Chess Problems

The following problems (and solutions) come from a Master of Basic Science project by University of Colorado at Denver student Kris Kulpa.

Domination problems.

In a domination problem you are asked for the minimum number of playing pieces (kings, queens, bishops or rooks) that can be placed on a given size board so that each "unoccupied" square is attacked by at least one piece. (Pieces may or may not attack one another).

For example, you can dominate a 3x7 board with 3 rooks (place one in each row, any column)

For domination problems, rooks are easy, on an nxm board you need the minimum of n and m. For other pieces, the question is more difficult.

1. Dominate a 3x12 board with 8 knights.
2. Dominate a 4x5 board with 4 knights.
3. Dominate a 3x6 board with 2 kings.
4. Dominate a 4x9 board with 6 kings.
5. Dominate a 5x7 board with 3 queens.
6. Dominate a 6x8 board with 4 queens.

   More difficult problems of this type:

1. Dominate a 6x8 board with 8 knights.
2. Dominate a 3x14 board with 11 knights.
3. Dominate a 5x12 board with 4 queens.
4. Dominate a 5x12 board with 10 bishops.

   Total Domination.

   A total domination is a domination as above but with the added restriction that every piece must be attacked by another piece.

   Easy:

1. Totally dominate a 4x9 board with 4 queens.
2. Totally dominate a 4x9 board with 10 knights.

   Slightly harder:

    

1. Totally dominate a 5x8 board with 4 queens.
2. Totally dominate a 4x6 board with 6 knights.
3. Totally dominate a 5x5 board with 6 bishops.

   Tougher yet: 

1. Totally dominate a 5x9 board with 4 queens.
2. Totally dominate a 6x7 board with 4 queens.

   Further information on these types of chessboard problems can be found in the articles:

   1. E.J. Cockayne, "Chessboard Domination Problems", Discrete Mathematics, 86(1990), pp. 13-20.

   2. M. Gardner, Mathematical Magic Show, W.H. Freeman & Co. , New York, 1977, pp. 127, 194-202.