Numb3rs Season 4 Episode 14: Checkmate

In this episode, Bishop and David play very quick games of chess, a variant of chess called speed chess.



Chess is a two-player game whose origins can be traced to a game played in India around 500 A.D. It is widely believed that Muslim traders were responsible for bringing the game to Europe, and by 1000 A.D., the game was extremely popular there. However, these earlier versions of the game had quite different rules and also had quite different play styles. As the game matured, additional moves were added to the basic imported ruleset.

As will be discussed below, the added rules included castling, en passant, and the option to move pawns two squares on their initial move.

By 1500 A.D., the game had morphed into the modern version we play today. The first chess tourney, as we know it, was played in 1851 in London, and chess has been an enduring figure in popular culture for many centuries.



Chess is a two-player zero sum board game. The two players are called "white" and "black". The two players alternate turns, where each turn consists of moving pieces of their color on an 8x8 square board. The white player always moves first. The core of the game is in how placing pieces in different positions on the board can help one player or the other to achieve the victory conditions. Usually, a move consists of the player moving one chosen piece from one square to another square, though there is one exceptional move (castling), detailed below, which consists of two pieces moving during one turn.

The board is square, and is made up of 64 smaller squares. 32 of these smaller squares are light squares and 32 of these smaller squares are dark. The smaller squares are arranged in a checkerboard pattern so that each light square has a dark square to its top side, bottom side, left side, and right side.

A chess board looks like:

Notice that there is light square on the bottom right-hand corner. This is important. If the chess board were rotated ninety degrees, there would be a black square in the bottom right, and it would not be a valid chess board.

Either the top or the bottom of the board is the white player's side and the opposing side of the board is the black player's side. When setting up the board, all of a players pieces are on the closest two rows to that player, and in fact completely fill up these rows. Which side each player is on also affects how the pawns move.


Each piece has a different type of legal move. On a player's turn, they may choose one piece and move it according to the legal moves for that piece.

No piece may move into a square already occupied by a piece of the same color.

With the exception of the knight, no piece may move over any pieces of either color.

However, a piece may move into a square occupied by a piece of the opposing color. The opposing color piece is taken off the board and out of play and has no effect on the rest of the game. This move is called "capturing" the opposing piece.

For every piece except one (the pawn), the valid moves for a piece are the same as the valid capture for the piece. For the most part, there is no purpose to distinguishing the two types of moves, yet they are distinct.


There are six types of pieces that can be placed on the squares of the chess board. Each piece has a white version controlled by the white player and a black version controlled by the black player.


The king can move one square up, down, left, right, or diagonally in any direction.

The king is simultaneously one of the weaker pieces in the game because of its limited movement and yet the most valuable, since threats of capturing the king are central to the winning conditions, as will be described below.


The queen can move in a straight line in any of the eight semi-cardinal directions (up, down, left, right, or any diagonal). The queen cannot move over any piece, and the queen cannot move into the same square as a piece of the same color. However the queen can capture any piece in a straight line from the queen, so long as the "line-of-sight" from the queen is unobstructed.

The queen is the most powerful piece on the board, and hence the most valuable, after the king.


The bishop can move as far as it is unobstructed along any diagonal.


The knight is special because it is the only piece that can move over (jump over) other pieces. Unlike Checkers, none of the "jumped" pieces are taken off the board - they are unaffected.

The knight moves in an "L" shape. He moves two squares in one cardinal direction (up, down, left, or right) and one square in a perpendicular cardinal direction. A knight near the center of the board has eight possible postions he can move to arranged like the following:


The rook is much like the queen or bishop. The rook can move as far as it is unobstructed up, down, left, or right.


The pawn's moves are the most complicated. The pawn's normal move is to move one space forward or up, from the point of view of its owner. So, in particular, the white pawns and the black pawns move in opposite directions, each moving forward from the point of view of two players who are facing each other. Pawns can only move forward, and can never move backwards.

Moving forward, the pawn cannot capture. If there is an opposing piece in front of the pawn, the pawn is blocked and cannot move forward. A pawn can only capture by moving one square forward and diagonally either to the left or to the right. A pawn may not execute this diagonal move unless there is an opposing piece in the destination square.

On a pawn's first move, and only on its first move, the player may choose to move the pawn two squares forward, so long as both the destination square, and the square one space forward are both vacant. A pawn may not capture while moving two spaces forward.

Piece Setup

Each player starts the game with one king, one queen, two bishops, two knights, two rooks, and eight pawns, for a total of sixteen pieces. These sixteen pieces fill up the first two rows closest to the side of each particular player.

For the white player, the pieces, from left to right on the first row closest to the white player are :rook, knight, bishop, queen, king, bishop, knight, and rook. For the black player, the pieces, from left to right on the first row closest to the white player are :rook, knight, bishop, king, queen, bishop, knight, and rook. The only difference between these two are the ordering of the king and queen. This is so that the kings and queens each face each other. In both cases, the queen is on the same color square as her own color. This is why it is important to ensure that there is a white square in the bottom right corner for each player.

For each player, the second row closest to them is completely filled with eight pawns.

A completely setup board before the first move looks like:


If, after moving, a player has the opportunity to capture the opposing king if the opposing player does nothing, then the player declares "check", and the opposing player's king is said to be "in check".

A player whose king is in check must get their king out of check by the end of his or her move. There are three ways to get out of check. One way is to capture the opposing piece that is threatening the king. Another way is to block the attack by placing another piece in between the attacking piece and the king (this will not work when the attacking piece is a knight). The last way is to move the king.


If a player is in check and has no move to get their king out of check, then the opposing player declares "checkmate" and wins the game. This is the only winning condition.


There are several ways to generate a stalemate, or a tie. If it is a player's turn and his king is not in check, but every legal move he can make puts his king in check, then the game ends in a tie.

If each player takes fifty consecutive turns where no piece is captured and no pawns move, the game is declared a draw. This rule is somewhat archaic as recently, forced mates with over 75 moves have been discovered with the help of computer searches.

The last way that a draw is reached is if a board position is reached for the third time.

One important consequence of these two rules is that no game can last forever. There must be some upper bound on the number of moves in a game of chess. Hence, there must also be a finite number of games.

Special moves

If all of the squares between the king and one of his rooks are vacant, then the king can move two spaces in the direction of the rook, and the rook is placed on the opposite side of the king. This move is called castling, and is differentiated into two subtypes, called queen's side castling and king's side castling, depending on which rook is employed. Castling may not be done if either the king or rook has been moved previously, if the king is currently in check, or if the king moves through check or ends the turn in check.
Pawn Promotion
If a pawn reaches the opposite side of the board, it is promoted to either a rook, bishop, knight, or queen, at the player's discretion.
En Passant
If a pawn is on the fifth row away from his owner and an opposing pawn in an adjacent column moves two squares in one move to come beside him, then the original pawn may capture the second pawn as if he had only moved forward one square. This capture must be done immediately after the second pawn moves forward two squares, or else the chance is lost.

Speed Chess

In all competitive chess, there are limits on the amount of time that a player has on their moves. Speed chess is a variant of chess where the amount of time given to a player is so small as to severely impact their gameplay. In speed chess, thinking about longer-term strategies is sacrificed for extremely quick, almost reflexive, reaction to shorter-term tactics.

Popular variants are bullet chess and blitz chess. In bullet chess, each player has either one or two minutes to make all of their moves. In blitz chess, each player has five minutes to make all of their moves.

When time is a concern, there is another win condition: if a player runs out of time during their turn, they lose the game and their opponent is the winner.

Speed chess matches are always played with a special electronic chess timer that each player taps after they move. The timer keeps track of the total amount of time spent on all of each player's moves.

Mathematics of Chess

Game Theory

Questions about Chess have motivated explorations of a branch of mathematics known as game theory. What makes chess so attractive is that there are a finite number of game states and they are easily described by words or characters.

Because Chess is a finite game with complete information, it is possible for either white or black to force a win or for both sides to be able to force a draw. Which one of these three actually holds for chess is not known, though it is all but impossible that black is able to force a win. Many people believe that if both sides play perfectly, then the result is a draw.

Chess has been a very fruitful domain for artificial intelligence researchers, trying to program computers to play chess. Chess is perceived by humans to be very difficult, though it is not nearly as hard as other games, such as Go. Chess artificial intelligence has more or less achieved its goal of besting human masters.

However, the programs themselves that consistently beat the best humans do not seem to be particularly intelligent. They mostly achieve their stunning victories by considering millions or billions of possible move combinations rather than understanding themes or recognizing patterns. There is ongoing research on how to design artificial intelligence to think about chess in a more fluid way, more like human masters do.


There are several questions one can ask about chess that are essentially counting how many of something there are. Unfortunately, practically none of the answers to these questions are known.

How many valid game positions are there? No one knows for sure, but Claude Shannon estimated that the answer is around .

How many different games are possible? Again no one knows, but Hardy estimated this at


How many rooks can you place on the board with none of them attacking each other? Seems easy? Try it with queens. This is known as the Queens Problem. Try it with the other pieces as well.

A knights tour consists of a knight moving around the board, landing on every square exactly once, and ending up where he started. See if you can find a knights tour on an 8x8 board.

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