Knight's tour

Knight's tour is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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.

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.

• N-queens problem

C++

Works with: C++0x

Uses Warnsdorff's rule and (iterative) backtracking if that fails.

<lang cpp>#include <iostream>

1. include <iomanip>
2. include <array>
3. include <string>
4. include <tuple>

using namespace std;

template<size_t N = 8> class Board { public:

 array<array<int, N>, N> data;
 array<pair<int, int>, 8> moves;
 int sx, sy;

 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)
 {
   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);
   }

   random_shuffle(counts.begin(), counts.end()); // Shuffle to randomly break ties
   sort(counts.begin(), counts.end()); // Lexicographic sort
   
   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 x = start[0] - 'a';
   int y = N - (start[1] - '0');
   data[y][x] = 1;
   sx = x;
   sy = y;

   array<tuple<int, int, int, array<int, 8>>, N*N> order;
   order[0] = make_tuple(sx, sy, 0, sortMoves(sx, sy));

   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)
       {
         get<2>(order[n]) = i+1;

         ++n;
         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;
 Board<8> b2;
 Board<31> b3; // Max size for <1000 squares

 b1.solve("c3");
 cout << b1 << endl;

 b2.solve("b5");
 cout << b2 << endl;

 b3.solve("a1");
 cout << b3 << endl;

}</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

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):

   knightmoves(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