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Case-lesson “Chess – combinatorial thinking”
Case-lesson “Chess – combinatorial thinking”

Category: Sport

Level (grade): 8-10

Subject: Chess

Objective: To give combinatorial thinking skills, cultivate the will to win and character.

What information is waiting for me here?

  • Why are chess players considered to be the smartest?
  • Is chess a kind of sport, science or art?
  • Is it possible to apply chess strategies in politics, economy and everyday life?
  • Who is a better chess player: a human or a computer?
  • Is chess really the most calm and balanced kind of sport?
10 scans of the subjects, phenomena and practices:

For many people chess is a model of action, as there are a lot of different pieces involved in the game, each of which has its own trajectory of motion. In addition, there is an opponent whose moves are difficult to predict accurately. And chess clock that forces you to think faster. This is why a chess player must be a far-sighted wise strategist and a cunning tactician simultaneously.

Is it possible to use the experience of chess “battles” in state leadership, corporate wars or family life?

Today we will go on a virtual tour in the world of chess, which has already become a sport-science-art simultaneously. This combination is the main combination of the game, which helps people find winning ways not only in sports but in business and life as well.

There is an ancient legend that ascribes the creation of chess to some Brahmin. For his invention, he asked the Indian king NOT for jewelry or princess but insignificant at first glance reward: as many wheat grains as will fit onto the board, if the first cell contains one grain, the second – two grains, the third – four grains, the fourth – 8, the fifth – already 64… The king was surprised and delighted that he asked for such a trifle. But after a while he realized how severely he had miscalculated! Wheat and chessboard problem is one of the classic mathematical problems. On each following cell the number of grains double.

This way the number of grains increases in geometric progression until the 64th cell:


What will result in 18.446.744.073.709.551.615 grains.

This sequence is also called exponential.

There are few variations of this problem. One of them is the “Second Half of the Chessboard» by Ray Kurzweil. He noted that at the time when the number of grains is significant on the first half of the chessboard their number on the second half will be much greater than the chessboard itself.

On the first half we will get 1 + 2 + 4 + 8 + 32 + ... + 2147483648 grains, i.e. only 4.294.697.296 grains (it is approximately 100.000 kg of grain if the weight of one grain is 25 g). It is almost the same amount as an annual rice harvest in India.

The number of grains on the second half will be 232 + 233 + 234 ... + 263 or 9.223.372.036.854.775.808 (230.584.300.921 tn).

Tasks like this are used in economics to develop strategies that take into account rapidly growing factors. That is, when the rise of one factor will affect the strategy of the entire enterprise.

Perhaps the Brahmin owned good combinatorial thinking, which enabled him to outwit the king, who could think only “linearly” and thus blundered.


On one hand, it can be natural symmetry, which appears in the course of a chess game, and on the other hand – the one that is used in chess problems or sketches. There are different types of symmetry. The most common ones are central and axial. On the chessboard the axial symmetry’s axis is introduced by a line that separates left and right flanks of the board or the upper and lower parts. The central symmetry requires pieces to be placed symmetrically relatively to the central square.


Once upon a time a novice player has figured out how to play a draw with a grandmaster. He’s challenged the grandmaster to a match, and began to make symmetrical moves regarding his moves. Grandmaster has made a move with a pawn, putting it 2 fields further, and the novice has done the same. Grandmaster has moved a knight and the newbie as well. This has been going on for quite a long time. Complete symmetry. Only our loser-inventor hadn’t taken into account that that when the grandmaster mated him, the mate in response became impossible.

Coordinate system


A chessboard also has coordinates. In professional competitions records are maintained using the notation of figures and these coordinates.

The geometry of a chessboard


During the game chess players think about combinations and calculate their options, examining the board, and ignorant players firstly speculate like “the pawn goes here, the king goes there”. Herewith, they frequently get confused and, finally, miscalculate. Still, the result of a match is easy to estimate with the help of the “rule of the square” at the end of the match: it is enough to figure out whether the black king is able to get to a pawn square in one move. If not, the white have an advantage.

If you read “Alice in Wonderland”, you could find not only the impetuous flight of the author's imagination, but also numerous life combinations, in which Alice had come into. And it’s all only because the author, Lewis Carroll, was a talented mathematician and chess player. In him, precisely, as in author, the genial combination of science-sport-art have been embodied.

Chess can also combine mathematics and composing activities. After all, there are chess compositions – creating problems using chess pieces and chessboard. Whoever is solving the problem is given a chess position and he or she needs to figure out a winning combination of moves in a certain direction (this way approaches inherent to combinatorial mathematics are necessary).

What “value” do pieces have? Since they are all different, sometimes they are “changed”, and you should be aware of their values to avoid miscalculation. For example, if you give up a queen for a pawn, you can quickly lose, and such exchange would be unequal. It is generally believed that the queen is worth 9 pawns, rook – 5 pawns, bishop and knight – 3 pawns each. Owing to these different “prices” a lot of combinations appear; this is why chess players calculate so many moves and choose the best combination.

In combinatorial thinking not only the “price” of pieces is included, but the evaluation of positions as well. At this stage a novice chess player starts pondering where and what piece to move. “Advanced” ones contemplate about how their opponents will answer to that. Experienced chess players calculate 3-5 moves ahead. And grandmasters calculate not moves and pieces, but what position they want to come to in 10-15 moves, and whether it will be a winning one. Life and business work similarly to this.



1. Compose a chess combination with mate in two and three moves.


According to the legend of grains on the chess board, the birthplace of chess is India.

The first name of chess was “chaturanga”. In the VI century chaturanga was borrowed by Arabs. In the Arab East chaturanga was slightly modified: the number of players fixed on two, both of them got two sets of pieces. Dice were abandoned in favor of doing strictly one move in rotation. The victory was considered to be achieved not by “taking” all of the opponent's pieces, but by mating or stalemating. Approximately in the year 820 chaturanga appeared in Rus, migrating there directly Persia through the Caucasus and Khazar Khaganate.

“Scene from the Life of the Russian Tsar” Schwarz V. G., 1865.

In the VIII-IX centuries, during the Arab conquest of the south Europe, chess came to Spain, and then, in a few decades – in Portugal, Italy and France.

Charles Webb

Roberto Raimondi

The game quickly gained the sympathy of Europeans. By the XV century it had been already known in all countries of Europe and Scandinavia. European masters had continued the improvement of the rules and, after all, had turned chess into a modern mind game.


How have chess and society’s attitude to them been changing?

During the archaeological excavations in Samarkand figurines made of ivory were found (VII - VIII cc.) – The most ancient chess. They portrayed the shah, horseback and foot warriors in miniature. But Islam prohibits certain images, so people have started to make the chess pieces in form of abstract shapes. And they have also begun to use a cheaper material – clay. This has contributed to the spreading of the game.

Iranian ceramic chess with stylized pieces.

In the XI - XII cc. chess was a common pastime in feudal nobility. The game was even included in the program of Knighthood training. Board games, widespread in those times Europe, were games of chance played for money. The Roman Catholic Church even tried to ban it as harmful gambling.

The first international chess tournament was held in London in 1851. Andersen became its winner. But the first official match for the world championship was held in 1886 in the United States between Steinitz and Zukertort. Steinitz became the first world champion. In 1924 the World Chess Federation (FIDE) was founded. Since 1927 the world championship competitions has started being conducted among women. The first champion was Vera Menchik.

The number of types of modern chess is growing. This is due to the fact that classical chess has unexpectedly begun to lose popularity. Partly this can be attributed to the creation of computer programs, what has downgraded the role of human intelligence. Besides, if earlier people used to fill their leisure time with chess, now it is replaced by computer games. And if in the previous times the prize fund of the match for the world champion title ranged from 2 to 5 million dollars, now it is several times less. In chess, players also experiment with timing. Earlier each player was given 2 minutes for a match; now rapid chess is played up to 30 minutes, and blitz – up to 5 minutes. Chess players who can quickly calculate their options and run the risk of using combinations have the advantage in short matches, since it is difficult for their opponents to unravel the course of the combinatorial thought of such players. Combinatorial thinking shows also in the fact that the player is trying to get the opponent in

to time trouble (lack of time) to force them to make mistakes. Or get into the zugzwang (moves limitations), to compel them to make unprofitable moves. Business chess is a team variation of the game, offered by a Moscow doctor Hrachya Hovakimyan in 1992.

According to the author of business chess, their advantages are:

• team discussion and collective decision-making process;

• increasing the dynamism of the game;

• visibility of all stages of chess players considerations, their evaluations of positions, calculation of options, choosing of alternatives, etc.

Battle chess is gaining popularity. It simulates a combat situation, when enemy’s dislocation and capabilities are unknown. Prior to the “battle” opponents arrange pieces on separated parts of a chessboard. Then they join the halves. Nowadays, there are portals for playing different types of chess, including “battle”, in the Internet in online-mode.

Playing is available, for instance, on chessok.net/chessonline/ or www.chess.com/?gclid=CIWP0cz8-MgCFSWzcgod8fEH2A


There are about 60 variations of chess. Their rules differ slightly.



Variations of chess game in the world

Spreading of chess in the world:



1. How had the rules of classical chess been changing since their invention and until these days?

Choose any variation of chess game and prepare a report on its national characteristics.


Few people realize how important are the material from which the pieces are made and their sizes. So many checkmates have been recorded due to the fact that a piece has fallen or has been touched at the wrong time! Therefore, the usual classical chess pieces are the result of gradual perfecting. On one hand unification of appearance, optimizing weight and size (the ratio of shape and weight, weight and size).

Chess pieces meet the following requirements: reliability and practicality, smooth surface with optimum roughness, low electrical conductivity, and low swelling ability.

After hundreds of tests and agreements with chess players, in 1849 Nathaniel Cook registered ownership of a new design of chess pieces, and called them “Staunton”, after the man who was actively contributing to their nascence, and later was handling the new pieces design’s advertising campaign, with emphasis on comfort, expressiveness and simplicity of new shapes. The new chess set was so popular that later it became a standard one, and today “Staunton” chess set is recommended for international competitions.

The accepted design of modern chess – Staunton’s design

Beside classic chess you can find other unusual designs: chess pieces made of plastic (polymer material), precious metals, alloys. They can be both a work of design art and elements of landscape design.



As at the incipience of the game, when the replacement of valuable pieces with the clay ones contributed to the spreading of chess game, in today's world, the use of plastic allowed to reduce the cost of chess sets. Plastic is an artificially created material formed from a polymer.

Polymers are natural and synthetic compounds whose molecules consist of a large number of repeating identical or different atomic groups. These groups are interconnected by chemical or coordination bonds into long linear or ramified chains. The structural units constituting polymers are called monomers.

The advantage of polymers lies in their high manufacturability, since during their production machining stages are excluded. Plastics are in the liquid shape at high temperatures (plasticity) and in the solid at normal temperatures. Here is an injection molding machine and an example of the chemical formula of the polymer chain:


In addition to the unification of chess a familiar attribute of the chess game – a chess clock – has also been changing.

The time control has been being used since the XIX century. Initially for this purpose people were using an hourglass. In 1883, Wilson built the first chess clock. Any chess clock has two time mechanisms, which turn on in rotation. There is also a special small red flag that falls when the time is out, which means losing of the game. However, in order to not let the time pressure become a major cause of losing, chess clocks are now designed so that each time you press the button 10 seconds for the next move are automatically added.

There are also clocks, in which clock hands go backwards so that players can see how much time they have left.


Try to do your homework on all subjects with the chess clock. For example, set the timing on 10 minutes for 2 exercises or 3 tasks and win this competition with time. With a frequent “time” training you will develop the ability to always be in time. This is how the sense of time is trained.

Chess competition is a fascinating event because it is a testament to high intelligence of players. The main institution, engaged in organizing and handling chess tournaments, is the International Chess Federation.

Can we consider chess a sport? Yes, because sport envisages physical and emotional development, training of spirit and persistence.

It is believed that sport should always be based on physical development. In chess, as a sport, the main idea is the development of the combinatorial and tactical thinking. Also, oddly enough, a chess player should be in a good physical shape to withstand many hours of training and competitions, a significant emotional load during the game. Consequently, in chess, in the core of development lies operability and self-organization.



1. Think about whether it is reasonable to include chess in the Olympic sports list? Give proofs.
In recent years, computers have made significant progress in battling humans. The first sensation was a victory of a chess computer “Deep Blue” over Garry Kasparov in 1997.


In October, 2002, Vladimir Kramnik played a draw with “Deep Fritz” program. Kramnik won the second and third games, and the computer the fifth and sixth games. The first, fourth, seventh and eighth games were played drawn.

The biggest victory of computers occurred in 2004-2006 yy. In 2004, two games against a world champion Ruslan Ponomariov were won by a chess computer “Hydra”. In 2005, the same computer “Hydra” beat Michael Adams in a match, consisting of six games. At that time Adams held the seventh line of the world ranking.

The algorithm of the computer has been adjusted to a certain person. At the same time a grandmaster didn’t know how the machine worked. Periodic restarts and changes in the program during the games indicate that the computer could not cope without a team of developers.

Unlike human chess computer calculates millions of operations per second. However, computers are still far from full reproduction of the human mind. The thing is, the computer simply enumerates all the possible moves and combinations, while human immediately rejects losing and weak ones. However, computer programs have already learned to filter moves and positions.

Does human have advantages over computer in chess?

Computers are prevalent over humans in short tactical maneuvers. Especially dangerous in such cases is the queen, which is perfect for short-term maneuvers. This is why in when playing against the computer, people often attempt to induce a program to exchange the queens. This occurs, for example, when a player deliberately worsens his position in the beginning of the game. The computer sees it as advantageous situation.

If the program evaluates its position as a superior one, then, most likely, it will exchange pieces. This is beneficial to human. Of course, programmers have learned about such “tricks”, and consider them in the latest versions of their programs.

On the other hand, players have to play against the computer with long complicated maneuvers. Such maneuvers are difficult for the program to predict and assess. So, we are talking about competition of “strategic vision” and “tactical steps”.

As an example – here’s an option how to beat the computer with “Kasparov program”.


It has long been known that chess helps to activate the cognitive functions, development of memory, attention, and areas of the brain that are responsible for making decisions. In addition, this game is quite emotional. It is no accident that chess is considered a type of sport that requires total concentration and involvement of all mental faculties.


All these functions are directly related to the part of the brain that is responsible for emotions.

Emotions, in their turn, influence the decision-making and memory functioning. The reverse statement is also true: the game of chess contributes to emotional and psychological changes that are not always visible to an outside observer.

But emotions of a chess player are usually associated with changes in parameters that can be controlled. This applies, in particular, to body temperature, nature of the heart beat. In the latter case the changes in heart rate and the occurrence of a cardiac arrhythmia are possible. Stress, occurring in the course of the game of chess, causes repeated changes in heart rate. The sensors, which record heart rate, allow instant receiving of information, which indicates the emotional state of the player during the game.

Chess combines the ability to affect both the psycho-emotional and physical components. On one hand, emotional loads promote the development of stress. But stress itself also triggers the physiological mechanism of adaptation.

Adaptation is a biological function of stress. That is, stress, as a special psycho-physiological state, provides protection of the body against threatening and destructive influences (mental or physical). This involves the nervous, hormonal and other body systems.

So, stress is a normal reaction of a healthy human, the defense mechanism of the biological system.


Literature and cinematography

The appearance of the chess pieces in books is often a kind of quality mark. Here are examples of fiction, where chess is one of the central plot figures.


“The Ultimate Secret” by Bernard Werber

The iconic French author invites readers to dive into the mysteries of human’s mind. A brilliant scientist plays a crucial game of chess with a computer brain. The world champion title is ahead. Mankind wins. Hall applauds. And suddenly the unexpected death of the winner occurs. Journalists begin their own investigation.

“The Luzhin Defense” by Vladimir Nabokov

Brilliant chess player Luzhin lives in a wonderful world of the ancient game, its harmony and strict logic have captured him. Life stops miraculously on an unfinished game…

“The Eight” by Katherine Neville

An ancient knowledge is buried within the walls of an ancient monastery in the south of France. The power concealed in them, goes beyond the laws of nature and human understanding. They are encoded in ancient chess pieces, and for their possession a bloody warfare between the sinister figures of the era of Great Terror begins.


1. What other books that mention the chess subject have you met? Make a short presentation on one of them.


Chess play an important role in many works of cinematography and other branches of art. In addition to a vast number of works, where the chess game in one form or another is mentioned in passing, there are some in which chess is the basis of the plot. Sometimes chess play an important role in the part of the plot, or it is just noticeably distinguished by an author.

Movie “Revolver”. Chess play the role of a landfill for disclosure of human psychology, his actions, and weaknesses in situations where there is a game, risk and competition. Movie “Searching for Bobby Fischer”. The film tells the story of a famous chess player Joshua Waitzkin’s (Pomeranz) childhood and of the conditions in which his stellar career had begun.


Movie “Knight Moves”. During the international chess tournament, someone is killing young women. A leader of the tournament – Peter Sanderson becomes a suspect. The police are trying to solve a complex problem by “playing a chess game”. Meanwhile, the killer is teasing Sanderson, giving him new puzzles with every new victim. The champion should solve them all. Movie “Harry Potter and the Philosopher’s Stone”. Magic chess (rules are the same but the pieces are animated by magic).


Are you familiar with the stills shown in the section? What other movies that mention chess, have you watched?


Chess is a kind of human activity, which is extremely rich in outstanding, extraordinary personalities.

World champions since 1886:


The world shows respect for chess players because they embody the power of intellect. And some players are famous not only for chess. For example, Vasily Smyslov was a highly professional baritone singer in the Bolshoi Theatre while being the world chess champion.

Mikhail Botvinnik was a professor of electrical engineering, who had optimized the electricity network of the country.

Garry Kasparov has graduated from a linguistic university and became a well-known politician and businessman. He has earned a lot of “chess” money and began to wonder where to invest them. By using his filigree combinatorial thinking he has opted for an “eternal and lucrative” global business – cargo air transportations. And he hasn’t been out-of-pocket.

Unlike other sports, chess champions are always “in the public eye”. People are interested in their biographies because they are rich in many intellectual victories, unexpected combinatorial moves and uncommon abilities of chess players.

For example, Capablanca is an outstanding Cuban chess player who was the world champion from 1921 to 1927.


Jose Raul learned to play chess very early, when he was four, by watching his father playing. At nine he began to attend a chess club in Havana weekly. In 1899, Juan Corzo became a Cuban national champion whom Capablanca has challenged. The match, which was handled in 1902, was won by a young confident Cuban.

In 1904, Capablanca went to New York, devoting most of his time playing chess in a prestigious Manhattan chess club. Being confident in his own strength, he was already making plans for the international chess career in Europe, where the best tournaments were held and where the strongest chess players of the world used to meet.

In August 1922 he easily won a large international tournament in London. During the tournament, Capablanca and 7 other leading players of that time agreed upon the rules for future matches for the world champion title. In November, Capablanca was ready to play with Akiba Rubinstein, who, however, failed to raise enough money to host the match. The year before, Alexander Alekhine tried to negotiate with American businessmen on financing of his match against Capablanca. They refused but did finance the tournament, which took place on March 16, 1924 in New York. On that tournament Capablanca won ten times, achieved nine draws and lost for the first time in eight years to Czechoslovak Richard Reti.

The following year, Capablanca played in the international tournament in Moscow, the first tournament, which was organized in the USSR. In that tournament the first Capablanca chess disadvantages flaunted themselves: he ignored the preparation for the game with an opponent, practiced infrequently, hadn’t studied the theory, because he trusted his experience and skills.


In chess, people earn money in different ways. Top grandmasters, with a coefficient higher than 2.700 Elo, receive tens or hundreds of thousands of dollars for participation in tournaments. However, ordinary players, on the contrary, spend money on the game by attending chess clubs, in some of which, for example, you have to purchase a subscription. All the more so, the clubs are present in most cities. On average, the monthly subscription costs 200 UAH per a person. But what does this cost consist of?


Which professions which are related to chess (excluding the game itself, of course), could provide a stable income? These may be chess journalists, programmers, organizers of chess tournaments. The most important thing is to learn to think combinatorially when there are a lot of pieces, rules, strategies and even two colors of pieces at your disposal. After all, one of the basic combinations in chess is represented by a slogan: “White move first and win”. We should aspire this in life.

Lesson summary


The contents


The results of the case-lesson can be complemented with findings of students



What three websites helped find important information?





To help student and couch:



Where can you get information for the case?






Location of the lesson:

The case-lesson takes place in the classroom.

It is possible to hold the lesson at museum or library.



Teams of boys and girls.

The score was:


  1.     What number of rice grains will appear on the 5th cell, if the

first one contains 3 grains?

  1.     How much will these grains weigh, if the weight of one

grain is 0,055 g?




Formulate 5 questions to the case (an individual task).



90 min (double lesson) 


The possibility to conduct the lesson with a student-understudy: 



The acquired knowledge and developed competence:

  •   Ability to find information on a topic and related to it quickly;
  •   Ability to use the acquired information creatively and practically;
  •   Receiving concrete mathematical knowledge (given in the scans)
  •   Ability to “construct” acquired knowledge basing on received data and its segments;
  •   Upgrade of intellectual and communicative skills.




The case contains links to sections and lessons which correspond

to the school program: geometric progression, symmetry,



Author of the basic case: Hrabowska Larysa Leonidivna


Participated in the case upgrading:


The End

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