What math is helpful for learning chess

Silke Pyroth

Breaking down an initial problem into sub-problems

This simple task has a lot of potential and can also be a challenge for adults. One possible strategy for solving this problem is to only look at parts of the chessboard at first. The laws found can then be transferred to the entire playing field.

The children of class 3a are excited when the teacher enters the classroom with a chessboard. “Do we want to play chess today?” Asks Tim. “No,” replies the teacher, “we want to find out how many squares there are on a chessboard”. “Oh, that's easy,” yells Yusuf, “that's 64 squares.” But this answer is wrong.
In most study groups - from 3rd grade to seminars with students - the question “How many squares can be seen on a chessboard?” Quickly yields two answers: “64” and “65”. Most children and adults alike react with astonishment when they say that there are more squares.
With great enthusiasm, the children are now asking how many squares there actually are. (KV 13) Different children quickly come up with the idea that several fields can combine to form a square and explain their discovery to the other children in the class. In the further course of the lesson, the students are allowed to work alone, in pairs or in small groups in order to exchange ideas about their approach. According to the principle of minimal help, the teacher holds back and does not provide any solution strategies.
In a 4th grade two different discoveries occurred almost simultaneously in this phase: One small group found out that there are squares made up of 2x2 small squares, another first discovered the squares made up of 7x7 fields. Regardless of whether a learning group initially sees one or more options, this is a good starting point for finding all the other square variants that are gradually systematically recorded on the blackboard. The reverse notation is also possible, i.e. starting with the large squares.
On the chessboard are:
  • 1x1 squares (one)
  • 2x2 squares (fours)
  • 3x3 squares (nines)
  • 4x4 squares (16s)
  • 5x5 squares (25s)
  • 6x6 squares (36s)
  • 7x7 squares (49ers)
  • 8x8 squares (64)
This approach has two advantages: Firstly, it is immediately apparent that all options are listed in full and none are missing. Second, the problem is broken down into several sub-problems that are easier to work on and on which all children can continue to work alone or in small groups in the following phase. To do this, you decide yourself whether you want to start with the smaller or larger squares and whether you want to share the work or work together.
The fact that they should not only find the squares that are next to each other, but also take into account the overlapping squares, proves to be difficult for some students (Fig. 1). Therefore it makes little sense to cut the chessboard templates completely into squares or to lay them out completely. It is particularly helpful if the children cut out correspondingly large squares from blank paper or a chessboard template and move them one by one on the chessboard. (Fig. 2)
It is also possible to use the foils enclosed with the material package with red frames for the corresponding square sizes or self-drawn frames on foils for moving. (Fig. 3) (KV 14)
In each case, the students should write down the number of squares they found. When moving, you have to record how often the respective square fits into a row - and ultimately onto the entire chessboard. Even weaker students often manage to count all possible positions in this way.
The comparison of the numbers for the individual square sizes is exciting. ...