Mathematical chess problem
Chess mathematics refers to the mathematical analysis of chess and associated problems.
The field is very extensive of the tasks as well as of the involved mathematical theories forth - " The literature on chess mathematics is unmanageable. " ( Evgeni J. Quack ) - and can only be described by a list of examples.
The mathematical models for chess problems come from graph theory and combinatorics.
Calculation of the rating and tournament schedules
The most important for chess players of application of mathematics is the calculation of the rating in the rating systems (see the article Elo and Ingo number). Creating mating plans for chess tournaments also requires the help of mathematical methods (see tournament format, slip system, Swiss system and Scheveningen ).
Even if the " mathematics of tournaments " and the rating systems are mentioned in Overviews, the area does not belong in the strict sense for chess mathematics, because these methods can in principle be applied to other board games or two sports.
Tasks that combine chess and mathematics
Ways the pieces on the chessboard
A typical task is the Springer problem: without a field to enter Find a way for the knight, leads him all over the board, twice. This type of tasks is also provided for generalized chess boards and chess pieces for fairy tales.
Constellations of the pieces on the chessboard
Often the viewing of the particular geometry of the chessboard is based. Many puzzles act not erect figures according to specified conditions:
Independence
How many characters of a certain type can be put on the chessboard so that no standing in an area where another, and how many ways are there for such a statement? Best known among these is the task conceived by the Bavarian Chess Champion Max Bezzel queens problem.
Guardian figures
How many characters of a certain type are needed to master all free fields of the chessboard? Such a set of figures called guardian figures. Are you master all fields on which these figures are, it is called dominant figures. However, if no field on which a figure stands dominated, they are called spanning figures.
In the case of the lady are both dominance and chucking five needed.
In total there are 4860 lineups of 5 guard ladies.
Relation
A different kind of chess mathematical tasks are the tasks Relations. This can either be a question that characters have a certain number of trains that can make them relative to each other, or have a certain relationship to one another and can change them.
One object of the first type would be similar to the following: In a legal position with three stones these possible moves have each other in the ratio of 1:2:3. After a train the stones have a ratio of 1:2:6. The solution here would be like in the following diagram.
The black king can by a2, b2 and b1 pull (3 possible moves ), the white king to d1, d2, e2, f2, f1 and castle (6 possible moves ) and the tower can pull along the h-file and after g1 and f1 (9 possible moves ). Thus, the ratio of black king, white king, white rook 1:2:3. After castling 1.0-0 has the black king nor the possible moves by a2 and b2 ( 2), the white king to f2, g2, h2, h1 (4 ) and the white tower along the f-file and the first row, except g1 and h1 according to (12). Thus, the ratio would now be 1:2:6.
Could be a function of the second type, however, these are ( Werner Keym, The Swallow, June 2004 ): The centers of the state fields of three stones ( in a legal position) form the vertices of a triangle. One can reduce its surface area by three different features of the white king on 1/3. What is the starting position?
The answer would be here WKE1 Th1 sKb3 with a surface area of 3 fields. After 1.Kd2, 1.Kf2 or 1.0-0, the area would be reduced to a field. A graphical solution would be as follows (see diagram, the explanation of the colors found on the bottom left ).