The Problem

A classic problem that we can solve using backtracking is the Eight Queens problem. This difficult problem deals with placing eight queens on a chessboard such that no two queens are attacking each other.

The game of chess is played on a board containing 64 squares of alternating color. Two players take turns moving a set of pieces on these squares. The object of the game is to capture the opponent's most important piece, the "king." While the king is the most important piece, the "queen" is the most powerful piece. In the game of chess, queens can "attack" or "capture" other pieces in two different ways. First, a queen can attack pieces on those squares that are in the same row or the same column as the queen. Second, a queen can attack pieces occupying the squares that run diagonally through the square that the queen occupies. Figure 2 shows the two different ways that queens attack. Note that the squares highlighted in red indicate the squares that the queen in the figure can attack.

Figure 2 Queens attack in two different ways

Combining these two methods together, we see (again in red) all the squares on a chessboard that a queen can attack in Figure 3. We consider the queen in Figure 3 to be "attacking" any piece that occupies a red square.

Figure 3 All the squares a queen can attack

A queen is considered the most powerful chess piece since it attacks the largest number of squares out of any other piece. Are queens so powerful that eight of them cannot be placed on a board without any two of them attacking each other? Spend some time and see if you can create a solution. If you do not have access to a chessboard, printing a copy of this page will give you, from the following figure, an empty board and eight queens.

Figure 4 An empty board and eight queens

Remember, when considering a solution to this problem you must make sure not to place two queens in the same row or in the same column. To complicate matters, you also cannot place two queens on the same diagonal. An example of a non-solution appears in Figure 5. This is a non-solution since the queen in the upper left corner attacks the queen in the lower right corner, and vice-versa.

Figure 5 A non-solution