The.Algorithm.Design.Manual.Springer-Verlag.1998

War Story: Covering Chessboards
Next: Heuristic Methods Up: Combinatorial Search and Heuristic Previous: Bandwidth Minimization
War Story: Covering Chessboards
Every researcher dreams of solving a classical problem, one that has remained open and unsolved for
over a hundred years. There is something romantic about communicating across the generations, being
part of the evolution of science, helping to climb another rung up the ladder of human progress. There is
also a pleasant sense of smugness that comes from figuring out how to do something that nobody else
could do before you.
There are several possible reasons why a problem might stay open for such a long period of time.
Perhaps the problem is so difficult and profound that it requires a uniquely powerful intellect to solve. A
second reason is technological - the ideas or techniques required to solve the problem may not have
existed when the problem was first posed. A final possibility is that no one may have cared enough about
the problem in the interim to seriously bother with it. Once, I was involved in solving a problem that had
been open for over a hundred years. Decide for yourself which reason best explains why.
Chess is a game that has fascinated mankind for thousands of years. In addition, it has inspired a number
of combinatorial problems of independent interest. The combinatorial explosion was first recognized in
the legend that the inventor of chess demanded as payment one grain of rice for the first square of the
board, and twice the amount of the ith square for the (i+1)st square, for a total of
36,893,488,147,419,103,231 grains. In beheading him, the wise king first established pruning as a
technique for dealing with the combinatorial explosion.
In 1849, Kling posed the question of whether all 64 squares on the board can be simultaneously
threatened by an arrangement of the eight main pieces on the chess board - the king, queen, two knights,
two rooks, and two oppositely colored bishops. Configurations that simultaneously threaten 63 squares
have been known for a long time, but whether this was the best possible remained an open problem. This
problem seemed ripe for solution by exhaustive combinatorial searching, although whether it was
solvable would depend upon the size of the search space.
Consider the 8 main pieces in chess (king, queen, two rooks, two bishops, two knights). How many ways
can they be positioned on a chessboard? The trivial bound is
positions. Anything much larger than about positions would be unreasonable to search on a modest
computer in a modest amount of time.
Getting the job done would require significant pruning. The first idea is to remove symmetries.
file:///E|/BOOK/BOOK2/NODE90.HTM (1 of 3) [19/1/2003 1:29:30]