Geometric Chess Puzzle Solutions

These puzzles will test your knowledge of chess and math. The original quiz is available at Geometric Chess Puzzles.

Start with a chess board and a large supply of chess pieces.

• In each puzzle you have a standard chessboard with algebraic coordinates, and 64 chess pieces of the same type.
• You need to place as many pieces as possible on the chessboard, so that no piece can capture another piece on the next move.
• You can place pieces on any squares you choose, but no more than one piece can be placed on each square.
• All puzzles have more than one solution.
• Two solutions are considered to be the same if they have the same pieces on the same squares, and each square is labeled with algebraic coordinates.
• After determining the maximum number of pieces that can be placed on the board in each puzzle, specify the number of different solutions which allow you to place that number of pieces on the board.
• Insert a chess diagram for each puzzle displaying one position which solves that puzzle.

In puzzles 1 to 4, there are 32 white pieces and 32 black pieces. These pieces move like standard chess pieces, and pieces of different colors can capture each other, but pieces of the same color cannot capture each other.

Puzzle 1) You have 32 white knights and 32 black knights. The number of white knights on the board must equal the number of black knights on the board. What is the maximum number of knights that can be placed on the board, so that no knight can capture another knight?

Answer 1) 24 white knights and 24 black knights is the maximum number that can be placed on the board. There are 24 different ways to place this number on the board. There are 6 primary solutions, shown in Diagrams 1 to 6. The other 18 solutions are obtained by taking the first 6 solutions, and rotating the positions 90, 180 or 270 degrees. It is not possible to place 25 knights of one color and 24 knights of the other color on the board.

Puzzle 2) You have 32 white bishops and 32 black bishops. The number of white bishops on the board must equal the number of black bishops on the board. What is the maximum number of bishops that can be placed on the board, so that no bishop can capture another bishop?

Answer 2) 32 white bishops and 32 black bishops is the maximum number that can be placed on the board. There are 2 different ways to place this number on the board. One way is to place all the white bishops on white squares and black bishops on black squares, shown in Diagram 7. The other way is to place all the white bishops on black squares and black bishops on white squares, shown in Diagram 8.

Puzzle 3) You have 32 white rooks and 32 black rooks. The number of white rooks on the board must equal the number of black rooks on the board. What is the maximum number of rooks that can be placed on the board, so that no rook can capture another rook?

Answer 3) 16 white rooks and 16 black rooks is the maximum number that can be placed on the board. There are 4900 different ways to place this number on the board. Two of these solutions are shown in Diagrams 9 and 10. It is not possible to place 17 rooks of one color and 16 rooks of the other color on the board.

To find a solution, select 4 rows and 4 columns on the chessboard, then color these 4 rows and 4 columns red. Place a white rook on each of the 16 squares which are an intersection of a red row and red column. Place a black rook on each of the 16 squares that are not red.

There are (8!)/[(4!)(4!)] = (8*7*6*5)/(4*3*2*1) = 70 different ways to choose 4 out of 8 rows. There are 70*70 = 4900 different ways to choose 4 out of 8 rows and 4 out of 8 columns.

Puzzle 4) You have 32 white queens and 32 black queens. The number of white queens on the board must equal the number of black queens on the board. What is the maximum number of queens that can be placed on the board, so that no queen can capture another queen?

Answer 4) 8 white queens and 8 black queens is the maximum number that can be placed on the board if the number of white queens is equal to the number of black queens. There are 176 different ways to place 8 white queens and 8 black queens on the board. There are 16 different ways to place 9 queens of one color and 8 queens of the other color on the board. It is not possible to place 9 queens of each color on the board.

There are 48 different ways to place 8 white queens and 8 black queens on the board, so that the 8 white queens are all on the same color squares, and the 8 black queens are all on squares of the opposite color as the white queens. There are 3 primary solutions, shown in Diagrams 11 to 13. The other 45 solutions can be created by modifying the first 3 solutions in one or more of these ways.

• Reverse the color of all queens on the board.
• Rotate the position by 90, 180 or 270 degrees.
• Create a horizontal mirror image of the position (so that material on squares a1 and h1 switch places).

There are 16 different ways to place 9 queens of one color and 8 queens of the other color on the board. Diagram 14 shows 8 white queens and 9 black queens. The other 15 ways to place 9 queens of one color and 8 queens of the other color on the board can be created by modifying the position from Diagram 14 in one or more of these ways.

• Reverse the color of all queens on the board.
• Rotate the position by 90, 180 or 270 degrees.
• Create a horizontal mirror image of the position (so that material on squares a1 and h1 switch places).

Diagram 14 has 9 black queens. 8 of the black queens are on black squares and one black queen is on e8, which is a white square. Leave the queen on e8 and remove one of the other black queens. This produces 8 different positions, each with 8 white queens and 8 black queens. These 8 positions can create 8*16 = 128 different positions by modifying each of the 8 positions in one or more of these ways.

• Reverse the color of all queens on the board.
• Rotate the position by 90, 180 or 270 degrees.
• Create a horizontal mirror image of the position (so that material on squares a1 and h1 switch places).

There are a total of 48+128 = 176 different ways to place 8 white queens and 8 black queens on the board, so that no queen can capture another queen.

In puzzles 5 to 8, all the pieces are white. These pieces move like standard chess pieces, except that white pieces can capture other white pieces.

Puzzle 5) You have 64 white knights. What is the maximum number of knights that can be placed on the board, so that no knight can capture another knight?

Answer 5) 32 knights is the maximum number that can be placed on the board. There are 2 different ways to place this number on the board. One way is to place all the knights on white squares, shown in Diagram 15. The other way is to place all the knights on black squares, shown in Diagram 16.

The bishop is the only chessman which always moves from a square of one color to a square of the same color. The knight is the only chessman which always moves from a square of one color to a square of the opposite color.

Puzzle 6) You have 64 white bishops. What is the maximum number of bishops that can be placed on the board, so that no bishop can capture another bishop?

Answer 6) 14 bishops is the maximum number that can be placed on the board. There are 4 different ways to place this number on the board, shown in Diagrams 17 to 20.

One main diagonal extends from square a1 to h8. There are 15 diagonals on the chessboard that are parallel to this main diagonal, including the main diagonal itself, the single square a8, and the single square h1. The other main diagonal extends from square a8 to h1, and there are 15 diagonals parallel to that main diagonal. Since bishops move diagonally, no more than one bishop can be placed on any diagonal in puzzle 6.

Puzzle 7) You have 64 white rooks. What is the maximum number of rooks that can be placed on the board, so that no rook can capture another rook?

Answer 7) 8 rooks is the maximum number that can be placed on the board. There are 40320 different ways to place this number on the board. One of these solutions is shown in Diagram 21.

To find a solution, every row and every column must have exactly one rook. There are 8! = 8*7*6*5*4*3*2*1 = 40320 different ways to place one rook on each row and each column.

Puzzle 8) You have 64 white queens. What is the maximum number of queens that can be placed on the board, so that no queen can capture another queen?

Answer 8) 8 queens is the maximum number that can be placed on the board. There are 4 different ways to place this number on the board, shown in Diagrams 22 to 25.

If all the queens from each of the 4 solutions are placed on the same chessboard, they produce the geometric pattern shown in Diagram 26.