Knight's tour
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).
You are encouraged to solve this task according to the task description, using any language you may know.
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.
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>
D
<lang d>import std.stdio, std.typecons, std.random, std.conv,
std.typetuple, std.range, std.algorithm;
int[N][N] knightTour(size_t N=8)(in string start) {
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; }
randomShuffle(counts[]); // shuffle to randomly break ties counts[].sort(); // lexicographic sort
int[8] result; copy(transversal(counts[], 1), result[]); return result; }
assert(start.length >= 2); immutable P 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" "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.Nanoseconds())) 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 h
Two 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>
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