Board<31> b3; // Max size for <1000 squares
b3.solve("a1");
cout << b3 << endl;
return 0;
}</lang>
Output:
23, 16, 11, 6, 21 10, 5, 22, 17, 12 15, 24, 1, 20, 7 4, 9, 18, 13, 2 25, 14, 3, 8, 19 63, 20, 3, 24, 59, 36, 5, 26 2, 23, 64, 37, 4, 25, 58, 35 19, 62, 21, 50, 55, 60, 27, 6 22, 1, 54, 61, 38, 45, 34, 57 53, 18, 49, 44, 51, 56, 7, 28 12, 15, 52, 39, 46, 31, 42, 33 17, 48, 13, 10, 43, 40, 29, 8 14, 11, 16, 47, 30, 9, 32, 41 275,112, 19,116,277,604, 21,118,823,770, 23,120,961,940, 25,122,943,926, 27,124,917,898, 29,126,911,872, 31,128,197,870, 33 18,115,276,601, 20,117,772,767, 22,119,958,851, 24,121,954,941, 26,123,936,925, 28,125,912,899, 30,127,910,871, 32,129,198 111,274,113,278,605,760,603,822,771,824,769,948,957,960,939,944,953,942,927,916,929,918,897,908,913,900,873,196,875, 34,869 114, 17,600,273,602,775,766,773,768,949,850,959,852,947,952,955,932,937,930,935,924,915,920,905,894,909,882,901,868,199,130 271,110,279,606,759,610,761,776,821,764,825,816,951,956,853,938,945,934,923,928,919,896,893,914,907,904,867,874,195,876, 35 16,581,272,599,280,607,774,765,762,779,950,849,826,815,946,933,854,931,844,857,890,921,906,895,886,883,902,881,200,131,194 109,270,281,580,609,758,611,744,777,820,763,780,817,848,827,808,811,846,855,922,843,858,889,892,903,866,885,192,877, 36,201 282, 15,582,269,598,579,608,757,688,745,778,819,754,783,814,847,828,807,810,845,856,891,842,859,884,887,880,863,202,193,132 267,108,283,578,583,612,689,614,743,756,691,746,781,818,753,784,809,812,829,806,801,840,835,888,865,862,203,878,191,530, 37 14,569,268,585,284,597,576,619,690,687,742,755,692,747,782,813,752,785,802,793,830,805,860,841,836,879,864,529,204,133,190 107,266,285,570,577,584,613,686,615,620,695,684,741,732,711,748,739,794,751,786,803,800,839,834,861,528,837,188,531, 38,205 286, 13,568,265,586,575,596,591,618,685,616,655,696,693,740,733,712,749,738,795,792,831,804,799,838,833,722,527,206,189,134 263,106,287,508,571,590,587,574,621,592,639,694,683,656,731,710,715,734,787,750,737,796,791,832,721,798,207,532,187,474, 39 12,417,264,567,288,509,572,595,588,617,654,657,640,697,680,713,730,709,716,735,788,727,720,797,790,723,526,473,208,135,186 105,262,289,416,507,566,589,512,573,622,593,638,653,682,659,698,679,714,729,708,717,736,789,726,719,472,533,184,475, 40,209 290, 11,418,261,502,415,510,565,594,513,562,641,658,637,652,681,660,699,678,669,728,707,718,675,724,525,704,471,210,185,136 259,104,291,414,419,506,503,514,511,564,623,548,561,642,551,636,651,670,661,700,677,674,725,706,703,534,211,476,183,396, 41 10,331,260,493,292,501,420,495,504,515,498,563,624,549,560,643,662,635,650,671,668,701,676,673,524,705,470,395,212,137,182 103,258,293,330,413,494,505,500,455,496,547,516,485,552,625,550,559,644,663,634,649,672,667,702,535,394,477,180,397, 42,213 294, 9,332,257,492,329,456,421,490,499,458,497,546,517,484,553,626,543,558,645,664,633,648,523,666,469,536,393,220,181,138 255,102,295,328,333,412,491,438,457,454,489,440,459,486,545,518,483,554,627,542,557,646,665,632,537,478,221,398,179,214, 43 8,319,256,335,296,345,326,409,422,439,436,453,488,441,460,451,544,519,482,555,628,541,522,647,468,631,392,219,222,139,178 101,254,297,320,327,334,411,346,437,408,423,368,435,452,487,442,461,450,445,520,481,556,629,538,479,466,399,176,215, 44,165 298, 7,318,253,336,325,344,349,410,347,360,407,424,383,434,427,446,443,462,449,540,521,480,467,630,391,218,223,164,177,140 251,100,303,300,321,316,337,324,343,350,369,382,367,406,425,384,433,428,447,444,463,430,539,390,465,400,175,216,169,166, 45 6,299,252,317,304,301,322,315,348,361,342,359,370,381,366,405,426,385,432,429,448,389,464,401,174,217,224,163,150,141,168 99,250,241,302,235,248,307,338,323,314,351,362,341,358,371,380,365,404,377,386,431,402,173,388,225,160,153,170,167, 46,143 240, 5, 98,249,242,305,234,247,308,339,232,313,352,363,230,357,372,379,228,403,376,387,226,159,154,171,162,149,142,151, 82 63, 2,239, 66, 97,236,243,306,233,246,309,340,231,312,353,364,229,356,373,378,227,158,375,172,161,148,155,152, 83,144, 47 4, 67, 64, 61,238, 69, 96, 59,244, 71, 94, 57,310, 73, 92, 55,354, 75, 90, 53,374, 77, 88, 51,156, 79, 86, 49,146, 81, 84 1, 62, 3, 68, 65, 60,237, 70, 95, 58,245, 72, 93, 56,311, 74, 91, 54,355, 76, 89, 52,157, 78, 87, 50,147, 80, 85, 48,145
C#
<lang csharp>using System; using System.Collections.Generic;
namespace prog { class MainClass { const int N = 8;
readonly static int[,] moves = { {+1,-2},{+2,-1},{+2,+1},{+1,+2}, {-1,+2},{-2,+1},{-2,-1},{-1,-2} }; struct ListMoves { public int x, y; public ListMoves( int _x, int _y ) { x = _x; y = _y; } }
public static void Main (string[] args) { int[,] board = new int[N,N]; board.Initialize();
int x = 0, // starting position y = 0;
List<ListMoves> list = new List<ListMoves>(N*N); list.Add( new ListMoves(x,y) );
do { if ( Move_Possible( board, x, y ) ) { int move = board[x,y]; board[x,y]++; x += moves[move,0]; y += moves[move,1]; list.Add( new ListMoves(x,y) ); } else { if ( board[x,y] >= 8 ) { board[x,y] = 0; list.RemoveAt(list.Count-1); if ( list.Count == 0 ) { Console.WriteLine( "No solution found." ); return; } x = list[list.Count-1].x; y = list[list.Count-1].y; } board[x,y]++; } } while( list.Count < N*N );
int last_x = list[0].x, last_y = list[0].y; string letters = "ABCDEFGH"; for( int i=1; i<list.Count; i++ ) { Console.WriteLine( string.Format("{0,2}: ", i) + letters[last_x] + (last_y+1) + " - " + letters[list[i].x] + (list[i].y+1) );
last_x = list[i].x; last_y = list[i].y; } }
static bool Move_Possible( int[,] board, int cur_x, int cur_y ) { if ( board[cur_x,cur_y] >= 8 ) return false;
int new_x = cur_x + moves[board[cur_x,cur_y],0], new_y = cur_y + moves[board[cur_x,cur_y],1];
if ( new_x >= 0 && new_x < N && new_y >= 0 && new_y < N && board[new_x,new_y] == 0 ) return true;
return false; } } }</lang>
CoffeeScript
This algorithm finds 100,000 distinct solutions to the 8x8 problem in about 30 seconds. It precomputes knight moves up front, so it turns into a pure graph traversal problem. The program uses iteration and backtracking to find solutions. <lang coffeescript> graph_tours = (graph, max_num_solutions) ->
# graph is an array of arrays
# graph[3] = [4, 5] means nodes 4 and 5 are reachable from node 3
#
# Returns an array of tours (up to max_num_solutions in size), where
# each tour is an array of nodes visited in order, and where each
# tour visits every node in the graph exactly once.
#
complete_tours = []
visited = (false for node in graph)
dead_ends = ({} for node in graph)
tour = [0]
valid_neighbors = (i) ->
arr = []
for neighbor in graph[i]
continue if visited[neighbor]
continue if dead_ends[i][neighbor]
arr.push neighbor
arr
next_square_to_visit = (i) ->
arr = valid_neighbors i
return null if arr.length == 0
# We traverse to our neighbor who has the fewest neighbors itself.
fewest_neighbors = valid_neighbors(arr[0]).length
neighbor = arr[0]
for i in [1...arr.length]
n = valid_neighbors(arr[i]).length
if n < fewest_neighbors
fewest_neighbors = n
neighbor = arr[i]
neighbor
while tour.length > 0
current_square = tour[tour.length - 1]
visited[current_square] = true
next_square = next_square_to_visit current_square
if next_square?
tour.push next_square
if tour.length == graph.length
complete_tours.push (n for n in tour) # clone
break if complete_tours.length == max_num_solutions
# pessimistically call this a dead end
dead_ends[current_square][next_square] = true
current_square = next_square
else
# we backtrack
doomed_square = tour.pop()
dead_ends[doomed_square] = {}
visited[doomed_square] = false
complete_tours
knight_graph = (board_width) ->
# Turn the Knight's Tour into a pure graph-traversal problem
# by precomputing all the legal moves. Returns an array of arrays,
# where each element in any subarray is the index of a reachable node.
index = (i, j) ->
# index squares from 0 to n*n - 1
board_width * i + j
reachable_squares = (i, j) ->
deltas = [
[ 1, 2]
[ 1, -2]
[ 2, 1]
[ 2, -1]
[-1, 2]
[-1, -2]
[-2, 1]
[-2, -1]
]
neighbors = []
for delta in deltas
[di, dj] = delta
ii = i + di
jj = j + dj
if 0 <= ii < board_width
if 0 <= jj < board_width
neighbors.push index(ii, jj)
neighbors
graph = []
for i in [0...board_width]
for j in [0...board_width]
graph[index(i, j)] = reachable_squares i, j
graph
illustrate_knights_tour = (tour, board_width) ->
pad = (n) ->
return " _" if !n?
return " " + n if n < 10
"#{n}"
console.log "\n------"
moves = {}
for square, i in tour
moves[square] = i + 1
for i in [0...board_width]
s =
for j in [0...board_width]
s += " " + pad moves[i*board_width + j]
console.log s
BOARD_WIDTH = 8 MAX_NUM_SOLUTIONS = 100000
graph = knight_graph BOARD_WIDTH tours = graph_tours graph, MAX_NUM_SOLUTIONS console.log "#{tours.length} tours found (showing first and last)" illustrate_knights_tour tours[0], BOARD_WIDTH illustrate_knights_tour tours.pop(), BOARD_WIDTH </lang>
output <lang> > time coffee knight.coffee 100000 tours found (showing first and last)
1 4 57 20 47 6 49 22 34 19 2 5 58 21 46 7 3 56 35 60 37 48 23 50 18 33 38 55 52 59 8 45 39 14 53 36 61 44 51 24 32 17 40 43 54 27 62 9 13 42 15 30 11 64 25 28 16 31 12 41 26 29 10 63
1 4 41 20 63 6 61 22 34 19 2 5 42 21 44 7 3 40 35 64 37 62 23 60 18 33 38 47 56 43 8 45 39 14 57 36 49 46 59 24 32 17 48 55 58 27 50 9 13 54 15 30 11 52 25 28 16 31 12 53 26 29 10 51
real 0m29.741s user 0m25.656s sys 0m0.253s </lang>
D
<lang d>import std.stdio, std.algorithm, std.random, std.range,
std.conv, std.typecons, std.typetuple;
int[N][N] knightTour(size_t N=8)(in string start) {
assert(start.length >= 2);
alias Tuple!(int,"x", int,"y") P;
// is enum slower than immutable here?
immutable P[8] moves = [P(2,1), P(1,2), P(-1,2), P(-2,1),
P(-2,-1), P(-1,-2), P(1,-2), P(2,-1)];
int[N][N] data;
int[8] sortMoves(in int x, in int y) {
int[2][8] counts;
foreach (i, ref d1; moves) {
int c = 0;
foreach (ref d2; moves) {
immutable p = P(x + d1.x + d2.x, y + d1.y + d2.y);
if (p.x >= 0 && p.x < N && p.y >= 0 && p.y < N &&
data[p.y][p.x] == 0)
c++;
}
//counts[i] = [c, i]; // slow
counts[i][0] = c;
counts[i][1] = i;
}
counts.randomShuffle(); // shuffle to randomly break ties
counts[].sort(); // lexicographic sort
int[8] result;
copy(transversal(counts[], 1), result[]);
return result;
}
immutable p0 = P(start[0] - 'a', N - to!int(start[1..$])); data[p0.y][p0.x] = 1;
Tuple!(int, int, int, int[8])[N*N] order; order[0] = tuple(p0.x, p0.y, 0, sortMoves(p0.x, p0.y));
int n = 0;
while (n < N*N-1) {
immutable int x = order[n][0];
immutable int y = order[n][1];
bool ok = false;
foreach (i; order[n][2] .. 8) {
immutable P d = moves[order[n][3][i]];
if (x+d.x < 0 || x+d.x >= N || y+d.y < 0 || y+d.y >= N)
continue;
if (data[y + d.y][x + d.x] == 0) {
order[n][2] = i + 1;
n++;
data[y + d.y][x + d.x] = n + 1;
order[n]= tuple(x+d.x,y+d.y,0,sortMoves(x+d.x,y+d.y));
ok = true;
break;
}
}
if (!ok) { // Failed. Backtrack.
data[y][x] = 0;
n--;
}
}
return data;
}
void main() {
foreach (i, side; TypeTuple!(5, 8, 31)) {
immutable form = "%" ~ text(text(side ^^ 2).length) ~ "d";
foreach (ref row; knightTour!side(["c3", "b5", "a1"][i]))
writefln(form, row);
writeln();
}
}</lang> Output:
[23, 16, 11, 6, 21] [10, 5, 22, 17, 12] [15, 24, 1, 20, 7] [ 4, 9, 18, 13, 2] [25, 14, 3, 8, 19] [63, 20, 3, 24, 59, 36, 5, 26] [ 2, 23, 64, 37, 4, 25, 58, 35] [19, 62, 21, 50, 55, 60, 27, 6] [22, 1, 54, 61, 38, 45, 34, 57] [53, 18, 49, 44, 51, 56, 7, 28] [12, 15, 52, 39, 46, 31, 42, 33] [17, 48, 13, 10, 43, 40, 29, 8] [14, 11, 16, 47, 30, 9, 32, 41] [275, 112, 19, 116, 277, 604, 21, 118, 823, 770, 23, 120, 961, 940, 25, 122, 943, 926, 27, 124, 917, 898, 29, 126, 911, 872, 31, 128, 197, 870, 33] [ 18, 115, 276, 601, 20, 117, 772, 767, 22, 119, 958, 851, 24, 121, 954, 941, 26, 123, 936, 925, 28, 125, 912, 899, 30, 127, 910, 871, 32, 129, 198] [111, 274, 113, 278, 605, 760, 603, 822, 771, 824, 769, 948, 957, 960, 939, 944, 953, 942, 927, 916, 929, 918, 897, 908, 913, 900, 873, 196, 875, 34, 869] [114, 17, 600, 273, 602, 775, 766, 773, 768, 949, 850, 959, 852, 947, 952, 955, 932, 937, 930, 935, 924, 915, 920, 905, 894, 909, 882, 901, 868, 199, 130] [271, 110, 279, 606, 759, 610, 761, 776, 821, 764, 825, 816, 951, 956, 853, 938, 945, 934, 923, 928, 919, 896, 893, 914, 907, 904, 867, 874, 195, 876, 35] [ 16, 581, 272, 599, 280, 607, 774, 765, 762, 779, 950, 849, 826, 815, 946, 933, 854, 931, 844, 857, 890, 921, 906, 895, 886, 883, 902, 881, 200, 131, 194] [109, 270, 281, 580, 609, 758, 611, 744, 777, 820, 763, 780, 817, 848, 827, 808, 811, 846, 855, 922, 843, 858, 889, 892, 903, 866, 885, 192, 877, 36, 201] [282, 15, 582, 269, 598, 579, 608, 757, 688, 745, 778, 819, 754, 783, 814, 847, 828, 807, 810, 845, 856, 891, 842, 859, 884, 887, 880, 863, 202, 193, 132] [267, 108, 283, 578, 583, 612, 689, 614, 743, 756, 691, 746, 781, 818, 753, 784, 809, 812, 829, 806, 801, 840, 835, 888, 865, 862, 203, 878, 191, 530, 37] [ 14, 569, 268, 585, 284, 597, 576, 619, 690, 687, 742, 755, 692, 747, 782, 813, 752, 785, 802, 793, 830, 805, 860, 841, 836, 879, 864, 529, 204, 133, 190] [107, 266, 285, 570, 577, 584, 613, 686, 615, 620, 695, 684, 741, 732, 711, 748, 739, 794, 751, 786, 803, 800, 839, 834, 861, 528, 837, 188, 531, 38, 205] [286, 13, 568, 265, 586, 575, 596, 591, 618, 685, 616, 655, 696, 693, 740, 733, 712, 749, 738, 795, 792, 831, 804, 799, 838, 833, 722, 527, 206, 189, 134] [263, 106, 287, 508, 571, 590, 587, 574, 621, 592, 639, 694, 683, 656, 731, 710, 715, 734, 787, 750, 737, 796, 791, 832, 721, 798, 207, 532, 187, 474, 39] [ 12, 417, 264, 567, 288, 509, 572, 595, 588, 617, 654, 657, 640, 697, 680, 713, 730, 709, 716, 735, 788, 727, 720, 797, 790, 723, 526, 473, 208, 135, 186] [105, 262, 289, 416, 507, 566, 589, 512, 573, 622, 593, 638, 653, 682, 659, 698, 679, 714, 729, 708, 717, 736, 789, 726, 719, 472, 533, 184, 475, 40, 209] [290, 11, 418, 261, 502, 415, 510, 565, 594, 513, 562, 641, 658, 637, 652, 681, 660, 699, 678, 669, 728, 707, 718, 675, 724, 525, 704, 471, 210, 185, 136] [259, 104, 291, 414, 419, 506, 503, 514, 511, 564, 623, 548, 561, 642, 551, 636, 651, 670, 661, 700, 677, 674, 725, 706, 703, 534, 211, 476, 183, 396, 41] [ 10, 331, 260, 493, 292, 501, 420, 495, 504, 515, 498, 563, 624, 549, 560, 643, 662, 635, 650, 671, 668, 701, 676, 673, 524, 705, 470, 395, 212, 137, 182] [103, 258, 293, 330, 413, 494, 505, 500, 455, 496, 547, 516, 485, 552, 625, 550, 559, 644, 663, 634, 649, 672, 667, 702, 535, 394, 477, 180, 397, 42, 213] [294, 9, 332, 257, 492, 329, 456, 421, 490, 499, 458, 497, 546, 517, 484, 553, 626, 543, 558, 645, 664, 633, 648, 523, 666, 469, 536, 393, 220, 181, 138] [255, 102, 295, 328, 333, 412, 491, 438, 457, 454, 489, 440, 459, 486, 545, 518, 483, 554, 627, 542, 557, 646, 665, 632, 537, 478, 221, 398, 179, 214, 43] [ 8, 319, 256, 335, 296, 345, 326, 409, 422, 439, 436, 453, 488, 441, 460, 451, 544, 519, 482, 555, 628, 541, 522, 647, 468, 631, 392, 219, 222, 139, 178] [101, 254, 297, 320, 327, 334, 411, 346, 437, 408, 423, 368, 435, 452, 487, 442, 461, 450, 445, 520, 481, 556, 629, 538, 479, 466, 399, 176, 215, 44, 165] [298, 7, 318, 253, 336, 325, 344, 349, 410, 347, 360, 407, 424, 383, 434, 427, 446, 443, 462, 449, 540, 521, 480, 467, 630, 391, 218, 223, 164, 177, 140] [251, 100, 303, 300, 321, 316, 337, 324, 343, 350, 369, 382, 367, 406, 425, 384, 433, 428, 447, 444, 463, 430, 539, 390, 465, 400, 175, 216, 169, 166, 45] [ 6, 299, 252, 317, 304, 301, 322, 315, 348, 361, 342, 359, 370, 381, 366, 405, 426, 385, 432, 429, 448, 389, 464, 401, 174, 217, 224, 163, 150, 141, 168] [ 99, 250, 241, 302, 235, 248, 307, 338, 323, 314, 351, 362, 341, 358, 371, 380, 365, 404, 377, 386, 431, 402, 173, 388, 225, 160, 153, 170, 167, 46, 143] [240, 5, 98, 249, 242, 305, 234, 247, 308, 339, 232, 313, 352, 363, 230, 357, 372, 379, 228, 403, 376, 387, 226, 159, 154, 171, 162, 149, 142, 151, 82] [ 63, 2, 239, 66, 97, 236, 243, 306, 233, 246, 309, 340, 231, 312, 353, 364, 229, 356, 373, 378, 227, 158, 375, 172, 161, 148, 155, 152, 83, 144, 47] [ 4, 67, 64, 61, 238, 69, 96, 59, 244, 71, 94, 57, 310, 73, 92, 55, 354, 75, 90, 53, 374, 77, 88, 51, 156, 79, 86, 49, 146, 81, 84] [ 1, 62, 3, 68, 65, 60, 237, 70, 95, 58, 245, 72, 93, 56, 311, 74, 91, 54, 355, 76, 89, 52, 157, 78, 87, 50, 147, 80, 85, 48, 145]
Go
<lang go>/* Adapted from "Enumerating Knight's Tours using an Ant Colony Algorithm" by Philip Hingston and Graham Kendal, PDF at http://www.cs.nott.ac.uk/~gxk/papers/cec05knights.pdf. */
package main
import (
"fmt" "math/rand" "sync" "time"
)
const boardSize = 8 const nSquares = boardSize * boardSize const completeTour = nSquares - 1
// task input: starting square. These are 1 based, but otherwise 0 based // row and column numbers are used througout the program. const rStart = 2 const cStart = 3
// pheromone representation read by ants var tNet = make([]float64, nSquares*8)
// row, col deltas of legal moves var drc = [][]int{{1, 2}, {2, 1}, {2, -1}, {1, -2},
{-1, -2}, {-2, -1}, {-2, 1}, {-1, 2}}
// get square reached by following edge k from square (r, c) func dest(r, c, k int) (int, int, bool) {
r += drc[k][0] c += drc[k][1] return r, c, r >= 0 && r < boardSize && c >= 0 && c < boardSize
}
// struct represents a pheromone amount associated with a move type rckt struct {
r, c, k int t float64
}
func main() {
fmt.Println("Starting square: row", rStart, "column", cStart)
// initialize board
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
for k := 0; k < 8; k++ {
if _, _, ok := dest(r, c, k); ok {
tNet[(r*boardSize+c)*8+k] = 1e-6
}
}
}
}
// waitGroups for ant release clockwork var start, reset sync.WaitGroup start.Add(1) // channel for ants to return tours with pheremone updates tch := make(chan []rckt)
// create an ant for each square
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
go ant(r, c, &start, &reset, tch)
}
}
// accumulator for new pheromone amounts tNew := make([]float64, nSquares*8)
// each iteration is a "cycle" as described in the paper
for {
// evaporate pheromones
for i := range tNet {
tNet[i] *= .75
}
reset.Add(nSquares) // number of ants to release
start.Done() // release them
reset.Wait() // wait for them to begin searching
start.Add(1) // reset start signal for next cycle
// gather tours from ants
for i := 0; i < nSquares; i++ {
tour := <-tch
// watch for a complete tour from the specified starting square
if len(tour) == completeTour &&
tour[0].r == rStart-1 && tour[0].c == cStart-1 {
// task output: move sequence in a grid.
seq := make([]int, nSquares)
for i, sq := range tour {
seq[sq.r*boardSize+sq.c] = i + 1
}
last := tour[len(tour)-1]
r, c, _ := dest(last.r, last.c, last.k)
seq[r*boardSize+c] = nSquares
fmt.Println("Move sequence:")
for r := 0; r < boardSize; r++ {
for c := 0; c < boardSize; c++ {
fmt.Printf(" %3d", seq[r*boardSize+c])
}
fmt.Println()
}
return // task only requires finding a single tour
}
// accumulate pheromone amounts from all ants
for _, move := range tour {
tNew[(move.r*boardSize+move.c)*8+move.k] += move.t
}
}
// update pheromone amounts on network, reset accumulator
for i, tn := range tNew {
tNet[i] += tn
tNew[i] = 0
}
}
}
type square struct {
r, c int
}
func ant(r, c int, start, reset *sync.WaitGroup, tourCh chan []rckt) {
rnd := rand.New(rand.NewSource(time.Now().UnixNano())) tabu := make([]square, nSquares) moves := make([]rckt, nSquares) unexp := make([]rckt, 8) tabu[0].r = r tabu[0].c = c
for {
// cycle initialization
moves = moves[:0]
tabu = tabu[:1]
r := tabu[0].r
c := tabu[0].c
// wait for start signal
start.Wait()
reset.Done()
for {
// choose next move
unexp = unexp[:0]
var tSum float64
findU:
for k := 0; k < 8; k++ {
dr, dc, ok := dest(r, c, k)
if !ok {
continue
}
for _, t := range tabu {
if t.r == dr && t.c == dc {
continue findU
}
}
tk := tNet[(r*boardSize+c)*8+k]
tSum += tk
// note: dest r, c stored here
unexp = append(unexp, rckt{dr, dc, k, tk})
}
if len(unexp) == 0 {
break // no moves
}
rn := rnd.Float64() * tSum
var move rckt
for _, move = range unexp {
if rn <= move.t {
break
}
rn -= move.t
}
// move to new square
move.r, r = r, move.r
move.c, c = c, move.c
tabu = append(tabu, square{r, c})
moves = append(moves, move)
}
// compute pheromone amount to leave
for i := range moves {
moves[i].t = float64(len(moves)-i) / float64(completeTour-i)
}
// return tour found for this cycle
tourCh <- moves
}
}</lang> Output:
Starting square: row 2 column 3 Move sequence: 64 33 36 3 54 49 38 51 35 4 1 30 37 52 55 48 32 63 34 53 2 47 50 39 5 18 31 46 29 20 13 56 62 27 44 19 14 11 40 21 17 6 15 28 45 22 57 12 26 61 8 43 24 59 10 41 7 16 25 60 9 42 23 58
Haskell
<lang Haskell> import System (getArgs) import Data.Char (ord, chr) import Data.List (minimumBy, (\\), intercalate, sort) import Data.Ord (comparing)
type Square = (Int, Int)
board :: [Square] board = [ (x,y) | x <- [1..8], y <- [1..8] ]
knightMoves :: Square -> [Square] knightMoves (x,y) = filter (flip elem board) jumps
where jumps = [ (x+i,y+j) | i <- jv, j <- jv, abs i /= abs j ]
jv = [1,-1,2,-2]
knightTour :: [Square] -> [Square] knightTour moves
| candMoves == [] = reverse moves
| otherwise = knightTour $ newSquare : moves
where newSquare = minimumBy (comparing (length . findMoves)) candMoves
candMoves = findMoves $ head moves
findMoves sq = knightMoves sq \\ moves
main :: IO () main = do
sq <- fmap (toSq . head) getArgs
printTour $ map toAlg $ knightTour [sq]
where toAlg (x,y) = [chr (x + 96), chr (y + 48)]
toSq [x,y] = ((ord x) - 96, (ord y) - 48)
printTour [] = return ()
printTour tour = do
putStrLn $ intercalate " -> " $ take 8 tour
printTour $ drop 8 tour
</lang>
Output:
e5 -> f7 -> h8 -> g6 -> h4 -> g2 -> e1 -> f3 g1 -> h3 -> g5 -> h7 -> f8 -> d7 -> b8 -> a6 b4 -> a2 -> c1 -> d3 -> b2 -> a4 -> b6 -> a8 c7 -> e8 -> g7 -> h5 -> f4 -> e6 -> d8 -> b7 c5 -> b3 -> a1 -> c2 -> a3 -> b1 -> d2 -> c4 a5 -> c6 -> a7 -> c8 -> d6 -> b5 -> d4 -> e2 c3 -> d1 -> f2 -> h1 -> g3 -> e4 -> f6 -> g8 h6 -> g4 -> h2 -> f1 -> e3 -> f5 -> e7 -> d5
Icon and Unicon
This implements Warnsdorff's algorithm using unordered sets.
- The board must be square (it has only been tested on 8x8 and 7x7). Currently the maximum size board (due to square notation) is 26x26.
- Tie breaking is selectable with 3 variants supplied (first in list, random, and Roth's distance heuristic).
- A debug log can be generated showing the moves and choices considered for tie breaking.
The algorithm doesn't always generate a complete tour. <lang Icon>link printf
procedure main(A) ShowTour(KnightsTour(Board(8))) end
procedure KnightsTour(B,sq,tbrk,debug) #: Warnsdorff’s algorithm
/B := Board(8) # create 8x8 board if none given /sq := ?B.files || ?B.ranks # random initial position (default) sq2fr(sq,B) # validate initial sq if type(tbrk) == "procedure" then
B.tiebreak := tbrk # override tie-breaker
if \debug then write("Debug log : move#, move : (accessibility) choices")
choices := [] # setup to track moves and choices every (movesto := table())[k := key(B.movesto)] := copy(B.movesto[k])
B.tour := [] # new tour repeat {
put(B.tour,sq) # record move
ac := 9 # accessibility counter > maximum
while get(choices) # empty choices for tiebreak
every delete(movesto[nextsq := !movesto[sq]],sq) do { # make sq unavailable
if ac >:= *movesto[nextsq] then # reset to lower accessibility count
while get(choices) # . re-empty choices
if ac = *movesto[nextsq] then
put(choices,nextsq) # keep least accessible sq and any ties
}
if \debug then { # move#, move, (accessibility), choices
writes(sprintf("%d. %s : (%d) ",*B.tour,sq,ac))
every writes(" ",!choices|"\n")
}
sq := B.tiebreak(choices,B) | break # choose next sq until out of choices
}
return B end
procedure RandomTieBreaker(S,B) # random choice return ?S end
procedure FirstTieBreaker(S,B) # first one in the list return !S end
procedure RothTieBreaker(S,B) # furthest from the center if *S = 0 then fail # must fail if [] every fr := sq2fr(s := !S,B) do {
d := sqrt(abs(fr[1]-1 - (B.N-1)*0.5)^2 + abs(fr[2]-1 - (B.N-1)*0.5)^2) if (/md := d) | ( md >:= d) then msq := s # save sq }
return msq end
record board(N,ranks,files,movesto,tiebreak,tour) # structure for board
procedure Board(N) #: create board N := *&lcase >=( 0 < integer(N)) | stop("N=",image(N)," is out of range.") B := board(N,[],&lcase[1+:N],table(),RandomTieBreaker) # setup every put(B.ranks,N to 1 by -1) # add rank #s every sq := !B.files || !B.ranks do # for each sq add
every insert(B.movesto[sq] := set(), KnightMoves(sq,B)) # moves to next sq
return B end
procedure sq2fr(sq,B) #: return numeric file & rank f := find(sq[1],B.files) | runerr(205,sq) r := integer(B.ranks[sq[2:0]]) | runerr(205,sq) return [f,r] end
procedure KnightMoves(sq,B) #: generate all Kn accessible moves from sq fr := sq2fr(sq,B) every ( i := -2|-1|1|2 ) & ( j := -2|-1|1|2 ) do
if (abs(i)~=abs(j)) & (0<(ri:=fr[2]+i)<=B.N) & (0<(fj:=fr[1]+j)<=B.N) then
suspend B.files[fj]||B.ranks[ri]
end
procedure ShowTour(B) #: show the tour write("Board size = ",B.N) write("Tour length = ",*B.tour) write("Tie Breaker = ",image(B.tiebreak))
every !(squares := list(B.N)) := list(B.N,"-") every fr := sq2fr(B.tour[m := 1 to *B.tour],B) do
squares[fr[2],fr[1]] := m
every (hdr1 := " ") ||:= right(!B.files,3) every (hdr2 := " +") ||:= repl((1 to B.N,"-"),3) | "-+"
every write(hdr1|hdr2) every r := 1 to B.N do {
writes(right(B.ranks[r],3)," |")
every writes(right(squares[r,f := 1 to B.N],3))
write(" |",right(B.ranks[r],3))
}
every write(hdr2|hdr1|&null) end</lang>
The following can be used when debugging to validate the board structure and to image the available moves on the board. <lang Icon>procedure DumpBoard(B) #: Dump Board internals write("Board size=",B.N) write("Available Moves at start of tour:", ImageMovesTo(B.movesto)) end
procedure ImageMovesTo(movesto) #: image of available moves every put(K := [],key(movesto)) every (s := "\n") ||:= (k := !sort(K)) || " : " do
every s ||:= " " || (!sort(movesto[k])|"\n")
return s end</lang>
Sample output:
Board size = 8
Tour length = 64
Tie Breaker = procedure RandomTieBreaker
a b c d e f g h
+-------------------------+
8 | 53 10 29 26 55 12 31 16 | 8
7 | 28 25 54 11 30 15 48 13 | 7
6 | 9 52 27 62 47 56 17 32 | 6
5 | 24 61 38 51 36 45 14 49 | 5
4 | 39 8 63 46 57 50 33 18 | 4
3 | 64 23 60 37 42 35 44 3 | 3
2 | 7 40 21 58 5 2 19 34 | 2
1 | 22 59 6 41 20 43 4 1 | 1
+-------------------------+
a b c d e f g hTwo 7x7 boards:
Board size = 7
Tour length = 33
Tie Breaker = procedure RandomTieBreaker
a b c d e f g
+----------------------+
7 | 33 4 15 - 29 6 17 | 7
6 | 14 - 30 5 16 - 28 | 6
5 | 3 32 - - - 18 7 | 5
4 | - 13 - 31 - 27 - | 4
3 | 23 2 - - - 8 19 | 3
2 | 12 - 24 21 10 - 26 | 2
1 | 1 22 11 - 25 20 9 | 1
+----------------------+
a b c d e f g
Board size = 7
Tour length = 49
Tie Breaker = procedure RothTieBreaker
a b c d e f g
+----------------------+
7 | 35 14 21 46 7 12 9 | 7
6 | 20 49 34 13 10 23 6 | 6
5 | 15 36 45 22 47 8 11 | 5
4 | 42 19 48 33 40 5 24 | 4
3 | 37 16 41 44 27 32 29 | 3
2 | 18 43 2 39 30 25 4 | 2
1 | 1 38 17 26 3 28 31 | 1
+----------------------+
a b c d e f g
J
Solution:
The Knight's tour essay on the Jwiki shows a couple of solutions including one using Warnsdorffs algorithm.
<lang j>NB. knight moves for each square of a (y,y) board
kmoves=: monad define
t=. (>,{;~i.y) +"1/ _2]\2 1 2 _1 1 2 1 _2 _1 2 _1 _2 _2 1 _2 _1
(*./"1 t e. i.y) <@#"1 y#.t
)
ktourw=: monad define
M=. >kmoves y
p=. k=. 0
b=. 1 $~ *:y
for. i.<:*:y do.
b=. 0 k}b
p=. p,k=. ((i.<./) +/"1 b{~j{M){j=. ({&b # ]) k{M
end.
assert. ~:p
(,~y)$/:p
)</lang>
Example Use: <lang j> ktourw 8 NB. solution for an 8 x 8 board
0 25 14 23 28 49 12 31
15 22 27 50 13 30 63 48 26 1 24 29 62 59 32 11 21 16 51 58 43 56 47 60
2 41 20 55 52 61 10 33
17 38 53 42 57 44 7 46 40 3 36 19 54 5 34 9 37 18 39 4 35 8 45 6
9!:37]0 64 4 4 NB. truncate lines longer than 64 characters and only show first and last four lines
ktourw 202 NB. 202x202 board -- this implementation failed for 200 and 201 0 401 414 405 398 403 424 417 396 419 43... 413 406 399 402 425 416 397 420 439 430 39... 400 1 426 415 404 423 448 429 418 437 4075... 409 412 407 446 449 428 421 440 40739 40716 43...
...
550 99 560 569 9992 779 786 773 10002 9989 78... 555 558 553 778 563 570 775 780 785 772 1000... 100 551 556 561 102 777 572 771 104 781 57... 557 554 101 552 571 562 103 776 573 770 10...</lang>
Lua
<lang lua>N = 8
moves = { {1,-2},{2,-1},{2,1},{1,2},{-1,2},{-2,1},{-2,-1},{-1,-2} }
function Move_Allowed( board, x, y )
if board[x][y] >= 8 then return false end
local new_x, new_y = x + moves[board[x][y]+1][1], y + moves[board[x][y]+1][2] if new_x >= 1 and new_x <= N and new_y >= 1 and new_y <= N and board[new_x][new_y] == 0 then return true end return false
end
board = {}
for i = 1, N do
board[i] = {}
for j = 1, N do
board[i][j] = 0
end
end
x, y = 1, 1
lst = {} lst[1] = { x, y }
repeat
if Move_Allowed( board, x, y ) then
board[x][y] = board[x][y] + 1
x, y = x+moves[board[x][y]][1], y+moves[board[x][y]][2]
lst[#lst+1] = { x, y }
else
if board[x][y] >= 8 then
board[x][y] = 0
lst[#lst] = nil
if #lst == 0 then
print "No solution found."
os.exit(1)
end
x, y = lst[#lst][1], lst[#lst][2]
end
board[x][y] = board[x][y] + 1
end
until #lst == N^2
last = lst[1] for i = 2, #lst do
print( string.format( "%s%d - %s%d", string.sub("ABCDEFGH",last[1],last[1]), last[2], string.sub("ABCDEFGH",lst[i][1],lst[i][1]), lst[i][2] ) )
last = lst[i]
end</lang>
Mathprog
While a little slower than using Warnsdorff this solution is interesting:
1. It shows that Hidato (see: http://rosettacode.org/wiki/Hidato) and Knights Tour are essentially the same problem.
2. It is possible to specify which square is used for any Knights Move.
<lang> /*Knights.mathprog
Find a Knights Tour
Nigel_Galloway January 11th., 2012
- /
param ZBLS; param ROWS; param COLS; param D := 2; 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-2,z+1] + BR[r-1,c+2,z+1] + BR[r-2,c-1,z+1] + BR[r-2,c+1,z+1] + BR[r+1,c+2,z+1] + BR[r+1,c-2,z+1] + BR[r+2,c-1,z+1] + BR[r+2,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 := 5; param COLS := 5; param ZBLS := 25; param Iz: 1 2 3 4 5 :=
1 . . . . . 2 . 19 2 . . 3 . . . . . 4 . . . . . 5 . . . . . ; end;
</lang>
Produces:
<lang> GLPSOL: GLPK LP/MIP Solver, v4.47 Parameter(s) specified in the command line:
--minisat --math Knights.mathprog
Reading model section from Knights.mathprog... Reading data section from Knights.mathprog... 62 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 2549 rows, 2106 columns, and 9349 non-zeros 575 covering inequalities 1924 partitioning equalities Solving CNF-SAT problem... Instance has 3356 variables, 10874 clauses, and 34549 literals
============================[MINISAT]=============================
| Conflicts | ORIGINAL | LEARNT | Progress | | | Clauses Literals | Limit Clauses Literals Lit/Cl | |
==================================================================
| 0 | 9000 32675 | 3000 0 0 0.0 | 0.000 % | | 101 | 6025 21551 | 3300 93 1620 17.4 | 57.688 % | | 251 | 6025 21551 | 3630 243 4961 20.4 | 57.688 % |
==================================================================
SATISFIABLE Objective value = 0.000000000e+000 Time used: 0.0 secs Memory used: 6.5 Mb (6775701 bytes)
1 12 7 18 3 6 19 2 13 8 11 22 15 4 17 20 5 24 9 14 23 10 21 16 25
Model has been successfully processed </lang>
and
<lang> /*Knights.mathprog
Find a Knights Tour
Nigel_Galloway January 11th., 2012
- /
param ZBLS; param ROWS; param COLS; param D := 2; 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-2,z+1] + BR[r-1,c+2,z+1] + BR[r-2,c-1,z+1] + BR[r-2,c+1,z+1] + BR[r+1,c+2,z+1] + BR[r+1,c-2,z+1] + BR[r+2,c-1,z+1] + BR[r+2,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 := 64; param Iz: 1 2 3 4 5 6 7 8 :=
1 . . . . . . . . 2 . . . . . . 48 . 3 . . . . . . . . 4 . . . . . . . . 5 . . . . . . . . 6 . . . . . . . . 7 . 58 . . . . . . 8 . . . . . . . . ; end;
</lang>
Produces:
<lang> GLPSOL: GLPK LP/MIP Solver, v4.47 Parameter(s) specified in the command line:
--minisat --math Knights.mathprog
Reading model section from Knights.mathprog... Reading data section from Knights.mathprog... 65 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 10466 rows, 9360 columns, and 55330 non-zeros 3968 covering inequalities 6370 partitioning equalities Solving CNF-SAT problem... Instance has 15056 variables, 46754 clauses, and 149794 literals
============================[MINISAT]=============================
| Conflicts | ORIGINAL | LEARNT | Progress | | | Clauses Literals | Limit Clauses Literals Lit/Cl | |
==================================================================
| 0 | 40512 143552 | 13504 0 0 0.0 | 0.000 % | | 100 | 32458 114610 | 14854 89 5138 57.7 | 46.633 % | | 250 | 32458 114610 | 16340 239 18544 77.6 | 46.633 % | | 475 | 27499 102956 | 17974 424 42212 99.6 | 46.892 % | | 813 | 27366 102490 | 19771 757 73184 96.7 | 51.541 % | | 1322 | 27366 102490 | 21748 1264 137991 109.2 | 52.245 % | | 2083 | 23226 92730 | 23923 2010 250286 124.5 | 53.620 % | | 3227 | 22239 90284 | 26315 3138 460582 146.8 | 53.620 % | | 4937 | 22239 90284 | 28947 4848 769486 158.7 | 53.620 % | | 7499 | 22206 90168 | 31842 7404 1258240 169.9 | 55.167 % | | 11346 | 21067 87284 | 35026 11248 2085553 185.4 | 55.167 % | | 17113 | 21067 87284 | 38528 17015 3625910 213.1 | 55.167 % | | 25763 | 21067 87284 | 42381 25665 5906283 230.1 | 55.167 % | | 38738 | 21051 87252 | 46619 38638 9316878 241.1 | 55.679 % | | 58199 | 21051 87252 | 51281 16434 3967196 241.4 | 55.685 % | | 87393 | 20707 86474 | 56410 45624 13013357 285.2 | 56.277 % | | 131184 | 20180 84834 | 62051 37252 8996727 241.5 | 56.542 % | | 196871 | 20180 84834 | 68256 49392 13807861 279.6 | 56.542 % | | 295399 | 20180 84834 | 75081 22688 5827696 256.9 | 56.542 % |
==================================================================
SATISFIABLE Objective value = 0.000000000e+000 Time used: 333.0 secs Memory used: 28.2 Mb (29609617 bytes)
51 24 31 6 49 26 33 64 30 5 50 25 32 63 48 43 23 52 7 4 27 44 15 34 8 29 60 45 62 47 42 17 59 22 53 28 3 16 35 14 54 9 56 61 46 39 18 41 21 58 11 38 19 2 13 36 10 55 20 57 12 37 40 1
Model has been successfully processed </lang>
Perl
Knight's tour using Warnsdorffs algorithm <lang perl>use strict; use warnings;
- Find a knight's tour
my @board;
- Choose starting position - may be passed in on command line; if
- not, choose random square.
my ($i, $j); if (my $sq = shift @ARGV) {
die "$0: illegal start square '$sq'\n" unless ($i, $j) = from_algebraic($sq);
} else {
($i, $j) = (int rand 8, int rand 8);
}
- Move sequence
my @moves = ();
foreach my $move (1..64) {
# Record current move push @moves, to_algebraic($i,$j); $board[$i][$j] = $move++;
# Get list of possible next moves my @targets = possible_moves($i,$j);
# Find the one with the smallest degree
my @min = (9);
foreach my $target (@targets) {
my ($ni, $nj) = @$target;
my $next = possible_moves($ni,$nj);
@min = ($next, $ni, $nj) if $next < $min[0];
}
# And make it ($i, $j) = @min[1,2];
}
- Print the move list
for (my $i=0; $i<4; ++$i) {
for (my $j=0; $j<16; ++$j) {
my $n = $i*16+$j;
print $moves[$n];
print ', ' unless $n+1 >= @moves;
}
print "\n";
} print "\n";
- And the board, with move numbers
for (my $i=0; $i<8; ++$i) {
for (my $j=0; $j<8; ++$j) {
# Assumes (1) ANSI sequences work, and (2) output
# is light text on a dark background.
print "\e[7m" if ($i%2==$j%2);
printf " %2d", $board[$i][$j];
print "\e[0m";
}
print "\n";
}
- Find the list of positions the knight can move to from the given square
sub possible_moves {
my ($i, $j) = @_;
return grep { $_->[0] >= 0 && $_->[0] < 8
&& $_->[1] >= 0 && $_->[1] < 8
&& !$board[$_->[0]][$_->[1]] } (
[$i-2,$j-1], [$i-2,$j+1], [$i-1,$j-2], [$i-1,$j+2],
[$i+1,$j-2], [$i+1,$j+2], [$i+2,$j-1], [$i+2,$j+1]);
}
- Return the algebraic name of the square identified by the coordinates
- i=rank, 0=black's home row; j=file, 0=white's queen's rook
sub to_algebraic {
my ($i, $j) = @_;
chr(ord('a') + $j) . (8-$i);
}
- Return the coordinates matching the given algebraic name
sub from_algebraic {
my $square = shift;
return unless $square =~ /^([a-h])([1-8])$/;
return (8-$2, ord($1) - ord('a'));
}</lang>
Sample output (start square c3):
Perl 6
<lang perl6>my @board;
my $I = 8; my $J = 8; my $F = $I*$J > 99 ?? "%3d" !! "%2d";
- Choose starting position - may be passed in on command line; if
- not, choose random square.
my ($i, $j);
if my $sq = shift @*ARGS {
die "$*PROGRAM_NAME: illegal start square '$sq'\n" unless ($i, $j) = from_algebraic($sq);
} else {
($i, $j) = (^$I).pick, (^$J).pick;
}
- Move sequence
my @moves = ();
for 1 .. $I * $J -> $move {
# Record current move
push @moves, to_algebraic($i,$j);
# @board[$i] //= []; # (uncomment if autoviv is broken)
@board[$i][$j] = $move;
# Find move with the smallest degree
my @min = (9);
for possible_moves($i,$j) -> @target {
my ($ni, $nj) = @target;
my $next = possible_moves($ni,$nj);
@min = $next, $ni, $nj if $next < @min[0];
}
# And make it
($i, $j) = @min[1,2];
}
- Print the move list
for @moves.kv -> $i, $m {
print ',', $i %% 16 ?? "\n" !! " " if $i; print $m;
} say "\n";
- And the board, with move numbers
for ^$I -> $i {
for ^$J -> $j {
# Assumes (1) ANSI sequences work, and (2) output
# is light text on a dark background.
print "\e[7m" if $i % 2 == $j % 2;
printf $F, @board[$i][$j];
print "\e[0m";
}
print "\n";
}
- Find the list of positions the knight can move to from the given square
sub possible_moves($i,$j) {
grep -> [$ni, $nj] { $ni ~~ ^$I and $nj ~~ ^$J and !@board[$ni][$nj] },
[$i-2,$j-1], [$i-2,$j+1], [$i-1,$j-2], [$i-1,$j+2],
[$i+1,$j-2], [$i+1,$j+2], [$i+2,$j-1], [$i+2,$j+1];
}
- Return the algebraic name of the square identified by the coordinates
- i=rank, 0=black's home row; j=file, 0=white's queen's rook
sub to_algebraic($i,$j) {
chr(ord('a') + $j) ~ ($I - $i);
}
- Return the coordinates matching the given algebraic name
sub from_algebraic($square where /^ (<[a..z]>) (\d+) $/) {
$I - $1, ord(~$0) - ord('a');
}</lang> (Output identical to Perl's above.)
PicoLisp
<lang PicoLisp>(load "@lib/simul.l")
- Build board
(grid 8 8)
- Generate legal moves for a given position
(de moves (Tour)
(extract
'((Jump)
(let? Pos (Jump (car Tour))
(unless (memq Pos Tour)
Pos ) ) )
(quote # (taken from "games/chess.l")
((This) (: 0 1 1 0 -1 1 0 -1 1)) # South Southwest
((This) (: 0 1 1 0 -1 1 0 1 1)) # West Southwest
((This) (: 0 1 1 0 -1 -1 0 1 1)) # West Northwest
((This) (: 0 1 1 0 -1 -1 0 -1 -1)) # North Northwest
((This) (: 0 1 -1 0 -1 -1 0 -1 -1)) # North Northeast
((This) (: 0 1 -1 0 -1 -1 0 1 -1)) # East Northeast
((This) (: 0 1 -1 0 -1 1 0 1 -1)) # East Southeast
((This) (: 0 1 -1 0 -1 1 0 -1 1)) ) ) ) # South Southeast
- Build a list of moves, using Warnsdorff’s algorithm
(let Tour '(b1) # Start at b1
(while
(mini
'((P) (length (moves (cons P Tour))))
(moves Tour) )
(push 'Tour @) )
(flip Tour) )</lang>
Output:
-> (b1 a3 b5 a7 c8 b6 a8 c7 a6 b8 d7 f8 h7 g5 h3 g1 e2 c1 a2 b4 c2 a1 b3 a5 b7 d8 c6 d4 e6 c5 a4 c3 d1 b2 c4 d2 f1 h2 f3 e1 d3 e5 f7 h8 g6 h4 g2 f4 d5 e7 g8 h6 g4 e3 f5 d6 e8 g7 h5 f6 e4 g3 h1 f2)
PostScript
You probably shouldn't send this to a printer. Solution using Warnsdorffs algorithm. <lang postscript>%!PS-Adobe-3.0 %%BoundingBox: 0 0 300 300
/s { 300 n div } def /l { rlineto } def
% draws a square /bx { s mul exch s mul moveto s 0 l 0 s l s neg 0 l 0 s neg l } def
% draws checker board /xbd { 1 setgray
0 0 moveto 300 0 l 0 300 l -300 0 l fill
.7 1 .6 setrgbcolor
0 1 n1 { dup 2 mod 2 n1 { 1 index bx fill } for pop } for
0 setgray
} def
/ar1 { [ exch { 0 } repeat ] } def /ar2 { [ exch dup { dup ar1 exch } repeat pop ] } def
/neighbors {
-1 2 0
1 2 0
2 1 0
2 -1 0
1 -2 0
-1 -2 0
-2 -1 0
-2 1 0
%24 x y add 3 mul roll
} def
/func { 0 dict begin mark } def /var { counttomark -1 1 { 2 add -1 roll def } for cleartomark } def
% x y can_goto -> bool /can_goto {
func /x /y var
x 0 ge
x n lt
y 0 ge
y n lt
and and and {
occupied x get y get 0 eq
} { false } ifelse
end
} def
% x y num_access -> number of cells reachable from (x,y) /num_access {
func /x /y var
/count 0 def
x y can_goto {
neighbors
8 { pop y add exch x add exch can_goto {
/count count 1 add def
} if
} repeat
count 0 gt { count } { 9 } ifelse
} { 10 } ifelse
end
} def
% a circle /marker { x s mul y s mul s 20 div 0 360 arc fill } def
% n solve -> draws board of size n x n, calcs path and draws it /solve {
func /n var
/n1 n 1 sub def
/c false def
8 n div setlinewidth
gsave
0 1 n1 { /x exch def c not {
0 1 n1 {
/occupied n ar2 def
c not {
/c true def
/y exch def
grestore xbd gsave
s 2 div dup translate
n n mul 2 sub -1 0 { /iter exch def
c {
0 setgray marker x s mul y s mul moveto
occupied x get y 1 put
neighbors
8 { pop y add exch x add exch 2 copy num_access 24 3 roll } repeat
7 { dup 4 index lt { 6 3 roll } if pop pop pop } repeat
9 ge iter 0 gt and { /c false def } if
/y exch def
/x exch def
.2 setgray x s mul y s mul lineto stroke
} if } for
% to be nice, draw box at final position
.5 0 0 setrgbcolor marker
y .5 sub x .5 sub bx 1 setlinewidth stroke
stroke
} if
} for } if } for showpage
grestore
end
} def
3 1 100 { solve } for
%%EOF</lang>
Prolog
Worlks with SWI-Prolog.
Knights tour using Warnsdorffs algorithm
<lang Prolog>% N is the number of lines of the chessboard knight(N) :- Max is N * N, length(L, Max), knight(N, 0, Max, 0, 0, L), display(N, 0, L).
% knight(NbCol, Coup, Max, Lig, Col, L), % NbCol : number of columns per line % Coup : number of the current move % Max : maximum number of moves % Lig/ Col : current position of the knight % L : the "chessboard"
% the game is over knight(_, Max, Max, _, _, _) :- !.
knight(NbCol, N, MaxN, Lg, Cl, L) :- % Is the move legal Lg >= 0, Cl >= 0, Lg < NbCol, Cl < NbCol,
Pos is Lg * NbCol + Cl, N1 is N+1, % is the place free nth0(Pos, L, N1),
LgM1 is Lg - 1, LgM2 is Lg - 2, LgP1 is Lg + 1, LgP2 is Lg + 2, ClM1 is Cl - 1, ClM2 is Cl - 2, ClP1 is Cl + 1, ClP2 is Cl + 2, maplist(best_move(NbCol, L), [(LgP1, ClM2), (LgP2, ClM1), (LgP2, ClP1),(LgP1, ClP2), (LgM1, ClM2), (LgM2, ClM1), (LgM2, ClP1),(LgM1, ClP2)], R), sort(R, RS), pairs_values(RS, Moves),
move(NbCol, N1, MaxN, Moves, L).
move(NbCol, N1, MaxN, [(Lg, Cl) | R], L) :- knight(NbCol, N1, MaxN, Lg, Cl, L); move(NbCol, N1, MaxN, R, L).
%% An illegal move is scored 1000 best_move(NbCol, _L, (Lg, Cl), 1000-(Lg, Cl)) :- ( Lg < 0 ; Cl < 0; Lg >= NbCol; Cl >= NbCol), !.
best_move(NbCol, L, (Lg, Cl), 1000-(Lg, Cl)) :- Pos is Lg*NbCol+Cl, nth0(Pos, L, V), \+var(V), !.
%% a legal move is scored with the number of moves a knight can make best_move(NbCol, L, (Lg, Cl), R-(Lg, Cl)) :- LgM1 is Lg - 1, LgM2 is Lg - 2, LgP1 is Lg + 1, LgP2 is Lg + 2, ClM1 is Cl - 1, ClM2 is Cl - 2, ClP1 is Cl + 1, ClP2 is Cl + 2, include(possible_move(NbCol, L), [(LgP1, ClM2), (LgP2, ClM1), (LgP2, ClP1),(LgP1, ClP2), (LgM1, ClM2), (LgM2, ClM1), (LgM2, ClP1),(LgM1, ClP2)], Res), length(Res, Len), ( Len = 0 -> R = 1000; R = Len).
% test if a place is enabled possible_move(NbCol, L, (Lg, Cl)) :- % move must be legal Lg >= 0, Cl >= 0, Lg < NbCol, Cl < NbCol, Pos is Lg * NbCol + Cl, % place must be free nth0(Pos, L, V), var(V).
display(_, _, []).
display(N, N, L) :-
nl,
display(N, 0, L).
display(N, M, [H | T]) :- writef('%3r', [H]), M1 is M + 1, display(N, M1, T). </lang>
Output :
?- knight(8). 1 16 31 40 3 18 21 56 30 39 2 17 42 55 4 19 15 32 41 46 53 20 57 22 38 29 48 43 58 45 54 5 33 14 37 52 47 60 23 62 28 49 34 59 44 63 6 9 13 36 51 26 11 8 61 24 50 27 12 35 64 25 10 7 true . ?- knight(20). 1 40 81 90 3 42 77 94 5 44 73 102 7 46 69 62 9 48 51 60 82 89 2 41 92 95 4 43 76 101 6 45 72 103 8 47 68 61 10 49 39 80 91 96 153 78 93 100 129 74 109 104 123 70 111 120 63 50 59 52 88 83 154 79 98 159 152 75 108 105 128 71 110 121 124 67 112 119 64 11 155 38 97 160 157 200 99 162 151 130 107 122 127 132 141 118 125 66 53 58 84 87 156 199 176 161 158 201 106 163 150 131 142 145 126 133 140 113 12 65 37 182 85 178 207 198 175 164 173 216 143 166 149 222 139 146 117 134 57 54 86 179 206 197 204 177 208 217 202 165 172 221 144 167 148 223 138 55 114 13 183 36 181 212 209 218 203 174 215 220 227 170 281 224 303 168 147 116 135 56 180 211 196 205 230 213 238 219 228 171 280 225 302 169 282 343 304 137 14 115 35 184 231 210 237 246 229 214 279 226 301 298 283 342 367 308 347 344 305 136 232 195 236 245 234 239 278 247 300 297 284 359 366 309 348 345 368 307 350 15 185 34 233 240 261 248 287 296 285 358 299 310 341 378 365 384 349 346 369 306 194 241 250 235 244 277 260 313 294 311 360 373 364 383 354 379 370 385 16 351 33 186 243 262 249 288 295 286 361 316 357 340 377 372 395 386 353 380 333 388 242 193 254 251 276 259 314 293 312 321 374 363 398 355 382 371 394 387 352 17 187 32 263 258 267 252 289 322 315 362 317 356 339 376 399 396 381 334 389 332 192 255 190 253 264 275 268 271 292 323 320 375 326 397 338 335 390 393 18 21 31 188 257 266 29 270 273 290 27 318 327 324 25 336 329 400 23 20 331 392 256 191 30 189 274 265 28 269 272 291 26 319 328 325 24 337 330 391 22 19 true .
Python
Knights tour using Warnsdorffs algorithm <lang python>import copy
boardsize=6 _kmoves = ((2,1), (1,2), (-1,2), (-2,1), (-2,-1), (-1,-2), (1,-2), (2,-1))
def chess2index(chess, boardsize=boardsize):
'Convert Algebraic chess notation to internal index format'
chess = chess.strip().lower()
x = ord(chess[0]) - ord('a')
y = boardsize - int(chess[1:])
return (x, y)
def boardstring(board, boardsize=boardsize):
r = range(boardsize)
lines =
for y in r:
lines += '\n' + ','.join('%2i' % board[(x,y)] if board[(x,y)] else ' '
for x in r)
return lines
def knightmoves(board, P, boardsize=boardsize):
Px, Py = P
kmoves = set((Px+x, Py+y) for x,y in _kmoves)
kmoves = set( (x,y)
for x,y in kmoves
if 0 <= x < boardsize
and 0 <= y < boardsize
and not board[(x,y)] )
return kmoves
def accessibility(board, P, boardsize=boardsize):
access = []
brd = copy.deepcopy(board)
for pos in knightmoves(board, P, boardsize=boardsize):
brd[pos] = -1
access.append( (len(knightmoves(brd, pos, boardsize=boardsize)), pos) )
brd[pos] = 0
return access
def knights_tour(start, boardsize=boardsize, _debug=False):
board = {(x,y):0 for x in range(boardsize) for y in range(boardsize)}
move = 1
P = chess2index(start, boardsize)
board[P] = move
move += 1
if _debug:
print(boardstring(board, boardsize=boardsize))
while move <= len(board):
P = min(accessibility(board, P, boardsize))[1]
board[P] = move
move += 1
if _debug:
print(boardstring(board, boardsize=boardsize))
input('\n%2i next: ' % move)
return board
if __name__ == '__main__':
while 1:
boardsize = int(input('\nboardsize: '))
if boardsize < 5:
continue
start = input('Start position: ')
board = knights_tour(start, boardsize)
print(boardstring(board, boardsize=boardsize))</lang>
- Sample runs
boardsize: 5 Start position: c3 19,12,17, 6,21 2, 7,20,11,16 13,18, 1,22, 5 8, 3,24,15,10 25,14, 9, 4,23 boardsize: 8 Start position: h8 38,41,18, 3,22,27,16, 1 19, 4,39,42,17, 2,23,26 40,37,54,21,52,25,28,15 5,20,43,56,59,30,51,24 36,55,58,53,44,63,14,29 9, 6,45,62,57,60,31,50 46,35, 8,11,48,33,64,13 7,10,47,34,61,12,49,32 boardsize: 10 Start position: e6 29, 4,57,24,73, 6,95,10,75, 8 58,23,28, 5,94,25,74, 7,100,11 3,30,65,56,27,72,99,96, 9,76 22,59, 2,63,68,93,26,81,12,97 31,64,55,66, 1,82,71,98,77,80 54,21,60,69,62,67,92,79,88,13 49,32,53,46,83,70,87,42,91,78 20,35,48,61,52,45,84,89,14,41 33,50,37,18,47,86,39,16,43,90 36,19,34,51,38,17,44,85,40,15 boardsize: 200 Start position: a1 510,499,502,101,508,515,504,103,506,5021 ... 195,8550,6691,6712,197,6704,201,6696,199 501,100,509,514,503,102,507,5020,5005,10 ... 690,6713,196,8553,6692,6695,198,6703,202 498,511,500,4989,516,5019,5004,505,5022, ... ,30180,8559,6694,6711,8554,6705,200,6697 99,518,513,4992,5003,4990,5017,5044,5033 ... 30205,8552,30181,8558,6693,6702,203,6706 512,497,4988,517,5018,5001,5034,5011,504 ... 182,30201,30204,8555,6710,8557,6698,6701 519,98,4993,5002,4991,5016,5043,5052,505 ... 03,30546,30183,30200,30185,6700,6707,204 496,4987,520,5015,5000,5035,5012,5047,51 ... 4,30213,30202,31455,8556,6709,30186,6699 97,522,4999,4994,5013,5042,5051,5060,505 ... 7,31456,31329,30184,30199,30190,205,6708 4986,495,5014,521,5036,4997,5048,5101,50 ... 1327,31454,30195,31472,30187,30198,30189 523,96,4995,4998,5041,5074,5061,5050,507 ... ,31330,31471,31328,31453,30196,30191,206 ... 404,731,704,947,958,1013,966,1041,1078,1 ... 9969,39992,39987,39996,39867,39856,39859 5,706,735,960,955,972,957,1060,1025,106 ... ,39978,39939,39976,39861,39990,297,39866 724,403,730,705,946,967,1012,971,1040,10 ... 9975,39972,39991,39868,39863,39860,39855 707, 4,723,736,729,956,973,996,1061,1026 ... ,39850,39869,39862,39973,39852,39865,298 402,725,708,943,968,945,970,1011,978,997 ... 6567,39974,39851,39864,36571,39854,36573 3,722,737,728,741,942,977,974,995,1010, ... ,39800,39849,36570,39853,36574,299,14088 720,401,726,709,944,969,742,941,980,975, ... ,14091,36568,36575,14084,14089,36572,843 711, 2,721,738,727,740,715,976,745,940,9 ... 65,36576,14083,14090,36569,844,14087,300 400,719,710,713,398,717,746,743,396,981, ... ,849,304,14081,840,847,302,14085,842,845 1,712,399,718,739,714,397,716,747,744,3 ... 4078,839,848,303,14082,841,846,301,14086
The 200x200 example warmed my study in its computation but did return a tour.
P.S. There is a slight deviation to a strict interpretation of Warnsdorffs algorithm in that as a conveniance, tuples of the length of the night moves followed by the position are minimized so knights moves with the same length will try and break the ties based on their minimum x,y position. In practice, it seems to give comparable results to the original algorithm.
Tcl
<lang tcl>package require Tcl 8.6; # For object support, which makes coding simpler
oo::class create KnightsTour {
variable width height visited
constructor {{w 8} {h 8}} {
set width $w set height $h set visited {}
}
method ValidMoves {square} {
lassign $square c r set moves {} foreach {dx dy} {-1 -2 -2 -1 -2 1 -1 2 1 2 2 1 2 -1 1 -2} { set col [expr {($c % $width) + $dx}] set row [expr {($r % $height) + $dy}] if {$row >= 0 && $row < $height && $col >=0 && $col < $width} { lappend moves [list $col $row] } } return $moves
}
method CheckSquare {square} {
set moves 0 foreach site [my ValidMoves $square] { if {$site ni $visited} { incr moves } } return $moves
}
method Next {square} {
set minimum 9 set nextSquare {-1 -1} foreach site [my ValidMoves $square] { if {$site ni $visited} { set count [my CheckSquare $site] if {$count < $minimum} { set minimum $count set nextSquare $site } elseif {$count == $minimum} { set nextSquare [my Edgemost $nextSquare $site] } } } return $nextSquare
}
method Edgemost {a b} {
lassign $a ca ra lassign $b cb rb # Calculate distances to edge set da [expr {min($ca, $width - 1 - $ca, $ra, $height - 1 - $ra)}] set db [expr {min($cb, $width - 1 - $cb, $rb, $height - 1 - $rb)}] if {$da < $db} {return $a} else {return $b}
}
method FormatSquare {square} {
lassign $square c r format %c%d [expr {97 + $c}] [expr {1 + $r}]
}
method constructFrom {initial} {
while 1 { set visited [list $initial] set square $initial while 1 { set square [my Next $square] if {$square eq {-1 -1}} { break } lappend visited $square } if {[llength $visited] == $height*$width} { return } puts stderr "rejecting path of length [llength $visited]..." }
}
method constructRandom {} {
my constructFrom [list \ [expr {int(rand()*$width)}] [expr {int(rand()*$height)}]]
}
method print {} {
set s " " foreach square $visited { puts -nonewline "$s[my FormatSquare $square]" if {[incr i]%12} { set s " -> " } else { set s "\n -> " } } puts ""
}
method isClosed {} {
set a [lindex $visited 0] set b [lindex $visited end] expr {$a in [my ValidMoves $b]}
}
}</lang> Demonstrating: <lang tcl>set kt [KnightsTour new] $kt constructRandom $kt print if {[$kt isClosed]} {
puts "This is a closed tour"
} else {
puts "This is an open tour"
}</lang> Sample output:
e6 -> f8 -> h7 -> g5 -> h3 -> g1 -> e2 -> c1 -> a2 -> b4 -> a6 -> b8 -> d7 -> b6 -> a8 -> c7 -> e8 -> g7 -> h5 -> g3 -> h1 -> f2 -> d1 -> b2 -> a4 -> c3 -> b1 -> a3 -> b5 -> a7 -> c8 -> e7 -> g8 -> h6 -> f7 -> h8 -> g6 -> h4 -> g2 -> f4 -> d5 -> f6 -> g4 -> h2 -> f1 -> e3 -> f5 -> d6 -> e4 -> d2 -> c4 -> a5 -> b7 -> d8 -> c6 -> e5 -> f3 -> e1 -> d3 -> c5 -> b3 -> a1 -> c2 -> d4 This is a closed tour
The above code supports other sizes of boards and starting from nominated locations: <lang tcl>set kt [KnightsTour new 7 7] $kt constructFrom {0 0} $kt print if {[$kt isClosed]} {
puts "This is a closed tour"
} else {
puts "This is an open tour"
}</lang> Which could produce this output:
a1 -> c2 -> e1 -> g2 -> f4 -> g6 -> e7 -> f5 -> g7 -> e6 -> g5 -> f7 -> d6 -> b7 -> a5 -> b3 -> c1 -> a2 -> b4 -> a6 -> c7 -> b5 -> a7 -> c6 -> d4 -> e2 -> g1 -> f3 -> d2 -> f1 -> g3 -> e4 -> f2 -> g4 -> f6 -> d7 -> e5 -> d3 -> c5 -> a4 -> b2 -> d1 -> e3 -> d5 -> b6 -> c4 -> a3 -> b1 -> c3 This is an open tour