The chakana is a stepped cross made up of an equal-armed cross indicating the cardinal points of the compass and a superimposed square. The square is suggested to represent the other two levels of existence; the three levels of existence are Hana Pacha, Kay Pacha, Ukhu or Urin Pacha. The hole through the centre of the cross is the Axis by means of which the shaman transits the cosmic vault to the other levels, it is said to represent Cusco, the center of the Incan empire, the Southern Cross constellation. The geometry of the symbol has a high degree of symmetry; the symbol can be drawn from a circle. A square is inscribed with the corners tangent to the circle; this forms the "middle step" of the ladder. A smaller square is made from the midpoints of the large square. Connecting the midpoints of the small square and extending the lines to the edge of the circle will form the arms of the cross, otherwise known as the "first" and "last" steps of the chakana. Lines are drawn from the points the lines exit the circle.
A small circle is made from the diameter of the cross lines. To proof the construction, a separate construction line is drawn from the point where the square corners with the cross rectangles to the same point on the opposite side. If the line is 27 degrees from the vertical, the chakana is properly drawn; the mestee historian Garcilaso de la Vega, el Ynga, reports about a holy cross of white and red marble or jasper, venerated in 16th-century Cusco. The cross had been kept in a royal house, in a sacred place or wak'a, but the Incas did not worship it, they admired it because of its beauty. The cross was square, measuring about two by two feet, its branches three inch wide, the edges squared and the surface brightly polished; the Incas began to venerate the holy cross, after they heard how Pedro de Candia had miraculously defied a lion and a tiger holding a cross. When the Spaniards captured the city, they transferred the cross to sacristy of the newly built cathedral, where De la Vega saw it in 1560.
He was surprised. As we know from Middle America, this may have been part of a deliberate strategy by mendicant friars, trying to adapt to indigenous cultural codes; as a rule, the veneration of the holy cross was a designed ecclesiastical enterprise, incorporating native symbols and reproducing them on sacral level. Most surviving Andean crosses do not predate the 16th century. Ongoing stories about indigenous crosses contributed to the idea of a'natural' religion that would have prepared the Indians for their inevitable conversion to Christianity; this Andean "cross" has been documented in Tiwanaku architecture dating back to 300 BC and AD 300. Other Andean cultures, whose presence in the Americas predates Western presence and date back 600 t 500 BC; the "chakana" is an invented tradition. Although the Chakana as the'Andean cross', presented as an Inca and pre-Inca symbol bearing cultural, spiritual, or mystical interpretations as expressed in this article, has wide popularity in contemporary Andean culture, its roots are no older than the late 20th century, the popular version, than 2003.
The current Chakana mythos as it impacts the New Age belief system and the Peruvian tourist-oriented economy initiates from the 2003 publication of the book Andean Awakening, authored by Jorge Luis Delgado, Mary Anne Male. This is the source of the myth, they are followed by such authors as Roger Calverley. The archaeoastronomer Carlos Milla Villena published his own, distinctly different speculative interpretations of the chacana as “andean, cross” in Génesis de la cultura andina,1983. Mainstream scientific and archaeological sources do not refer to or support the word chakana as the cross-and-box design - rather, it is in Runasimi, the traditional language of the Inca peoples, derived from chaka,'bridge', means'to cross over', or'a crossing'. Among chroniclers such as the Jesuit missionary and naturalist José de Acosta, 1590, it is applied to the group of stars identified as the Belt of Orion; the chroniclers do not refer to the chakana, or chacana, as the “andean cross,” or as reflecting a symbolical or semiotic tradition.
The twelve-cornered design itself appears in pre-contact artifacts such as textiles and ceramics from such cultures as the Wari and Tiwanaku, but with no particular emphasis and no key or guide to a means of interpretation. Some Andeans and Andeanists regard the imposition of this modern “invented tradition"” as cultural appropriation. Wiphala Axis Mundi Soledad Cachuan: Mitología Inca, Buenos Aires 2008 Drury: The Elements of Shamanism, Element Books, 1989. Mariano Cueva: Historia de la iglesia en Mexico, vol. 1, Mexico 1928, pp. 82–86 Wilbert Escobedo Araoz: La cruz cuadrada andina, chacana. Javier Lajo, Filosofía indígena inka: la Tawachakana www.ancientworlds.net/aw/Article/941062 members.fortunecity.es/chakana/chakana.ht
Dots is an abstract strategy game, played by two or more people on a sheet of squared paper. The game is superficially similar to Go, except that pieces are not taken, the primary target of dots is capturing enemy dots by surrounding them with a continuous line of one's own dots. Once surrounded, dots are not playable. Dots is played on a grid of some finite size 39x32 but arbitrary sizes can be used. Players take turns by placing a dot of their own color on empty intersections of the grid. If a newly placed dot completes a closed chain of dots of the same color which encloses at least one of the enemy dots all the area inside it is surrounded. To form a chain dots must be adjacent to each other either vertically, horizontally or diagonally. Surrounded enemy dots are added to the score of the player. All enclosed dots and empty intersections are excluded from further play and can't be used to make new surrounds. To mark a newly surrounded area, the surrounding player must draw a boundary line through all his dots that are part of enclosing chain.
Note, that players can't surround areas. As a consequence the enemy can use empty intersections inside it to complete his own enclosing chain. However, if one places a dot into empty area surrounded by the opponent and cannot use it to surround this dot can be captured by the opponent. There is more than one way of choosing an enclosing chain of dots; when played with pens and paper, players are free to choose one. When game is played on a computer, to simplify user input, programs automatically surround minimum area. In some cases this can be tactically exploited to one's advantage; the goal of the game is to capture more dots than opponent. At early stages of game formation Dots was played on piece of paper with two pens until one of the players surrendered; however this informal way of ending the game is inapplicable in competitive play. Nowadays always a grounding rule is used, described here. Continuous groups of dots that touch the border of the field can't be captured no matter how many moves opponent make.
These dots are said to be «grounded». At any moment of the game either player can stop the game, a way to say «All the dots that I want to preserve are grounded. You can take everything else». After that his opponent is allowed to make as many moves as he want to capture all the remaining dots that he can; the game whoever captured most dots wins. In lost situation good players surrender before his opponent has to explicitly apply this rule. If one of the players only places his dots touching the border, none of them could be captured. To prevent forced draw situations, the game is played either from initial position or with first several moves restricted to some area around the center of the field; the most popular initial position is a cross. Other popular ones are four crosses placed randomly. General trend is: the more crosses are placed on the field, more active, less probable to end in a draw, more challenging the game for both players will be
In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e; the definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs there may be multiple dual graphs, depending on the choice of planar embedding of the graph; the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, is in turn generalized algebraically by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, duality for graphs embedded onto non-planar two-dimensional surfaces.
However, these notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph. The term "dual" is used because the property of being a dual graph is symmetric, meaning that if H is a dual of a connected graph G G is a dual of H; when discussing the dual of a graph G, the graph G itself may be referred to as the "primal graph". Many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, simple graphs are dual to 3-edge-connected graphs. Graph duality can help explain the structure of mazes and of drainage basins. Dual graphs have been applied in computer vision, computational geometry, mesh generation, the design of integrated circuits; the unique planar embedding of a cycle graph divides the plane into only two regions, the inside and outside of the cycle, by the Jordan curve theorem.
However, in an n-cycle, these two regions are separated from each other by n different edges. Therefore, the dual graph of the n-cycle is a multigraph with two vertices, connected to each other by n dual edges; such a graph is called a dipole graph. Conversely, the dual to an n-edge dipole graph is an n-cycle. According to Steinitz's theorem, every polyhedral graph must be planar and 3-vertex-connected, every 3-vertex-connected planar graph comes from a convex polyhedron in this way; every three-dimensional convex polyhedron has a dual polyhedron. Whenever two polyhedra are dual, their graphs are dual. For instance the Platonic solids come in dual pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron, the tetrahedron dual to itself. Polyhedron duality can be extended to duality of higher dimensional polytopes, but this extension of geometric duality does not have clear connections to graph-theoretic duality. A plane graph is said to be self-dual; the wheel graphs provide an infinite family of self-dual graphs coming from self-dual polyhedra.
However, there exist self-dual graphs that are not polyhedral, such as the one shown. Servatius & Christopher describe two operations and explosion, that can be used to construct a self-dual graph containing a given planar graph, it follows from Euler's formula that every self-dual graph with n vertices has 2n − 2 edges. Every simple self-dual planar graph contains at least four vertices of degree three, every self-dual embedding has at least four triangular faces. Many natural and important concepts in graph theory correspond to other natural but different concepts in the dual graph; because the dual of the dual of a connected plane graph is isomorphic to the primal graph, each of these pairings is bidirectional: if concept X in a planar graph corresponds to concept Y in the dual graph concept Y in a planar graph corresponds to concept X in the dual. The dual of a simple graph need not be simple: it may have self-loops or multiple edges connecting the same two vertices, as was evident in the example of dipole multigraphs being dual to cycle graphs.
As a special case of the cut-cycle duality discussed below, the bridges of a planar graph G are in one-to-one correspondence with the self-loops of the dual graph. For the same reason, a pair of parallel edges in a dual multigraph corresponds to a 2-edge cutset in the primal graph. Therefore, a planar graph is only if its dual has no 1 - or 2-edge cutsets; the simple planar graphs whose duals are simple are the 3-edge-connected simple planar graphs. This class of graphs includes, but is not the same as, the class of 3-vertex-connected simple planar graphs. For instance, the figure showing a self-dual graph is 3-edge-connected but is not 3-vertex-connected; because the dual graph depends on a particular embedding, the dual graph of a p
Bolivia the Plurinational State of Bolivia is a landlocked country located in western-central South America. The capital is Sucre; the largest city and principal industrial center is Santa Cruz de la Sierra, located on the Llanos Orientales a flat region in the east of Bolivia. The sovereign state of Bolivia is a constitutionally unitary state, divided into nine departments, its geography varies from the peaks of the Andes in the West, to the Eastern Lowlands, situated within the Amazon Basin. It is bordered to the north and east by Brazil, to the southeast by Paraguay, to the south by Argentina, to the southwest by Chile, to the northwest by Peru. One-third of the country is within the Andean mountain range. With 1,098,581 km2 of area, Bolivia is the fifth largest country in South America, the 27th largest in the world and the largest landlocked country in the Southern Hemisphere; the country's population, estimated at 11 million, is multiethnic, including Amerindians, Europeans and Africans.
The racial and social segregation that arose from Spanish colonialism has continued to the modern era. Spanish is the official and predominant language, although 36 indigenous languages have official status, of which the most spoken are Guarani and Quechua languages. Before Spanish colonization, the Andean region of Bolivia was part of the Inca Empire, while the northern and eastern lowlands were inhabited by independent tribes. Spanish conquistadors arriving from Cuzco and Asunción took control of the region in the 16th century. During the Spanish colonial period Bolivia was administered by the Royal Audiencia of Charcas. Spain built its empire in large part upon the silver, extracted from Bolivia's mines. After the first call for independence in 1809, 16 years of war followed before the establishment of the Republic, named for Simón Bolívar. Over the course of the 19th and early 20th century Bolivia lost control of several peripheral territories to neighboring countries including the seizure of its coastline by Chile in 1879.
Bolivia remained politically stable until 1971, when Hugo Banzer led a coup d'état which replaced the socialist government of Juan José Torres with a military dictatorship headed by Banzer. Banzer's regime cracked down on leftist and socialist opposition and other forms of dissent, resulting in the torture and deaths of a number of Bolivian citizens. Banzer was ousted in 1978 and returned as the democratically elected president of Bolivia from 1997 to 2001. Modern Bolivia is a charter member of the UN, IMF, NAM, OAS, ACTO, Bank of the South, ALBA and USAN. For over a decade Bolivia has had one of the highest economic growth rates in Latin America, it is a developing country, with a medium ranking in the Human Development Index, a poverty level of 38.6%, one of the lowest crime rates in Latin America. Its main economic activities include agriculture, fishing and manufacturing goods such as textiles, refined metals, refined petroleum. Bolivia is rich in minerals, including tin and lithium. Bolivia is named after Simón Bolívar, a Venezuelan leader in the Spanish American wars of independence.
The leader of Venezuela, Antonio José de Sucre, had been given the option by Bolívar to either unite Charcas with the newly formed Republic of Peru, to unite with the United Provinces of Rio de la Plata, or to formally declare its independence from Spain as a wholly independent state. Sucre opted to create a brand new state and on 6 August 1825, with local support, named it in honor of Simón Bolívar; the original name was Republic of Bolívar. Some days congressman Manuel Martín Cruz proposed: "If from Romulus comes Rome from Bolívar comes Bolivia"; the name was approved by the Republic on 3 October 1825. In 2009, a new constitution changed the country's official name to "Plurinational State of Bolivia" in recognition of the multi-ethnic nature of the country and the enhanced position of Bolivia's indigenous peoples under the new constitution; the region now known as Bolivia had been occupied for over 2,500 years. However, present-day Aymara associate themselves with the ancient civilization of the Tiwanaku culture which had its capital at Tiwanaku, in Western Bolivia.
The capital city of Tiwanaku dates from as early as 1500 BC when it was a small, agriculturally based village. The community grew to urban proportions between AD 600 and AD 800, becoming an important regional power in the southern Andes. According to early estimates, the city covered 6.5 square kilometers at its maximum extent and had between 15,000 and 30,000 inhabitants. In 1996 satellite imaging was used to map the extent of fossilized suka kollus across the three primary valleys of Tiwanaku, arriving at population-carrying capacity estimates of anywhere between 285,000 and 1,482,000 people. Around AD 400, Tiwanaku went from being a locally dominant force to a predatory state. Tiwanaku expanded its reaches into the Yungas and brought its culture and way of life to many other cultures in Peru and Chile. Tiwanaku was not a violent culture in many respects. In order to expand its reach, Tiwanaku exercised great political astuteness, creating colonies, fostering trade agree
John Horton Conway
John Horton Conway is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus. Conway was born in the son of Cyril Horton Conway and Agnes Boyce, he became interested in mathematics at a early age. By the age of eleven his ambition was to become a mathematician. After leaving sixth form, Conway entered Caius College, Cambridge to study mathematics. Conway, a "terribly introverted adolescent" in school, interpreted his admission to Cambridge as an opportunity to transform himself into a new person: an "extrovert", he was awarded his Bachelor of Arts degree in 1959 and began to undertake research in number theory supervised by Harold Davenport.
Having solved the open problem posed by Davenport on writing numbers as the sums of fifth powers, Conway began to become interested in infinite ordinals. It appears that his interest in games began during his years studying the Cambridge Mathematical Tripos, where he became an avid backgammon player, spending hours playing the game in the common room, he was awarded his doctorate in 1964 and was appointed as College Fellow and Lecturer in Mathematics at the University of Cambridge. After leaving Cambridge in 1986, he took up the appointment to the John von Neumann Chair of Mathematics at Princeton University. Conway is known for the invention of the Game of Life, one of the early examples of a cellular automaton, his initial experiments in that field were done with pen and paper, long before personal computers existed. Since the game was introduced by Martin Gardner in Scientific American in 1970, it has spawned hundreds of computer programs, web sites, articles, it is a staple of recreational mathematics.
There is an extensive wiki devoted to cataloging the various aspects of the game. From the earliest days it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data display. At times Conway has said he hates the Game of Life–largely because it has come to overshadow some of the other deeper and more important things he has done; the game did help launch a new branch of mathematics, the field of cellular automata. The Game of Life is now known to be Turing complete. Conway's career is intertwined with mathematics popularizer and Scientific American columnist Martin Gardner; when Gardner featured Conway's Game of Life in his Mathematical Games column in October 1970, it became the most read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, over the years Gardner had written about recreational aspects of Conway's work. For instance, he discussed Conway's game of Sprouts and his angel and devil problem.
In the September 1976 column he reviewed Conway's book On Numbers and Games and introduced the public to Conway's surreal numbers. Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, Conway himself has been a featured speaker at these events, discussing various aspects of recreational mathematics. Conway is known for his contributions to combinatorial game theory, a theory of partisan games; this he developed with Elwyn Berlekamp and Richard Guy, with them co-authored the book Winning Ways for your Mathematical Plays. He wrote the book On Numbers and Games which lays out the mathematical foundations of CGT, he is one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, peg solitaire, Conway's soldiers, he came up with the angel problem, solved in 2006. He invented a new system of numbers, the surreal numbers, which are related to certain games and have been the subject of a mathematical novel by Donald Knuth.
He invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation. Much of this is discussed in the 0th part of ONAG. In the mid-1960s with Michael Guy, son of Richard Guy, Conway established that there are sixty-four convex uniform polychora excluding two infinite sets of prismatic forms, they discovered the grand antiprism in the only non-Wythoffian uniform polychoron. Conway has suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. In the theory of tessellations, he devised the Conway criterion which describes rules for deciding if a prototile will tile the plane, he investigated lattices in higher dimensions, was the first to determine the symmetry group of the Leech lattice. In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials.
Conway further developed tangle theory and invented a system of notation for tabulating knots, nowadays known as Conway notation, while correcting a number of errors in the 19th century knot tables and extending them to include all but four of the non-alternating primes with 11 crossings. See Topology Proceedings 7 118, he was the primary author of the ATLAS of Finite Groups giving prope
Combinatorial game theory
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
Elwyn Ralph Berlekamp was an American mathematician known for his work in computer science, coding theory and combinatorial game theory. He was a professor emeritus of EECS at the University of California, Berkeley. Berlekamp was the inventor of an algorithm to factor polynomials, was one of the inventors of the Berlekamp–Welch algorithm and the Berlekamp–Massey algorithms, which are used to implement Reed–Solomon error correction. Berlekamp had been active in money management. In 1986, he began information-theoretic studies of commodity and financial futures. Berlekamp was born in Ohio, his family moved to Northern Kentucky, where Berlekamp graduated from Ft. Thomas Highlands high school in Ft. Thomas, Campbell county, Kentucky. While an undergraduate at the Massachusetts Institute of Technology, he was a Putnam Fellow in 1961, he completed his bachelor's and master's degrees in electrical engineering in 1962. Continuing his studies at MIT, he finished his Ph. D. in electrical engineering in 1964.
Berlekamp taught electrical engineering at the University of California, Berkeley from 1964 until 1966, when he became a mathematics researcher at Bell Labs. In 1971, Berlekamp returned to Berkeley as professor of mathematics and EECS, where he served as the advisor for over twenty doctoral students, he was a member of the National Academy of Engineering and the National Academy of Sciences. He was elected a Fellow of the American Academy of Arts and Sciences in 1996, became a fellow of the American Mathematical Society in 2012. In 1991, he received the IEEE Richard W. Hamming Medal, in 1993, the Claude E. Shannon Award. In 1998, he received a Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society, he was on its board for many years. In the mid-1980s, he was president of Cyclotomics, Inc. a corporation that developed error-correcting code technology. He co-authored the book Winning Ways for your Mathematical Plays with John Horton Conway and Richard K. Guy, leading to his recognition as one of the founders of combinatorial game theory.
He has studied various games, including dots and boxes and Geese, Go. Berlekamp and co-author David Wolfe describe methods for analyzing certain classes of Go endgames in the book Mathematical Go. In 1989, Berlekamp purchased the largest interest in a trading company named Axcom Trading Advisors. After the firm's futures trading algorithms were rewritten, Axcom's Medallion Fund had a return of 55%, net of all management fees and transaction costs; the fund has subsequently continued to realize annualized returns exceeding 30% under management by James Harris Simons and his Renaissance Technologies Corporation. Berlekamp had a son with his wife Jennifer, he lived in California. He died in April 2019 at the age of 78. Block coding with noiseless feedback. Thesis, Massachusetts Institute of Technology, Dept. of Electrical Engineering, 1964. Algebraic Coding Theory, New York: McGraw-Hill, 1968. Revised ed. Aegean Park Press, 1984, ISBN 0-89412-063-8. Winning Ways for your Mathematical Plays. 1st edition, New York: Academic Press, 2 vols.
1982. 1, hardback: ISBN 0-12-091150-7, paperback: ISBN 0-12-091101-9. 2, hardback: ISBN 0-12-091152-3, paperback: ISBN 0-12-091102-7. 2nd edition, Massachusetts: A. K. Peters Ltd. 4 vols. 2001–2004. Mathematical Go. Wellesley, Massachusetts: A. K. Peters Ltd. 1994. ISBN 1-56881-032-6; the Dots-and-Boxes Game. Natick, Massachusetts: A. K. Peters Ltd. 2000. ISBN 1-56881-129-2. Elwyn Berlekamp home page at the University of California, Berkeley. Elwyn Berlekamp at the Mathematics Genealogy Project