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On Chess: Chess And Mathematics

Grandmaster Pepe Cuenca with students at John Burroughs School.
Crystal Fuller | St. Louis Chess Club
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As a professional chess player and Ph.D. in applied mathematics, I have always been fascinated by the relationship between the two disciplines.

What does chess require? Concentration, planning, patience, self-control (playing fast does not pay off), conduct rules, mistake learning, etc. Therefore, learning chess might affect the ability to concentrate, memory, other types of executive functions, as well as increasing intelligence and problem-solving skills.

The relationship between chess and mathematics in seen in a number of ways:

  • Chess promotes thinking skills of higher order
  • Analysis of positions has a lot in common with mathematical problems
  • Correlation: to decide what piece is best to sacrifice at a certain point
  • Introduces a coordinates system
  • Introduces geometric concepts (files, rows, diagonals)
  • Requires constant calculation
  • Develops visual memory
  • Spatial reasoning skills
  • Capacity to predict and anticipate consequences

Geometry and chess

The final stage of a chess game — the endgame — is very important. Geometry plays a very important role. 

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Credit St. Louis Chess Club
Figure 1. Rule of the square

Even during the Middle Ages, good chess players employed simple geometric rules to figure out, with a simple glance at the chess board, what would be the result of the encounter. As an example, we can talk about the famous “rule of the square.” (Figure 1).

In this figure, we can appreciate that the white player has a King and a pawn versus a King. The rule of the square is made to know whether if the black King can stop the passed pawn or if this pawn will promote. This way, chessmasters, drawing a simple mental square on the board, know the game result without needing to calculate move by move.

Similar to this, we have the “rule of Bahr.” (Figure 2).

Two arrows are drawn diagonally. If they meet, white wins the game. If they do not, the game will end up in a draw.

Mathematical problems on the board

figure2.png
Credit St. Louis Chess Club
Figure 2. Rule of Bahr

During the Renaissance, the popularity of chess raised exponentially in Europe. Different chess schools were created, and personalities and scientists form that era began to be interested in the game. This growth motivated many mathematicians to solve mathematical problems over the board.

The Eight Queens Puzzle 

Proposed by Max Bezzel in Germany in 1848, the eight queens puzzle (Figure 3) is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other. Cantor and Gauss were two important scientists interested in this problem. 

This problem was generalized to a NxN chess board. The correct solution was found in 1972 with the help of computers and backtracking; 92 solutions were found in total where 12 of them are linear independent (Ramirez, 2004).

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Credit St. Louis Chess Club
Figure 3. The eight queens puzzle

The Knight’s tour problem

The object of the puzzle is to find a sequence of moves that allow the knight to visit every square on the board exactly once. It is a direct mathematical problem, related to the Hamiltonian path problem in graph theory. It appeared for the first time in arabic manuscripts in the 9th century and was very popular among mathematicians from the 18th century due to all possible different solutions. Euler presented a very famous solution in the Berlin Academy of Science in 1759 based on the premise “divide and conquer.”

Chess as a tool for education

It is still an open question whether learning chess can improve students' scores on mathematical tests. In 2015, an experiment was tried in Aarhus, Denmark (Kamilla Gumede et al, 2015) in which researcehrs tried to measure this with different students.

Boruch (2011) and Berkman (2004) also discuss explicitly the relation between chess and mathematics.

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Bart (2014) concluded that to play chess properly, one needs understanding and evaluation of positions taking into account piece mobility patterns, requiring fluent intelligence and the ability of concentration. To evaluate possible moves, critical thinking is required, and that is why it increases cognitive skills.

In chess, patterns develope very often. One gets used to certain types of positions, or tactical motifs. This can help finding sequences of numbers and shapes in certain mathematical exercises.

Pepe Cuenca is a Spanish chess Grand Master, a civil engineer and PhD in applied mathematics.

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