A mathematical chess problem is a mathematical problem which is formulated using a chessboard and chess pieces. These problems belong to recreational mathematics. The most well-known problems of this kind are the eight queens puzzle and the knight's tour problem, which have connection to graph theory and combinatorics. Many famous mathematicians studied mathematical chess problems, such as, Thabit, Euler, Legendre and Gauss. Besides finding a solution to a particular problem, mathematicians are usually interested in counting the total number of possible solutions, finding solutions with certain properties, as well as generalization of the problems to N×N or M×N boards.

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Problems

1. The chess knight stands in the lower left corner of the board. Can he through a) 4; b) 5; c) 2013 moves to return to the original field?

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2. Is it possible to place the pieces in the cells of an 8×8 chessboard (in each cell there is no more than one piece) so that there are equal numbers of pieces in all verticals, but not equally in any two horizontals?
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3. On a 4x4 chessboard there is a piece - a “flying rook”, which moves in the same way as an ordinary rook, but cannot move to a square adjacent to the previous one in one move. Can she go around the entire board in 16 moves, standing on each square once, and return to the original square?

4. What is the largest number of a) rooks; b) can kings be placed on a chessboard so that they do not beat each other?
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5. Each cell of a 7x7x7 triangular board contains a beetle. At one point, each beetle crawled onto the cell next to it. a) Prove that at least one cell was free. b) What is the smallest number of cells that could be free? c) Come up with such a “crawling” of beetles so that as many cells as possible are empty.
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6. Arrange 32 knights on a chessboard so that each of them beats exactly two others.
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7. Is it possible to cut a chessboard into dominoes such that no two dominoes form a 2x2 square?

8. Is it possible to cut a chessboard into 15 vertical and 17 horizontal dominoes?

9. Prove that the number of ways to place 8 queens on a chessboard is even.

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10. Given a cube 4x4x4. Arrange 16 rooks in it so that they do not attack each other.
eleven.

11. On a 5x5 chessboard, the maximum number of knights was placed so that they did not beat each other. Prove that this arrangement is the only one.
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12. What is the largest number of squares that can be marked on a chessboard so that from each of them it is possible to move from each of them to any other marked square in exactly two moves of a chess knight?

13. Can you find a path for the knight to visit all 25 squares of a 5×5 chessboard exactly once?
Try drawing the board and numbering each square in the order the knight visits it. There is no need to return to the starting square - this is an open tour.

14. Can you find a sequence of moves for the knight to visit all 64 squares exactly once on a standard chessboard?
You can start from any square. Try to find either:

- open tour (the knight visits each square once, but does not return to the starting square).

- a closed tour (the knight returns to its starting square at the end).

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Where do you hold your classes?

We hold our classes online or on-site on Saturdays at our branch in Pimlico Academy, London.
You can find our timetable here.

What do you need to start learning online?

For lessons you only need a computer or phone with a microphone, camera and Internet access. Wherever you are - in London, Nottingham, New York or Bali - online lessons will be at hand.

When can I take the trial lesson?

You can get acquainted with the school at any time convenient for you. To do this, just leave a request and sign up for a lesson.

What should I expect from the trial lesson?

The trial lesson is a 30-minute online session designed to get a sense of how your child approaches mathematical thinking and problem solving. (In practice, it often runs a bit longer if the student is engaged!)

We typically explore a range of fun and challenging problems drawn from competitions. We adapt the difficulty based on how the student responds, aiming to make it both accessible and stimulating.

After the session, we’ll have a quick conversation with the parent to share observations and suggest a personalised path forward.

I can't attend class, what should I do?

It is OK, it happens! Students have the opportunity to cancel a lesson up to 8 hours before the scheduled time without loss of payment. So you can reschedule it for a convenient time, and the teacher will have the opportunity to

I don't have much free time, will I have time to study?

Learning can take place at your own pace. We will select a convenient schedule and at any time we will help you change the schedule, take a break or adjust the program.

How long is one lesson?

All classes last 1 hour.

Meet our team

Our teachers will tell you how to prepare for exams, help you cope with difficult tasks and win the Olympiad
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They will tell you about the pitfalls of exams and the most common mistakes, and explain how to avoid them

George Ionitsa

Founder &
Maths and Coding Coach

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