Mathematics In The Game Of Chess Maths Essay

INTRODUCTION:

Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. Each player begins the game with sixteen pieces: one king, one queen, two rooks, two knights, two bishops, and eight pawns. The object of the game is to checkmate the opponent’s king, whereby the king is under immediate attack (in “check”) and there is no way to remove or defend it from attack on the next move. The game’s present form emerged in Europe during the second half of the 15th century, an evolution of an older Indian game, Shatranj. Theoreticians have developed extensive chess strategies and tactics since the game’s inception. Computers have been used for many years to create chess-playing programs, and their abilities and insights have contributed significantly to modern chess theory. One, Deep Blue, was the first machine to beat a reigning World Chess Champion when it defeated Garry Kasparov in 1997.

Matches between individuals took place as early as the 9th century. The tradition of organized competitive chess started during the 16th century. The first official World Chess Champion, Wilhelm Steinitz, claimed his title in 1886; the current World Champion is Viswanathan Anand from India. In addition to the World Championship, there is also the WomenHYPERLINK “http://en.wikipedia.org/wiki/Women’s_World_Chess_Championship”‘HYPERLINK “http://en.wikipedia.org/wiki/Women’s_World_Chess_Championship”s World Championship, the Junior World Championship, the World Senior Championship, the Correspondence Chess World Championship, the World Computer Chess Championship, and Blitz and Rapid World Championships (see fast chess). The Chess Olympiad is a popular competition among teams from different nations. Online chess has opened amateur and professional competition to a wide and varied group of players. Chess is a recognized sport of the International Olympic Committee and international chess competition is sanctioned by the FIDE. Today, chess is one of the world’s most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments. Some other popular forms of chess are fast chess and computer chess. There are also many chess variants which have different rules, different pieces, different boards, etc.

History

Iranian chess set, glazed fritware, 12th century. New York Metropolitan Museum of Art.

Knights Templar playing chess, Libro de los juegos, 1283.

Chess is commonly believed to have originated in North-West India during the Gupta empire, where its early form in the 6th century was known as caturaá¹…ga (Sanskrit: four divisions [of the military] – infantry, cavalry, elephants, and chariotry, represented by the pieces that would evolve into the modern pawn, knight, bishop, and rook, respectively). The earliest evidence of Chess is found in the neighboring Sassanid Persia around 600 where the game came to be known under the name chatrang. Chatrang is evoked inside three epic romances written in Pahlavi (Medium Persian). Chatrang was taken up by the Muslim world after the Islamic conquest of Persia (633-644) where it was then named shatranj, with the pieces largely retaining their Persian names. In Spanish “shatranj” was rendered as ajedrez, in Portuguese as xadrez, and in Greek as zatrikion (which directly comes from Persian chatrang), but in the rest of Europe it was replaced by versions of the Persian shāh (“king”), which was familiar as an exclamation and became the English words “check” and “chess”. Murray theorized that this change happened from Muslim traders coming to European seaports with ornamental chess kings as curios before they brought the game of chess. The game reached Western Europe and Russia by at least three routes, the earliest being in the 9th century. By the year 1000 it had spread throughout Europe. Introduced into the Iberian Peninsula by the Moors in the 10th century, it was described in a famous 13th-century manuscript covering shatranj, backgammon, and dice named the Libro de los juegos. Another theory contends that chess arose from the game xiangqi (Chinese Chess) or one of its predecessors, although this has been contested.

Get Help With Your Essay

If you need assistance with writing your essay, our professional essay writing service is here to help!
Find out more about our Essay Writing Service

___________________________________________________

Mathematics In The Game Of Chess

Legend has it that the game was invented by a mathematician in India who elicited a huge reward for its creation. The King of India was so impressed with the game that he asked the mathematician to name a prize as reward. Not wishing to appear greedy, the mathematician asked for one grain of rice to be placed on the first square of the chess board, two grains on the second, four on the third and so on. The number of grains of rice should be doubled each time.

The King thought that he’d got away lightly, but little did he realise the power of doubling to make things big very quickly. By the sixteenth square there was already a kilo of rice on the chess board. By the twentieth square his servant needed to bring in a wheelbarrow of rice. He never reached the 64th and last square on the board. By that point the rice on the board would have totalled a staggering 18,446,744,073,709,551,615 grains.

Playing chess has strong resonances with doing mathematics. There are simple rules for the way each chess piece moves but beyond these basic constraints, the pieces can roam freely across the board. Mathematics also proceeds by taking self-evident truths (called axioms) about properties of numbers and geometry and then by applying basic rules of logic you proceed to move mathematics from its starting point to deduce new statements about numbers and geometry. For example, using the moves allowed by mathematics the 18th-century mathematician Lagrange reached an endgame that showed that every number can be written as the sum of four square numbers, a far from obvious fact. For example, 310 = 172 +42 + 22 + 12.

Some mathematicians have turned their analytic skills on the game of chess itself. A classic problem called the Knight’s Tour asks whether it is possible to use a knight to jump around the chess board visiting each square once only. The first examples were documented in a 9th-century Arabic manuscript. It is only within the past decade that mathematical techniques have been developed to count exactly how many such tours are possible.

It isn’t just mathematicians and chess players who have been fascinated by the Knight’s Tour. The highly styled Sanskrit poem Kavyalankara presents the Knight’s Tour in verse form. And in the 20th century, the French author Georges Perec’s novel Life: A User’s Manual describes an apartment with 100 rooms arranged in a 10×10 grid. In the novel the order that the author visits the rooms is determined by a Knight’s Tour on a 10×10 chessboard.

Mathematicians have also analysed just how many games of chess are possible. If you were to line up chessboards side by side, the number of them you would need to reach from one side of the observable universe to the other would require only 28 digits. Yet Claude Shannon, the mathematician credited as the father of the digital age, estimated that the number of unique games you could play was of the order of 10120 (a 1 followed by 120 0s). It’s this level of complexity that makes chess such an attractive game and ensures that at the Olympiad in Russia in 2010, local spectators will witness games of chess never before seen by the human eye, even if the winning team turns out to have familiar names.

________________________________________________________

Rules

The official rules of chess are maintained by the World Chess Federation. Along with information on official chess tournaments, the rules are described in the FIDE Handbook, section Laws of Chess.[2]

Setup

Pieces at the start of a game

A

b

C

d

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

A

b

C

D

e

f

g

h

Initial position: first row: rook, knight, bishop, queen, king, bishop, knight, and rook; second row: pawns

Chess is played on a square board of eight rows (called ranks and denoted with numbers 1 to 8) and eight columns (called files and denoted with letters a to h) of squares. The colors of the sixty-four squares alternate and are referred to as “light squares” and “dark squares”. The chessboard is placed with a light square at the right hand end of the rank nearest to each player, and the pieces are set out as shown in the diagram, with each queen on its own color.

The pieces are divided, by convention, into white and black sets. The players are referred to as “White” and “Black”, and each begins the game with sixteen pieces of the specified color. These consist of one king, one queen, two rooks, two bishops, two knights and eight pawns.

___________________________________________________________________________

Movement

White always moves first. After the initial move, the players alternately move one piece at a time (with the exception of castling, when two pieces are moved). Pieces are moved to either an unoccupied square, or one occupied by an opponent’s piece, capturing it and removing it from play. With the sole exception of en passant, all pieces capture opponent’s pieces by moving to the square that the opponent’s piece occupies. A player may not make any move which would put or leave his king under attack. If the player to move has no legal moves, the game is over; it is either a checkmate-if the king is under attack-or a stalemate-if the king is not.

Each chess piece has its own style of moving. In the diagrams, the dots mark the squares where the piece can move if no other pieces (including one’s own piece) are on the squares between the piece’s initial position and its destination.

The king moves one square in any direction, the king has also a special move which is called castling and also involves a rook.

The rook can move any number of squares along any rank or file, but may not leap over other pieces. Along with the king, the rook is also involved during the king’s castling move.

The bishop can move any number of squares diagonally, but may not leap over other pieces.

The queen combines the power of the rook and bishop and can move any number of squares along rank, file, or diagonal, but it may not leap over other pieces.

The knight moves to any of the closest squares which are not on the same rank, file or diagonal, thus the move forms an “L”-shape two squares long and one square wide. The knight is the only piece which can leap over other pieces.

The pawn may move forward to the unoccupied square immediately in front of it on the same file, or on its first move it may advance two squares along the same file provided both squares are unoccupied, or it may move to a square occupied by an opponent’s piece, which is diagonally in front of it on an adjacent file, capturing that piece. The pawn has two special moves, the en passant capture, and pawn promotion.

Moves of a king

a

B

c

D

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

D

e

f

g

h

Moves of a rook

a

B

c

D

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

D

e

f

g

h

Moves of a bishop

a

B

c

D

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

D

e

f

g

h

Moves of a queen

a

B

c

D

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

D

e

f

g

h

Moves of a knight

a

B

c

D

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

D

e

f

g

h

Moves of a pawn

a

B

c

D

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

D

e

f

g

h

* Pawns can optionally move two squares forward instead of one on their first move only. They capture diagonally (black x’s); they cannot capture with their normal move (black circles). Pawns are also involved in the special move en passant (below).

Check

When a king is under immediate attack by one or two of the opponent’s pieces, it is said to be in check. A response to a check is a legal move if it results in a position where the king is no longer under direct attack (i.e. not in check). This can involve capturing the checking piece, interposing a piece between the checking piece and the king (which is possible only if the attacking piece is a queen, rook, or bishop and there is a square between it and the king), or moving the king to a square where it is not under attack. Castling is not a permissible response to a check. The object of the game is to checkmate the opponent; this occurs when the opponent’s king is in check, and there is no legal way to remove it from attack.

End of the game

Although the objective of the game is to checkmate the opponent, chess games do not have to end in checkmate-either player may resign if the situation looks hopeless. It is considered bad etiquette to continue playing when in a truly hopeless position. If it is a timed game a player may run out of time and lose, even with a much superior position. Games also may end in a draw (tie). A draw can occur in several situations, including draw by agreement, stalemate, threefold repetition of a position, the fifty-move rule, or a draw by impossibility of checkmate (usually because of insufficient material to checkmate). As checkmate from some positions cannot be forced in less than 50 moves (see e.g. pawnless chess endgame and two knights endgame), the fifty-move rule is not applied everywhere,[6] particularly in correspondence chess.

White is in checkmate

a

B

c

D

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

d

e

f

g

h

White is in checkmate. He cannot escape from being attacked by the Black king and bishops.

Stalemate

a

B

c

d

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

d

e

f

g

h

Stalemate if Black is to move. The position is not checkmate, and since Black cannot move, the game is a draw.

Time control

A modern digital chess clock

Besides casual games without any time restriction, chess is also played with a time control, mostly by club and professional players. If a player’s time runs out before the game is completed, the game is automatically lost (provided his opponent has enough pieces left to deliver checkmate). The duration of a game ranges from long games played up to seven hours to shorter rapid chess games lasting usually 30 minutes or one hour per game. Even shorter is blitz chess with a time control of three to fifteen minutes for each player, or bullet chess (under three minutes). In tournament play, time is controlled using a game clock which has two displays, one for each player’s remaining time.

________________________________________________________

Notation for recording moves

Naming the squares in algebraic chess notation

Chess games and positions are recorded using a special notation, most often algebraic chess notation. Abbreviated (or short) algebraic notation generally records moves in the format abbreviation of the piece moved – file where it moved – rank where it moved, e.g. Qg5 means “queen moves to the g-file and 5th rank (that is, to the square g5). If there are two pieces of the same type that can move to the same square, one more letter or number is added to indicate the file or rank from which the piece moved, e.g. Ngf3 means “knight from the g-file moves to the square f3”. The letter P indicating a pawn is not used, so that e4 means “pawn moves to the square e4”.

If the piece makes a capture, “x” is inserted before the destination square, e.g. Bxf3 means “bishop captures on f3”. When a pawn makes a capture, the file from which the pawn departed is used in place of a piece initial, and ranks may be omitted if unambiguous. For example, exd5 (pawn on the e-file captures the piece on d5) or exd (pawn on e-file captures something on the d-file).

“ScholarHYPERLINK “http://en.wikipedia.org/wiki/Scholar’s_mate”‘HYPERLINK “http://en.wikipedia.org/wiki/Scholar’s_mate”s mate”

If a pawn moves to its last rank, achieving promotion, the piece chosen is indicated after the move, for example e1Q or e1=Q. Castling is indicated by the special notations 0-0 for kingside castling and 0-0-0 for queenside castling. A move which places the opponent’s king in check usually has the notation “+” added. Checkmate can be indicated by “#” (occasionally “++”, although this is sometimes used for a double check instead). At the end of the game, “1-0” means “White won”, “0-1” means “Black won” and “½-½” indicates a draw.

Chess moves can be annotated with punctuation marks and other symbols. For example “”!” indicates a good move, “!!” an excellent move, “?” a mistake, “??” a blunder, “!?” an interesting move that may not be best or “?!” a dubious move, but not easily refuted.[1]

For example, one variant of a simple trap known as the ScholarHYPERLINK “http://en.wikipedia.org/wiki/Scholar’s_mate”‘HYPERLINK “http://en.wikipedia.org/wiki/Scholar’s_mate”s mate, animated in the picture to the right, can be recorded:

e4 e5

Qh5?! Nc6

Bc4 Nf6??

Qxf7# 1-0

________________________________________________________

Strategy and tactics

Chess strategy consists of setting and achieving long-term goals during the game – for example, where to place different pieces – while tactics concentrate on immediate manoeuvre. These two parts of chess thinking cannot be completely separated, because strategic goals are mostly achieved by the means of tactics, while the tactical opportunities are based on the previous strategy of play.

A game of chess is normally divided into three phases: opening, typically the first 10 to 25 moves, when players move their pieces into useful positions for the coming battle; middlegame, usually the fiercest part of the game; and endgame, when most of the pieces are gone, kings typically take a more active part in the struggle, and pawn promotion is often decisive.

Opening

A chess opening is the group of initial moves of a game (the “opening moves”). Recognized sequences of opening moves are referred to as openings and have been given names such as the Ruy Lopez or Sicilian Defence. They are catalogued in reference works such as the Encyclopaedia of Chess Openings. There are dozens of different openings, varying widely in character from quiet positional play (e.g. the Réti Opening) to very aggressive (e.g. the Latvian Gambit). In some opening lines, the exact sequence considered best for both sides has been worked out to more than 30 moves. Professional players spend years studying openings, and continue doing so throughout their careers, as opening theory continues to evolve.

The fundamental strategic aims of most openings are similar:

Development: To place (develop) the pieces (particularly bishops and knights) on useful squares where they will have an optimal impact on the game.

Control of the center: Control of the central squares allows pieces to be moved to any part of the board relatively easily, and can also have a cramping effect on the opponent.

King safety: Keeping the king safe from dangerous possibilities. A correct timing for castling can often enhance this.

Pawn structure: Players strive to avoid the creation of pawn weaknesses such as isolated, doubled or backward pawns, and pawn islands – and to force such weaknesses in the opponent’s position.

Most players and theoreticians consider that White, by virtue of the first move, begins the game with a small advantage. This initially gives White the initiative. Black usually strives to neutralize White’s advantage and achieve equality, or to develop dynamic counterplay in an unbalanced position.

Middlegame

The middlegame is the part of the game which starts after the opening. There is no clear line between the opening and the middlegame, but typically the middlegame will start when most pieces have been developed. (Similarly, there is no clear transition from the middlegame to the endgame, see start of the HYPERLINK “http://en.wikipedia.org/wiki/Chess_endgame#The_start_of_the_endgame”endgame.) Because the opening theory has ended, players have to form plans based on the features of the position, and at the same time to take into account the tactical possibilities in the position. The middlegame is also the phase in which most combinations occur. Combinations are a series of tactical moves executed to achieve some gain. Middlegame combinations are often connected with an attack against the opponent’s king; some typical patterns have their own names, for example the BodenHYPERLINK “http://en.wikipedia.org/wiki/Boden’s_Mate”‘HYPERLINK “http://en.wikipedia.org/wiki/Boden’s_Mate”s Mate or the Lasker-Bauer combination.

Specific plans or strategic themes will often arise from particular groups of openings which result in a specific type of pawn structure. For example, the minority attack, that is the attack of queenside pawns against an opponent who has more pawns on the queenside. The study of openings should therefore be connected with the preparation of plans that are typical of the resulting middlegames.

Another important strategic question in the middlegame is whether and how to reduce material and transform into an endgame (i.e. simplify). For example, minor material advantages can generally be transformed into victory only in an endgame, and therefore the stronger side must choose an appropriate way to achieve an ending. Not every reduction of material is good for this purpose; for example, if one side keeps a light-squared bishop and the opponent has a dark-squared one, the transformation into a bishops and pawns ending is usually advantageous for the weaker side only, because an endgame with bishops on opposite colors is likely to be a draw, even with an advantage of a pawn, or sometimes with a two-pawn advantage.

_____________________________________________________________________________________

Endgame

a

B

c

D

e

f

g

h

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

a

B

c

d

e

f

g

h

An example of zugzwang: the side which is to make a move is at a disadvantage.

The endgame (or end game or ending) is the stage of the game when there are few pieces left on the board. There are three main strategic differences between earlier stages of the game and endgame:

During the endgame, pawns become more important; endgames often revolve around attempting to promote a pawn by advancing it to the eighth rank.

The king, which has to be protected in the middlegame owing to the threat of checkmate, becomes a strong piece in the endgame. It is often brought to the center of the board where it can protect its own pawns, attack the pawns of opposite color, and hinder movement of the opponent’s king.

Zugzwang, a disadvantage because the player has to make a move, is often a factor in endgames but rarely in other stages of the game. For example, the diagram on the right is zugzwang for both sides, as with Black to move he must play 1…Kb7 and let White promote a pawn after 2.Kd7; and with White to move he must allow a draw by 1.Kc6 stalemate or lose his last pawn by any other legal move.

Endgames can be classified according to the type of pieces that remain on board. Basic checkmates are positions in which one side has only a king and the other side has one or two pieces and can checkmate the opposing king, with the pieces working together with their king. For example, king and pawn endgames involve only kings and pawns on one or both sides and the task of the stronger side is to promote one of the pawns. Other more complicated endings are classified according to the pieces on board other than kings, such as the “rook and pawn versus rook endgame.

Origins of the modern game (1000-1850)

A tactical puzzle from LucenaHYPERLINK “http://en.wikipedia.org/wiki/Luis_Ramirez_de_Lucena”‘HYPERLINK “http://en.wikipedia.org/wiki/Luis_Ramirez_de_Lucena”s 1497 book

Around 1200, rules of shatranj started to be modified in southern Europe, and around 1475, several major changes made the game essentially as it is known today. These modern rules for the basic moves had been adopted in Italy and Spain. Pawns gained the option of advancing two squares on their first move, while bishops and queens acquired their modern abilities. The queen replaced the earlier vizier chess piece towards the end of the 10th century and by the 15th century, had become the most powerful piece; consequently modern chess was referred to as “Queen’s Chess” or “Mad Queen Chess”. These new rules quickly spread throughout western Europe, with the exception of the rules about stalemate, which were finalized in the early 19th century. To distinguish it from its predecessors, this version of the rules is sometimes referred to as western chess or international chess.

Writings about the theory of how to play chess began to appear in the 15th century. The Repetición de Amores y Arte de Ajedrez (Repetition of Love and the Art of Playing Chess) by Spanish churchman Luis Ramirez de Lucena was published in Salamanca in 1497. Lucena and later masters like Portuguese Pedro Damiano, Italians Giovanni Leonardo Di Bona, Giulio Cesare Polerio and Gioachino Greco or Spanish bishop Ruy López de Segura developed elements of openings and started to analyze simple endgames.

François-André Danican Philidor, 18th-century French chess Master

In the 18th century the center of European chess life moved from the Southern European countries to France. The two most important French masters were François-André Danican Philidor, a musician by profession, who discovered the importance of pawns for chess strategy, and later Louis-Charles Mahé de La Bourdonnais who won a famous series of matches with the Irish master Alexander McDonnell in 1834. Centers of chess activity in this period were coffee houses in big European cities like Café de la Régence in Paris and SimpsonHYPERLINK “http://en.wikipedia.org/wiki/Simpson’s-in-the-Strand”‘HYPERLINK “http://en.wikipedia.org/wiki/Simpson’s-in-the-Strand”s Divan in London.

As the 19th century progressed, chess organization developed quickly. Many chess clubs, chess books and chess journals appeared. There were correspondence matches between cities; for example the London Chess Club played against the Edinburgh Chess Club in 1824. Chess problems became a regular part of 19th-century newspapers; Bernhard Horwitz, Josef Kling and Samuel Loyd composed some of the most influential problems. In 1843, von der Lasa published his and BilguerHYPERLINK “http://en.wikipedia.org/wiki/Paul_Rudolf_von_Bilguer”‘HYPERLINK “http://en.wikipedia.org/wiki/Paul_Rudolf_von_Bilguer”s Handbuch des Schachspiels (Handbook of Chess), the first comprehensive manual of chess theory.

________________________________________________________

Competitive play

Contemporary chess is an organized sport with structured international and national leagues, tournaments and congresses. Chess’s international governing body is FIDE (Fédération Internationale des Échecs). Most countries have a national chess organization as well (such as the US Chess Federation and English Chess Federation), which in turn is a member of FIDE. FIDE is a member of the International Olympic Committee, but the game of chess has never been part of the Olympic Games; chess does have its own Olympiad, held every two years as a team event.

The current World Chess Champion Viswanathan Anand (left) playing chess against his predecessor Vladimir Kramnik.

The current World Chess Champion is Viswanathan Anand of India. The reigning Women’s World Champion is Alexandra Kosteniuk from Russia but the world’s highest rated female player, Judit Polgár, has never participated in the WomenHYPERLINK “http://en.wikipedia.org/wiki/Women’s_World_Chess_Championship”‘HYPERLINK “http://en.wikipedia.org/wiki/Women’s_World_Chess_Championship”s World Chess Championship, instead preferring to compete with the leading men and maintaining a ranking among the top male players.

Other competitions for individuals include the World Junior Chess Championship, the European Individual Chess Championship and the National Chess Championships. Invitation-only tournaments regularly attract the world’s strongest players and these include Spain’s Linares event, Monte Carlo’s Melody Amber tournament, the Dortmund Sparkassen meeting, Sofia’s M-tel Masters and Wijk aan Zee’s Corus tournament.

Regular