Knight's tour

Knight's tour
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.

• N-queens problem

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>

1. include <stdlib.h>
2. include <string.h>
3. 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; } } } }

1. 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++

Works with: C++11

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>
5. 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>

==[[:Category:|]] [[Category:]] { 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