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Box-making Game
A box-making game (often called just a box game) is a biased positional game where two players alternately pick elements from a family of pairwise-disjoint sets ("boxes"). The first player - called ''BoxMaker'' - tries to pick all elements of a single box. The second player - called ''BoxBreaker'' - tries to pick at least one element of all boxes. The box game was first presented by Paul Erdős and Václav Chvátal. It was solved later by Hamidoune and Las-Vergnas. Definition A box game is defined by: * A family of ''n'' pairwise-disjoint sets, A_1,\ldots,A_n, of different sizes. The sets are often called "boxes" and the elements are called "balls". * Two integers, ''p'' and ''q''. The first player, ''BoxMaker'', picks ''p'' balls (from the same or different boxes). Then the second player, ''BoxBreaker'', breaks ''q'' boxes. And so on. BoxMaker wins if he has managed to pick all balls in at least one box, before BoxBreaker managed to break this box. BoxBreaker wins if he has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biased Positional Game
A biased positional game is a variant of a positional game. Like most positional games, it is described by a set of ''positions/points/elements'' (X) and a family of subsets (\mathcal), which are usually called the ''winning-sets''. It is played by two players who take turns picking elements until all elements are taken. While in the standard game each player picks one element per turn, in the biased game each player takes a different number of elements. More formally, for every two positive integers ''p'' and ''q'', a (p:q)-positional game is a game in which the first player picks ''p'' elements per turn and the second player picks ''q'' elements per turn. The main question of interest regarding biased positional games is what is their ''threshold bias'' - what is the bias in which the winning-power switches from one player to the other player. Example As an example, consider the ''triangle game''. In this game, the elements are all edges of a complete graph on ''n'' vertices, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Václav Chvátal
Václav (Vašek) Chvátal () is a Professor Emeritus in the Department of Computer Science and Software Engineering at Concordia University in Montreal, Quebec, Canada and a Visiting Professor at Charles University in Prague. He has published extensively on topics in graph theory, combinatorics, and combinatorial optimization. Biography Chvátal was born in 1946 in Prague and educated in mathematics at Charles University in Prague, where he studied under the supervision of Zdeněk Hedrlín. He fled Czechoslovakia in 1968, three days after the Soviet invasion, and completed his Ph.D. in Mathematics at the University of Waterloo, under the supervision of Crispin St. J. A. Nash-Williams, in the fall of 1970. Subsequently, he took positions at McGill University (1971 and 1978-1986), Stanford University (1972 and 1974-1977), the Université de Montréal (1972-1974 and 1977-1978), and Rutgers University (1986-2004) before returning to Montreal for the Canada Research Chair in Combi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
