Knights Tour

( Knight's Tour ) One of the more interesting puzzlers for chess buffs is the Knight's Tour problem. The question is this: Can the chess piece called the knight move around an empty chessboard and touch each of the 64 squares once and only once? We study this intriguing problem in depth in this exercise.

The knight makes L-shaped moves (over two in one direction and then over one in a perpendicular direction). Thus, from a square in the middle of an empty chessboard, the knight can make eight different moves (numbered 0 through 7) as shown in the following.

Draw an 8-by-8 chessboard on a sheet of paper and attempt a Knight's Tour by hand. Put a 1 in the first square you move to, a 2 in the second square, a 3 in the third, etc. Before starting the tour, estimate how far you think you will get, remembering that a full tour consists of 64 moves. How far did you get? Was this close to your estimate?

Now let us develop a program that will move the knight around a chessboard. The board is represented by an 8-by-8 two-dimensional array board. Each of the squares is initialized to zero. We describe each of the eight possible moves in terms of both their horizontal and vertical components. For example, a move of type 0, consists of moving two squares horizontally to the right and one square vertically upward. Move 2 consists of moving one square horizontally to the left and two squares vertically upward. Horizontal moves to the left and vertical moves upward are indicated with negative numbers. The eight moves may be described by two one-dimensional arrays, horizontal and vertical, as follows:

horizontal[] = {2, 2, 1, 1, -2, -2, -1 -1}

vertical[] = {1, -1, 2, -2, 1, -1, 2, -2}

Let the variables currentRow and currentColumn indicate the row and column of the knight's current position. To make a move of type moveNumber, where moveNumber is between 0 and 7, your program uses the statements

currentRow += vertical[ moveNumber ];

currentColumn += horizontal[ moveNumber ];

Keep a counter that varies from 1 to 64. Record the latest count in each square the knight moves to. Remember to test each potential move to see if the knight has already visited that square, and, of course, test every potential move to make sure that the knight does not land off the chessboard. Now write a program to move the knight around the chessboard. Run the program. How many moves did the knight make?

After attempting to write and run a Knight's Tour program, you have probably developed some valuable insights. We will use these to develop a heuristic (or strategy) for moving the knight. Heuristics do not guarantee success, but a carefully developed heuristic greatly improves the chance of success. You may have observed that the outer squares are more troublesome than the squares nearer the center of the board. In fact, the most troublesome, or inaccessible, squares are the four corners.

Intuition may suggest that you should attempt to move the knight to the most troublesome squares first and leave open those that are easiest to get to, so when the board gets congested near the end of the tour, there will be a greater chance of success.

We may develop an "accessibility heuristic" by classifying each square according to how accessible it is and then always moving the knight to the square (within the knight's L-shaped moves, of course) that is most inaccessible. We label a two-dimensional array accessibility with numbers indicating from how many squares each particular square is accessible. On a blank chessboard, each center square is rated as 8, each corner square is rated as 2 and the other squares have accessibility numbers of 3, 4 or 6 as follows:

2 3 4 4 4 4 3 2

3 4 6 6 6 6 4 3

4 6 8 8 8 8 6 4

4 6 8 8 8 8 6 4

4 6 8 8 8 8 6 4

4 6 8 8 8 8 6 4

3 4 6 6 6 6 4 3

2 3 4 4 4 4 3 2

Now write a version of the Knight's Tour program using the accessibility heuristic. At any time, the knight should move to the square with the lowest accessibility number. In case of a tie, the knight may move to any of the tied squares. Therefore, the tour may begin in any of the four corners. [Note: As the knight moves around the chessboard, your program should reduce the accessibility numbers as more and more squares become occupied. In this way, at any given time during the tour, each available square's accessibility number will remain equal to precisely the number of squares from which that square may be reached.] Run this version of your program. Did you get a full tour? Now modify the program to run 64 tours, one starting from each square of the chessboard. How many full tours did you get?

[Extra credit: 20 points] Write a version of the Knight's Tour program which, when encountering a tie between two or more squares, decides what square to choose by looking ahead to those squares reachable from the "tied" squares. Your program should move to the square for which the next move would arrive at a square with the lowest accessibility number.

For your convenience, the following is the header file for my KnightTour class (KnightTour.h), you are welcome to design your own class.

#ifndef KNIGHT_H

#define KNIGHT_H

class KnightTour

{

public:

//constructor

KnightTour(int x = 0, int y = 0);

void initialize();

void getKightTour();

void updateAccess(int x, int y);

void nextMove();

bool inBoard(int x, int y);

void printBoard();

void printAccess();

private:

int board[8][8];

int access[8][8];

int currentx, currenty;

const static int horizontal[8];

const static int vertical[8];

};

#endif

The Pseudo-code for getKnightTour() member function is as follows:

Call initialize() member function to set all board entries to zeros and reset the access matrix

set step to 1

set board[current][currenty] to step

while(step is less than 64)

{

update the access matrix (e.g. updateAccess())

call nextMove() to find the next square to move to

increment step

set board[currentx][currenty] to step

}

print the board matrix show all move numbers

The pseudo-code for nextMove() member function is as follows:

set minx to 0

set miny to 0

set minAccess to 9

for(k = 0 to 7)

{

set testx to currentx + vertical[k]

set testy to currenty + horizontal[k]

if( (testx, testy) is in board and board[testx][testy] is still zero)

{

if(access[testx][testy] < minAccess)

{

set minAccess to access[testx][testy]

set minx to testx

set miny to testy

}

}

}

set currentx to minx

set currenty to miny