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Longest Uncrossed Knight's Path
The longest uncrossed (or nonintersecting) knight's path is a mathematical problem involving a knight (chess), knight on the standard 8×8 chessboard or, more generally, on a square ''n''×''n'' board. The problem is to find the longest path (graph theory), path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins. Known solutions The longest open paths on an ''n''×''n'' board are known only for ''n'' ≤ 9. Their lengths for ''n'' = 1, 2, …, 9 are: :0, 0, 2, 5, 10, 17, 24, 35, 47 The longest closed paths are known only for ''n'' ≤ 10. Their lengths for ''n'' = 1, 2, …, 10 are: : 0, 0, 0, 4, 8, 12, 24, 32, 42, 54 Generalizations The problem can be further generalized to rectangular ''n''×''m'' boards, or even to boards in the shape of any polyomino. The probl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knight (chess)
The knight (♘, ♞) is a piece in the game of chess, represented by a horse's head and neck. It moves two squares vertically and one square horizontally, or two squares horizontally and one square vertically, jumping over other pieces. Each player starts the game with two knights on the b- and g-, each located between a rook and a bishop. Movement Compared to other chess pieces, the knight's movement is unique: it moves two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of a capital L). When moving, the knight can jump over pieces to reach its destination. Knights capture in the same way, replacing the enemy piece on the square and removing it from the board. A knight can have up to eight available moves at once. Knights and pawns are the only pieces that can be moved in the chess starting position. Value Knights and bishops, also known as , have a value of about three pawns. Bishops u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chessboard
A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the board is oriented such that each player's near-right corner square is a light square. The columns of a chessboard are known as ', the rows are known as ', and the lines of adjoining same-coloured squares (each running from one edge of the board to an adjacent edge) are known as '. Each square of the board is named using algebraic, descriptive, or numeric chess notation; algebraic notation is the FIDE standard. In algebraic notation, using White's perspective, files are labeled ''a'' through ''h'' from left to right, and ranks are labeled ''1'' through ''8'' from bottom to top; each square is identified by the file and rank which it occupies. The a- through d-files comprise the , while the e- through h-files comprise the . History and evo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path (graph Theory)
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipathGraph Structure Theory: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, Held June 22 to July 5, 1991p.205/ref>) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See e.g. Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). Korte et al. (1990) cover more advanced algorithmic topics concerning paths in graphs. Definitions Walk, trail, and path * A walk is a finite or infinite sequence of edges which joins a sequence of vertices. : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in '' Fairy Chess Review'' between the years 1937 to 1957, under the name of "dissection problems." The name ''polyomino'' was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 "Mathematical Games" column in ''Scientific American''. Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes. In statistical physics, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chess Piece
A chess piece, or chessman, is a game piece that is placed on a chessboard to play the game of chess. It can be either white or black, and it can be one of six types: king, queen, rook, bishop, knight, or pawn. Chess sets generally come with sixteen pieces of each color. Additional pieces, usually an extra queen per color, may be provided for use in promotion. Number of pieces Each player begins with sixteen pieces (but see the subsection below for other usage of the term ''piece''). The pieces that belong to each player are distinguished by color: the lighter colored pieces are referred to as "white" and the player that owns them as "White", whereas the darker colored pieces are referred to as "black" and the player that owns them as "Black". In a standard game, each of the two players begins with the following sixteen pieces: * 1 king * 1 queen * 2 rooks * 2 bishops * 2 knights * 8 pawns Usage of the term ''piece'' The word "piece" has three meanings, depending on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fairy Chess Piece
A fairy chess piece, variant chess piece, unorthodox chess piece, or heterodox chess piece is a chess piece not used in conventional chess but incorporated into certain chess variants and some chess problems. Compared to conventional pieces, fairy pieces vary mostly in the way they move, but they may also follow special rules for capturing, promotions, etc. Because of the distributed and uncoordinated nature of unorthodox chess development, the same piece can have different names, and different pieces can have the same name in various contexts. Most are symbolised as inverted or rotated icons of the standard pieces in diagrams, and the meanings of these "wildcards" must be defined in each context separately. Pieces invented for use in chess variants rather than problems sometimes instead have special icons designed for them, but with some exceptions (the princess, empress, and occasionally amazon), many of these are not used beyond the individual games for which they were invente ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camel (chess)
The camel or long knight is a fairy chess piece with an elongated knight move.Piececlopedia: Camel by Hans Bodlaender, '''' It can jump three squares horizontally and one square vertically or three squares vertically and one square horizontally, regardless of intervening pieces. Therefore, it is a (1,3)-leaper. History and nomenclature The camel is a very old piece, appearing in some early |
Giraffe (chess)
The giraffe is a fairy chess piece with an elongated knight move.Piececlopedia: Giraffe by Hans Bodlaender, '''' It can jump four squares vertically and one square horizontally or four squares horizontally and one square vertically, regardless of intervening pieces; thus, it is a (1,4)- leaper. Movement History According to[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zebra (chess)
The zebra is a fairy chess piece that moves like a stretched knight. It jumps three squares horizontally and two squares vertically or three squares vertically and two squares horizontally, regardless of intervening pieces; thus, it is a (2,3)-leaper. A lame zebra, which moves one step orthogonally and then two steps diagonally outwards and can be blocked by intervening pieces, appears as the elephant in janggi. Movement Value The zebra by itself is worth just below two pawns (appreciably less than a knight) due to its restricted freedom of movement on an 8×8 board. Its larger move is the main reason why it is weaker than a camel on an 8×8 board, even though the camel is colorbound and the zebra is not. A king, a bishop, and a zebra can force checkmate on a bare king; a king, a knight, and a zebra cannot; and a king, a camel, and a zebra cannot. The rook versus zebra endgame is a win for the rook. (All endgame statistics mentioned are for the 8×8 board.) As a component ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knight's Tour
A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed (or re-entrant); otherwise, it is open. The knight's tour problem is the mathematical problem of finding a knight's tour. Creating a program to find a knight's tour is a common problem given to computer science students. Variations of the knight's tour problem involve chessboards of different sizes than the usual , as well as irregular (non-rectangular) boards. Theory The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time. Histor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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