Combinatorial game theory is a branch of mathematics and theoretical computer science that studies sequential games with perfect information. Study has been confined to two-player games that have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition. CGT has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are changing. Scholars will define what they mean by a "game" at the beginning of a paper, these definitions vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field. Combinatorial games include well-known games such as chess, Go, which are regarded as non-trivial, tic-tac-toe, considered as trivial in the sense of being "easy to solve".
Some combinatorial games may have an unbounded playing area, such as infinite chess. In CGT, the moves in these and other games are represented as a game tree. Combinatorial games include one-player combinatorial puzzles such as Sudoku, no-player automata, such as Conway's Game of Life, Game theory in general includes games of chance, games of imperfect knowledge, games in which players can move and they tend to represent real-life decision making situations. CGT has a different emphasis than "traditional" or "economic" game theory, developed to study games with simple combinatorial structure, but with elements of chance. CGT has contributed new methods for analyzing game trees, for example using surreal numbers, which are a subclass of all two-player perfect-information games; the type of games studied by CGT is of interest in artificial intelligence for automated planning and scheduling. In CGT there has been less emphasis on refining practical search algorithms, but more emphasis on descriptive theoretical results.
An important notion in CGT is that of the solved game. For example, tic-tac-toe is considered a solved game, as it can be proven that any game will result in a draw if both players play optimally. Deriving similar results for games with rich combinatorial structures is difficult. For instance, in 2007 it was announced that checkers has been weakly solved—optimal play by both sides leads to a draw—but this result was a computer-assisted proof. Other real world games are too complicated to allow complete analysis today, although the theory has had some recent successes in analyzing Go endgames. Applying CGT to a position attempts to determine the optimum sequence of moves for both players until the game ends, by doing so discover the optimum move in any position. In practice, this process is torturously difficult unless the game is simple, it can be helpful to distinguish between combinatorial "mathgames" of interest to mathematicians and scientists to ponder and solve, combinatorial "playgames" of interest to the general population as a form of entertainment and competition.
However, a number of games fall into both categories. Nim, for instance, is a playgame instrumental in the foundation of CGT, one of the first computerized games. Tic-tac-toe is still used to teach basic principles of game AI design to computer science students. CGT arose in relation to the theory of impartial games, in which any play available to one player must be available to the other as well. One such game is nim. Nim is an impartial game for two players, subject to the normal play condition, which means that a player who cannot move loses. In the 1930s, the Sprague–Grundy theorem showed that all impartial games are equivalent to heaps in nim, thus showing that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs. In the 1960s, Elwyn R. Berlekamp, John H. Conway and Richard K. Guy jointly introduced the theory of a partisan game, in which the requirement that a play available to one player be available to both is relaxed.
Their results were published in their book Winning Ways for your Mathematical Plays in 1982. However, the first work published on the subject was Conway's 1976 book On Numbers and Games known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was a fruit of the collaboration between Berlekamp and Guy. Combinatorial games are by convention, put into a form where one player wins when the other has no moves remaining, it is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the sum of two games, a game where each player may choose to move either in one game or the other at any point in the game, a player wins when his opponent has no move in either game; this way of combining games leads to a powerful mathematical structure. John Conway states in ONAG that the inspira
Events from the year 1565 in Sweden Monarch – Eric XIV January - The Swedes sack the Danish provinces of Scania and Halland under Klas Horn. 21 May - Swedish victory in the Sea Battle in Pomerania. 4 June - Action of 4 June 1565 9 June - The Teutonic Order declares war against Sweden in Livonia. 7 July - Action of 7 July 1565 July - The Danes burns Swedish Lödöse. 13 August - Swedish victory at the Battle of Obermühlenberg. 28 August - Danish Varberg is conquered by the Swedes. 8 September - Princess Cecilia of Sweden arrives on her official visit to England. Among the persons in her retinue is Helena Snakenborg. 20 October - Battle of Axtorna - Sweden is struck by the plague. 10 November - Laurentius Paulinus Gothus, theologian and archbishop
Hermann Vallendor was born in Offenburg, the Grand Duchy of Baden on 13 April 1894. He was an engineering student in Mannheim before World War. Vallendor joined the 114th Infantry Regiment of the German Army on 16 October 1914, he was promoted to Gefreiter on 19 May 1915. He was awarded the Iron Cross Second Class on 29 July 1915. A promotion to Vizefeldwebel followed on 5 December 1915, he was raised from the enlisted ranks, being commissioned as a leutnant in the reserves on 24 December 1915. He transferred to aviation duty and began pilot's training with FEA 5 on 16 October 1916; when he completed pilot's school, he exited training for Armee Flugpark 2 on 19 February 1917. In May 1917, he went to FA 23 to fly two-seaters, he soon left that detachment, to begin attending fighter pilot's school on 24 June 1917. He was posted to a fighter unit, Jagdstaffel 2 on 5 July 1917; as was pretty much customary in German military aviation of the time, Vallendor had his aircraft emblazoned in his personal insignia.
In his case, the marking was a huge white "V" on the fuselage. When he used a Fokker Triplane, he had the "V" painted on its top wing. Vallendor received the Order of the Zähringer Lion from his native Baden on 14 December 1917, he achieved his first aerial victory on 3 February 1918. He was awarded the Iron Cross First Class on 7 March 1918 and went on to score five more victories before war's end. See Aerial victory standards of World War I After war's end, Vallendor moved to Buenos Aires, Argentina in 1920 to work for Deutsche Bank, he went on to Montevideo, Uruguay in 1921. When he died on 15 November 1974 in Montevideo, he was buried in the British Cemetery there, in Plot E248. Above the Lines: The Aces and Fighter Units of the German Air Service, Naval Air Service and Flanders Marine Corps, 1914–1918. Norman Franks, Frank W. Bailey, Russell Guest. Grub Street, 1993. ISBN 0-948817-73-9, ISBN 978-0-948817-73-1. Jagdstaffel 2 Boelcke: Von Richthofen's Mentor: Volume 26 of Aviation Elite Units: Volume 26 of Osprey Aviation Elite.
Greg VanWyngarden. Osprey Publishing, 2007. ISBN 1-84603-203-2, ISBN 978-1-84603-203-5
Hou Junji was a Chinese general and official who served as a chancellor during the reign of Emperor Taizong in the Tang dynasty. He is best known for leading the Tang military campaigns against the Tuyuhun kingdoms. In 643, he was implicated in a plot by the crown prince, Li Chengqian, to overthrow Emperor Taizong, was executed, it is not known when Hou Junji was born, little is known about his family background other than that he was from Bin Prefecture. It was said that he always wanted to appear impressive, he favored using bow and arrows, while he never achieved greatness in archery, he became known for his fighting abilities. At a point early in the reign of Emperor Gaozu, still trying to reunify China after the collapse of Sui Dynasty, Hou came to serve under Emperor Gaozu's son, the major general Li Shimin the Prince of Qin. While serving under Li Shimin, for his accomplishments, Hou was created the Viscount of Quanjiao, he became a close associate of Li Shimin offering Li his strategies.
By 626, Li Shimin was locked in an intense rivalry with his older brother Li Jiancheng the Crown Prince, he feared that Li Jiancheng would kill him. Hou, along with Li Shimin's brother-in-law Zhangsun Wuji, Zhangsun's uncle Gao Shilian, the general Yuchi Gong, advised Li Shimin to act first and ambush Li Jiancheng and another brother who supported Li Jiancheng, Li Yuanji the Prince of Qi. Li Shimin agreed, in 626 ambushed Li Jiancheng and Li Yuanji, killing them. During the subsequent battles between Li Shimin's forces and Li Jiancheng's and Li Yuanji's forces, Hou led Li Shimin's forces; when the dust had settled, Li Shimin forced Emperor Gaozu to make him the crown prince, yield the throne to him. Late In 626, when Emperor Taizong ranked the contributions of the generals and officials in order to grant them fiefs, Emperor Taizong ranked five of them—Hou Junji, Zhangsun Wuji, Fang Xuanling, Du Ruhui, Yuchi Gong to be contributors of the highest grade, Hou was created the Duke of Lu. In 630, Emperor Taizong made Hou the minister of defense and gave him the additional designation of Canyi Chaozheng, making him a de facto chancellor.
In 634, Emperor Taizong, sending the senior general Li Jing to command the campaign against Tuyuhun's Busabo Khan Murong Fuyun, made Hou and Li Daozong the Prince of Rencheng Li Jing's assistants on the campaign. By spring 635, Tang forces achieved initial victories, but Tuyuhun forces burned the grazing grass to cut the food supplies to Tang horses. Most Tang generals wanted to withdraw, but Hou advocated continued advance, Li Jing agreed allowing complete victory, as Murong Fuyun was killed by his subordinates, allowing his son Murong Shun, whom Tang supported, to become khan. Around the new year 636, after Murong Shun was assassinated by his subordinates, Emperor Taizong sent Hou with an army to try to secure the throne for Murong Shun's son Murong Nuohebo. In 637, as part of Emperor Taizong's scheme to bestow prefectures on his relatives and great generals and officials as their permanent domains, Hou's title was changed to Duke of Chen, he was given the post of prefect of Chen Prefecture, to be inherited by his heirs.
Soon, with many objections to the system, the strongest of which came from Zhangsun Wuji, Emperor Taizong cancelled the scheme, although Hou's title remained Duke of Chen. In 638, Tufan's Songtsen Gampo, after hearing that the rulers of Tujue and Tuyuhun were all able to marry Tang princesses, requested to marry one as well, but was rebuffed by Emperor Taizong. In anger, he launched a major attack on Tang. Emperor Taizong sent Hou to counterattack, assisted by other generals Zhishi Sili, Niu Jinda, Liu Jian. Niu was subsequently able to defeat Tufan forces at Song Prefecture, Songtsen Gampo, in fear, but still requested to marry a Tang princess. Around the new year of 640, Qu Wentai, the king of Gaochang, formed an alliance with Western Tujues who are hostile to Tang. Emperor Taizong sent Hou, assisted by Xue Wanjun; when Hou arrived at Gaochang, Qu Wentai was succeeded by his son Qu Zhisheng. Rejecting a proposal by some of his generals to ambush the Gaochang nobles when they were burying Qu Wentai, he put Gaochang's capital under siege, forcing Qu Zhisheng to surrender.
Emperor Taizong annexed Gaochang territory except for three cities, which Gaochang had seized from Yanqi, therefore were returned to Yanqi after the king of Yanqi met with Hou to congratulate him), Hou returned to the Tang capital Chang'an with Qu Zhisheng and his subordinates as captives. Upon Hou's return to Chang'an, however, he found himself in trouble, as it was alleged that Hou had taken for himself treasures from the Gaochang imperial treasury and forced certain Gaochang captives to be his slaves; the other generals, seeing Hou's example did and he was in no position to stop them. Emperor Taizong, when he found out about these events, put Hou and some of his generals under arrest less than 10 days after their return to Chang'an. However, upon the advice of the official Cen Wenben, Emperor Taizong released Hou. Hou Junji was resentful that, despite his great achievement, he was put under arrest, albeit briefly. In the spring of 643, when fellow general Zhang Liang was sent out of the
Sophie Huet was a French journalist. She was a political journalist for Le Figaro, the first woman to serve as the president of the Association of Parliamentary Journalists. Sophie Huet was born on 20 January 1953 in Paris. Huet began her career as a journalist for L'Aurore in 1976, she covered politics in 1977 and the French Parliament in 1978. She joined Le Figaro in 1980. Huet served as the president of the Association of Parliamentary Journalists from 2006 to 2017, she was the first woman to serve in this capacity. Huet was the author of three books, she became an officer of the Legion of Honour in 2010. Huet was married twice, she first married François Montrognon de Salvert, followed by Lucien Neuwirth. Huet died on 29 July 2017. Huet, Sophie. Tout ce que vous direz pourra être retenu contre vous: ou les petites phrases du septennat. Paris: J. Picollec. ISBN 9782864770206. OCLC 8628135. Huet, Sophie. La Communication politique. Paris: Presses universitaires de France. ISBN 9782130378730. OCLC 251608514.
Huet, Sophie. Quand ils faisaient la guerre. Paris: Plon. ISBN 9782259026130. OCLC 243770657
Didier Cohen is an American-born Model, DJ/Producer. Cohen was discovered by an agent from Wilhelmina Models walking down Sunset Boulevard in Los Angeles, he moved to Australia and did a model campaign with Miranda Kerr. He has been featured in campaigns for Dolce & Gabbana and Adidas, was the face of Industrie with Miranda Kerr for two years. Didier took part in being a judge on Australia’s Next Top Model, as well as other TV shows. Cohen now resides in Los Angeles California where he is a Producer, he co-produced ‘Elevate’ on the Spider-Man-into the Spideverse movie soundtrack. Didier Cohen battles with Lyme Disease, he was engaged to Chadwick Model Jade Cara. Cohen has appeared in three short films, directed by James Demitri Cohen competed in and was eliminated from The Celebrity Apprentice Australia, where he was raising money for his charity, Youth Off The Streets, his most recent TV appearance was as a mentor for the ninth season of Australia's Next Top Model