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Steinhaus Chessboard Theorem
The Steinhaus chessboard theorem is the following theorem, due to Hugo Steinhaus:Consider a chessboard on which some cells contain landmines. Then, either the king can cross the board from left to right without meeting a mined square, or the rook can cross the board from top to bottom moving only on mined squares. Two-dimensional variants Gale proved a variant of the theorem in which the tiles on the chessboard are hexagons, as in the game of Hex. In this variant, there is no difference between king moves and rook moves. Kulpa, Socha and Turzanski prove a generalized variant of the chessboard theorem, in which the board can be partitioned into arbitrary polygons, rather than just squares. They also give an algorithm for finding either a king route or a rook route. n-dimensional variants Tkacz and Turzanski generalize the chessboard theorem to an n-dimensional board:Consider a grid of n-dimensional cubes. Color each cube with one of ''n'' colors 1,...,''n''. Then, there exists a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugo Steinhaus
Hugo Dyonizy Steinhaus ( , ; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz University in Lwów (now Lviv, Ukraine), where he helped establish what later became known as the Lwów School of Mathematics. He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach–Steinhaus theorem. After World War II Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war. Author of around 170 scientific articles and books, Steinhaus has left his legacy and contribution in many branches of mathematics, such as functional analysis, geometry, mathematical logic, and trigonometry. Notably he is regarded as one of the early founde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chessboard
A chessboard is a game board 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 constitute the , and the e- through h-files constitute the ; the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Land Mine
A land mine, or landmine, is an explosive weapon often concealed under or camouflaged on the ground, and designed to destroy or disable enemy targets as they pass over or near it. Land mines are divided into two types: anti-tank mines, which are designed to disable tanks or other vehicles; and anti-personnel mines, designed to injure or kill people. Land mines are typically pressure activated, exploding automatically when stepped on by a person or driven over by a vehicle, though alternative detonation mechanisms are sometimes used. A land mine may cause damage by direct blast effect, by fragments that are thrown by the blast, or by both. Land mines are typically laid throughout an area, creating a ''minefield'' which is dangerous to cross. The use of land mines is controversial because of their indiscriminate nature and their potential to remain dangerous many years after a conflict has ended, harming civilians and the economy. With pressure from a number of campaign gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
King (chess)
The king (♔, ♚) is the most important chess piece, piece in the game of chess. It may move to any adjoining square; it may also perform, in tandem with the Rook (chess), rook, a special move called ''castling''. If a player's king is threatened with capture, it is said to be ''in Check (chess), check'', and the player must remove or evade the threat of immediately, such as by moving it away from the attacked square. If this cannot be done, the king is said to be in checkmate, resulting in a loss for that player. A player cannot make any move that places their own king in check. Despite this, the king can become a strong offensive piece in the Chess endgame, endgame or, rarely, the Chess middlegame, middlegame. In Algebraic notation (chess), algebraic notation, the king is abbreviated by the letter ''K'' among English speakers. The white king starts the game on e1; the black king starts on e8. Unlike all other pieces, each player can have only one king, and the kings are never ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rook (chess)
The rook (; ♖, ♜) is a piece in the game of chess. It may move any number of squares horizontally or vertically without jumping, and it may an enemy piece on its path; it may participate in castling. Each player starts the game with two rooks, one in each corner on their side of the board. Formerly, the rook (from ) was alternatively called the ''tower'', ''marquess'', ''rector'', and ''comes'' (''count'' or ''earl''). The term "castle" is considered to be informal or old-fashioned. Placement and movement The white rooks start on the squares a1 and h1, while the black rooks start on a8 and h8. The rook moves horizontally or vertically, through any number of unoccupied squares. The rook cannot jump over pieces. The rook may capture an enemy piece by moving to the square on which the enemy piece stands, removing it from play. The rook also participates with the king in a special move called castling, wherein it is transferred to the square crossed by the king after th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gale
David Gale (December 13, 1921 – March 7, 2008) was an American mathematician and economist. He was a professor emeritus at the University of California, Berkeley, affiliated with the departments of mathematics, economics, and industrial engineering and operations research. He has contributed to the fields of mathematical economics, game theory, and convex analysis. Personal life Gale graduated with a Bachelor of Arts from Swarthmore College, obtained a M.A. from the University of Michigan in 1947, and earned his Ph.D. in Mathematics at Princeton University in 1949. He taught at Brown University from 1950 to 1965 and then joined the faculty at the University of California, Berkeley. Gale lived in Berkeley, California, and Paris, France with his partner Sandra Gilbert, feminist literary scholar and poet. He has three daughters and two grandsons. Contributions Gale's contributions to mathematical economics include an early proof of the existence of competitive equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hex (board Game)
Hex (also called Nash) is a two player abstract strategy board game in which players attempt to connect opposite sides of a rhombus-shaped board made of hexagonal cells. Hex was invented by mathematician and poet Piet Hein in 1942 and later rediscovered and popularized by John Nash. It is traditionally played on an 11×11 rhombus board, although 13×13 and 19×19 boards are also popular. The board is composed of hexagons called ''cells'' or ''hexes''. Each player is assigned a pair of opposite sides of the board, which they must try to connect by alternately placing a stone of their color onto any empty hex. Once placed, the stones are never moved or removed. A player wins when they successfully connect their sides together through a chain of adjacent stones. Draws are impossible in Hex due to the topology of the game board. Despite the simplicity of its rules, the game has deep strategy and sharp tactics. It also has profound mathematical underpinnings related to the Brouwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall's Marriage Theorem
In mathematics, Hall's marriage theorem, proved by , is a theorem with two equivalent formulations. In each case, the theorem gives a necessity and sufficiency, necessary and sufficient condition for an object to exist: * The Combinatorics, combinatorial formulation answers whether a Finite set, finite collection of Set (mathematics), sets has a transversal (combinatorics), transversal—that is, whether an element can be chosen from each set without repetition. Hall's condition is that for any group of sets from the collection, the total unique elements they contain is at least as large as the number of sets in the group. * The Graph theory, graph theoretic formulation answers whether a finite bipartite graph has a perfect matching—that is, a way to match each vertex from one group uniquely to an adjacent vertex from the other group. Hall's condition is that any subset of vertices from one group has a neighbourhood (graph theory), neighbourhood of equal or greater size. Combinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
