An Investigation into the Eight Queens Problem

Introduction

The eight queens problem relies on a very simple chess-based concept. However, no chess knowledge, other than this background, is required. Chess is a game plajghyed on a 8 by 8 grid of 64 squares. The grid is checkered with two colors, usually white and black, the bottom right hand corner being white. The players sit opposite of each other and have sixteen chess pieces that they use to attempt to win the game.1 The objective of the game is to use your sixteen chess pieces to checkmate the king?that is, having the king in a situation where he is unable to move without being in threat of being captured by an opponent?s piece. Each chess piece has certain chess rules which governs the way each chess piece may move. One of the pieces is called the Queen. The Queen is ultimate piece of the chess board being able to move in any direction and for any number of squares.2

How many ways can eight queens be placed on a chess board so that no other queen will attack another?3 A queen can attack any other queen along the eight paths which extend from queen. These paths are vertically, horizontally, and diagonally.4

Development

Some observations must be made before attempting to solve this problem. Since there is a 8 by 8 chess board and 8 queens, then there must be a queen on each row. Also, a chess board has four black, and four white squares in each row. That would mean that half of the queens must be placed on black squares and half of the queens must be placed on the white squares.5

Each chess square will be labeled in the method shown in figure 1.

Figure 1

Each coordinate on the chess board such as (4,5) or (x,y) will be referred to as being S. Again, a queen can attack anyone diagonally, vertically, or horizontally.6 Mathematically, a queens diagonal dominance can be described by using the coordinates and adding A or B to x and y, and A being a variable in the domain 1-1.7

( X + A, Y + A ) right diagonal dominance of the queen

( X + B, Y- B) left diagonal dominance

( X, A ) vertical dominance of the queen?s domain

( Y, A ) horizontal dominance of the queen?s domain.

The first queen can be placed on the coordinate (1,1). With the above rules, the following is true in Table 1:

S A (X+A) (Y+A) (X,A) (A,Y)

(1,1) 1 2 2 (1,1) (1,1)

2 3 3 (1,2) (2,1)

3 4 4 (1,3) (3,1)

4 5 5 (1,4) (4,1)

5 6 6 (1,5) (5,1)

6 7 7 (1,6) (6,1)

7 8 8 (1,7) (7,1)

8 9 9 (1,8) (8,1)

Table 1

When any other queens are placed on the chess board, it cannot be placed on any of the coordinates listed in Table 1 because it would fall into this queen?s dominance. This would be true for any other queen that will be placed on the board.8

The second queen will have six squares from which it can be placed since two of them have been dominated by the first queen. The queen will be unable to be placed in the second row in the coordinates (2,2) or (1,2). The second queen is able to be placed on the squares with the coordinates (3,2), (4,2), (5,2), (6,2), (7,2), and (8,2). There are many factors that must be considered before placing the next queen. One goal is to leave as many squares open that do not vertically align with any other squares on any other rows. For example, if a queen is placed on (3,2) then it would leave four squares open on rows 3, 4, 5, 6, and 7. Because there are only 8 squares across a chess board, four being left open in five rows would result in vertically aligned rows.9 (Figure 2)

Many of the squares would be eliminated quickly because many are aligned vertically together resulting in less squares made available than 8 because once a queen is placed in any of the remaining "free" squares, then the "free" vertically above and below that queen would be dominated along with the diagonal pathways that the queen would dominate also. Since this is true, then the next queen must leave the minimum number of spaces greater than one that do not align vertically and increase in the number of squares available for each ascending row. Keep in mind that half of the queens must be placed on white, and half of the queens must be placed on black. Table 2 shows the resulting "dominated" spaces for each possible placement of the second