Chess math
Chess mathematics describes the mathematical examination of chess and related problems, mostly as a special sub-area of entertainment mathematics . Mathematical models for chess problems often come from graph theory or combinatorics .
Calculation of the playing strength and tournament plans
The most important application of mathematics for chess players is the calculation of the playing strength in the rating systems (see the articles Elo number , DWZ or Ingo number ). The creation of pairing plans for chess tournaments also requires the use of mathematical methods (see tournament form , slide system , Swiss system and Scheveningen system ).
Even if the "mathematics of tournaments" and the rating systems are mentioned in overall presentations, the area in the narrower sense does not belong to chess mathematics, because these methods can in principle be applied to other board games or two-person sports .
Exercises that combine chess and math
Paths of the pieces on the chessboard
A typical task is the knight problem : Find a way for the knight that leads him over the whole board without entering a field twice. These types of tasks are also set for generalized chess boards and fairy tale chess pieces .
Lineups of pieces on the chessboard
Often the consideration is based on the special geometry of the chessboard . Many puzzle tasks are about setting up figures according to set conditions:
independence
How many pieces of a certain kind can be placed on the chessboard so that none is in the sphere of influence of another, and how many possibilities are there for such a configuration? The best known such task is the women 's problem devised by the Bavarian chess master Max Bezzel .
Guardian figures
How many pieces of a certain kind do you need to dominate all free squares on the chessboard? Such a set of figures is called guard figures . If they also dominate all of the fields on which the figures stand, one speaks of dominating figures . If, on the other hand, no field on which a figure stands is dominated, they are called spanning figures .
In the case of the queen, five are required for both dominance and tension.
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Five dominant ladies
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Five exciting ladies
There are a total of 4860 lists of five guard ladies.
relation
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Another type of chess math problem is the relation problem. It can either mean that figures have a certain number of moves that they can make relative to one another, or have a certain position to one another and can change them.
A task of the first type would be roughly the following: ( Werner Keym, Die Schwalbe, April 1987 ): In a legal position with three stones, these move possibilities in a ratio of 1: 2: 3 to each other. After a white and a black move, the stones have a ratio of 2: 1: 3. The only solution is as in the following diagram.
The black king can move to a7, b7 and b8 (3 possible moves), the white king to d1, d2, e2, f2, f1 and castle (6 possible moves) and the rook can move along the h-file and to g1 or f1 (9 options). So the ratio of black king: white king: rook 3: 6: 9 is 1: 2: 3. After the moves 1. 0–0 and Ka8 – b7 the black king has 8 possible moves, the white king 4 and the rook 12. Now the ratio 8: 4: 12 is 2: 1: 3.
A task of the second type, on the other hand, could be as follows ( Werner Keym, Die Schwalbe, June 2004 ): The centers of the standing fields of three stones (in legal position) form the corner points of a triangle . You can reduce its area to 1/3 each by three different moves of the white king. What is the starting position?
The answer here would be wKe1 Th1 sKb3 with an area of 3 fields. After 1. Kd2, 1. Kf2 or 1. 0–0 the area would be reduced to one square. A graphic solution would look like this (see diagram, the explanation of the colors can be found at the bottom left).
Individual evidence
- ↑ Evgeni J. Gik: Chess and Mathematics. Thum Verlag, Frankfurt am Main 1987, ISBN 978-3-87144-987-1 , pp. 169-189.
Web links
literature
- Eero Bonsdorff, Karl Fabel , Olavi Rllhlmaa: Chess and Numbers . Entertaining chess math . 3. Edition. Walter Rau Verlag, Düsseldorf 1978, ISBN 978-3-7919-0118-3 .
- Evgeni J. Gik: Chess and Mathematics . Thum Verlag, Frankfurt am Main 1987, ISBN 978-3-87144-987-1 .
- John J. Watkins: Across the Board. The Mathematics of Chess Problems . Princeton University Press, Princeton 2004, ISBN 0-691-11503-6 .