How to solve the knight's tour 

Introduction

In the following we will see a method that is easy to learn and can be used to easily construct a knight's tour in less than 5 minutes. The tours so constructed will have the additional property that it is possible for the knight to step from the final square to the initial square, thus forming a closed path or a cycle.

Broadly, our strategy is to stay near the edges of the board as much as possible, so that towards the end, the squares that are remaining form a well-connected subgraph at the center of the board, enabling us to complete the last few moves by trial and error.

Step 1

• Start from one of the edge squares (but not one of the corner squares).
• Keep close to the edges of the board, and avoid the 16 central squares unless there is no alternative (e.g. move 13 of figure 1).
• If you are one move away from a corner square, you must necessarily move to the corner square (See figure 1).
• Follow these rules until all edge squares and all squares bordering an edge square have been visited. You should be in a position similar to figure 2.

Figure 1. If you can move to a corner square, you must do so.


Figure 2. Step 1 completed.

Step 2

• 2a) Try to complete the tour by inspection. This is often not possible. If that is the case, read on.
• 2b) Arbitrarily extend the path to get a cycle (i.e. without visiting all the remaining squares). In figure 3, the path has been extended to a cycle of length 54.
• 2c) Find pairs of squares along the tour such that
  • they are one move away from each other.
  • each is one move away from some unvisited square.
• 2d) Use such pairs to complete the tour by inserting new squares between the pair of squares. In figure 3, this has been done using the pairs (5,6) and (8,9).
• 2e) Renumber the squares from 1 to 64 (see figure 4).

Figure 3. The sequence 5->6 has been extended to 5->5a->5b->5c->5d->6 and 8->9 has been extended to 8->8a->8b->8c->8d->8e->8f->9 to complete the tour.


Figure 4. Completed and properly numbered tour.

A heuristic

One popular heuristic for solving the knight's tour problem is the Warnsdorff rule. According to this rule, at each step the knight must move to that square which has the lowest degree (the degree of a square is the number of squares to which the knight can move from that square). The rationale for this is that towards the end, we will be left with squares having a high degree, and so there is a smaller chance of arriving at a square from which there is no move. The Warnsdorff rule is particularly suitable for writing computer programs to solve the knight's tour. Below is an example C program which uses this heuristic.

Example C program for solving the knight's tour using the Warnsdorff rule

 #include <stdio.h>
 #include <stdlib.h>

 
 /* The dimensions of the chess board */
 #define BWID 8
 #define BHEIT 8
 /* The moves of the knight */
 #define JWID 1
 #define JHEIT 2

 
 /* Macros for convenience */
 #define isinboard(board,x,y) ((x)>=0 && (y)>=0 && (x)<BWID && (y)<BHEIT)
 #define isfree(board,x,y) (isinboard(board,x,y) && board[y * BWID + x] < 0)

 
 /* These arrays define the set of neighbors of a square */
 static int incx[8] = {JWID, JWID, JHEIT, JHEIT, -JWID, -JWID, -JHEIT, -JHEIT};
 static int incy[8] = {JHEIT, -JHEIT, JWID, -JWID, JHEIT, -JHEIT, JWID, -JWID};

 
 /* The number of neighbors of a square to which we can move */
 int get_nbr_cnt(int *board, int x, int y)
 {
     int i, cnt = 0;
     for (i=0; i<8; i++)
         if (isfree(board, x+incx[i], y+incy[i]))
             cnt++;
     return cnt;
 }

 
 /* Make a move according to the smallest degree heuristic */
 int make_move(int *board, int *x, int *y)
 {
     int start, cnt, i, best = -1, nbrs, min_nbrs = 9, newx, newy;
     start = random() % 8;
     for (cnt=0; cnt<8; cnt++)
     {
         i = (start+cnt) % 8;
         newx = *x + incx[i];
         newy = *y + incy[i];
         if (isfree(board, newx, newy) && 
                 (nbrs = get_nbr_cnt(board, newx, newy)) < min_nbrs)
         {
             best = i;
             min_nbrs = nbrs;
         }
     }
     if (min_nbrs == 9)
         return 0;
     newx = *x + incx[best];
     newy = *y + incy[best];
     board[newy * BWID + newx] = board[*y * BWID + *x] + 1;
     *x = newx;
     *y = newy;
     return 1;
 }

 
 /* Print the solved position */
 void print_board(int *board)
 {
     int i, j;
     for(i=0; i<BWID; i++)
     {
         for(j=0; j<BHEIT; j++)
             printf ("%d\t", board[j * BWID + i]);
         printf ("\n");
     }
 }

 
 /* Check if 2 squares are neighbors of each other */
 int is_nbr (int x1, int y1, int x2, int y2)
 {
     int i;
     for (i=0; i<8; i++)
         if (x1 + incx[i] == x2 && y1 + incy[i] == y2)
             return 1;
     return 0;
 }

     
 /* Generate a path with a random starting square and each square 
  * chosen according to the smallest degree rule. If a hamiltonian
  * cycle results, print the solution and return 1. Else return 0.
  */
 int gen_sol()
 {
     /* board[y * BWID + x] represents the move number in which
        the square (x, y) was visited. It is -1 if the square has
        not been visited so far */
     int board[BWID*BHEIT];
     /* x and y represent the coordinates of the square being
        currently visited */
     int i, x, y, startx, starty;
     for(i=0; i<BWID*BHEIT; i++)
         board[i] = -1;
     startx = x = random() % BWID;
     starty = y = random() % BHEIT;
     board[y * BWID + x] = 1;
     for(i=0; i<BWID*BHEIT-1; i++)
         if (make_move(board, &x, &y) == 0)
             return 0;
     if (!is_nbr (x, y, startx, starty))
         return 0;
     print_board(board);
     return 1;
 }

 
 /* Loop endlessly until a solution is found */
 int main(void)
 {
     while (!gen_sol())
         ;
     return 0;
 }

Note that the above program isn't restricted to 8x8 chessboards; the parameters BWID and BHEIT control the width and the height respectively.

One solution obtained using the above program is shown below.

Figure 5. Computer solution using the Warnsdorff heuristic.

External links

• Warnsdorff's rule (http://w1.859.telia.com/~u85905224/knight/eWarnsd.htm)
• History of knight's tours (http://www.ktn.freeuk.com/1a.htm)

with an opera-glass." "Ah, that's a different affair. There isn't much going on in Ansbach, right angles with them, and nothing which went on in the streets was they lay full of sun. At such times the Marches were puzzled by the forgotten to put.html">put them down, or thought it not worth while in the brief them; and some other sights remained baffling to the last. Once a man after him with his musket on his shoulder, passed under their windows; Another time a pair went by on the way to the railway station: a young woman lugging a heavy carpet bag, who left them to the lasting question his mother. Women do not do everything in Ansbach, however, the sacristans being men, courtly past of the city by visiting the funeral chapel of the margraves there was something of the neighborly interest in the curiosity of of Ansbach, who willingly left his broom to guide them to the house of sacristan, when he put his velvet skull-cap out of an upper window and saying "Gleich!" instead of "Subito!" The architecture of the houses was Gumpertus, is Gothic, with the two unequal towers which seem distinctive the dwellings are Gothic too, and might be in Hamburg; but at the St. are of a sort of Teutonized renaissance. The rococo margraves and margravines used of course to worship in St. crypt of the church, in caskets of bronze and copper and marble, with last. Their courtly coffins are ranged in a kind of hemicycle, with the knowledge of their greatness. On one of these a kneeling figurine in plays tenderly with the fate of a little princess, who died in her first .

 On wordlookup.net  

All is still licensed under the GNU FDL.
It uses material from the wikipedia.