Neurobics

Chess Problems

Russell Dear

There are innumerable conundrums related to the game of chess and they feature in many books on recreational mathematics. Some of the conundrums are concerned with the chess pieces and how they move, others with the configuration of the board. They come loosely in four categories:

(1) End Games: Puzzles related to the rules of chess itself. For example, when the pieces are placed in various positions on the board, how many moves does it take for one player to checkmate the other?

(2) Placement Problems: For example; what is the largest number of chess pieces of a particular type that can be placed on the board so that no two can attack each other? Solve the problem for knights. Another placement problem is to find the smallest number of pieces of a particular type that can be placed on the board so that all unoccupied squares are under attack by at least one of the pieces. Again, you might like to solve this problem for knights.

(3) Tour Problems: To find a path of a particular chess piece that allows it to visit each square once and only once. The knight's tour alone has generated bookloads of literature on the subject.

(4) Problems and games related to the board: These often take the form of puzzles on different size chess boards. A fascinating chessboard cutting problem is to find how many different ways a board can be cut in half along the lines of the board (the edges of squares) so that the two halves fit on top of one another exactly. A four-by-four "chessboard" can be dissected like this in just six ways and a six-by-six in 255 ways. I'm not sure of the number of solutions of the full size eight by eight board but it will be very large. You might like to solve the problem for the five-by-five board with its centre square missing.

Here are some related problems you might like to think about.

(5) One Check Chess: The game is played exactly like conventional chess except that the game is won by the first player to place his or her opponent in check (not checkmate). Played intelligently white, who has the first move, can always win. The game can be over in five moves using only knights.

(6) A Chinese friend was describing an early version of the game of chess. The board size was the same as today but one of the pieces was different. It was called a gudgeon. It moved like a rook , parallel to the sides of the board, but only one square at a time. Apparently an old Chinese puzzle was to determine the shortest path made by a gudgeon when visiting all the squares on the board. See if you can work it out.

Russell Dear is a Mathematician living in Invercargill