Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line; the game starts with the players drawing a "ground" line and several line segments such that each line segment is connected to the ground, either directly at an endpoint, or indirectly, via a chain of other segments connected by endpoints. Any number of segments may meet at a point and thus. On his turn, a player "cuts" any line segment of his choice; every line segment no longer connected to the ground by any path "falls". According to the normal play convention of combinatorial game theory, the first player, unable to move loses. Hackenbush boards can consist of infinitely many line segments; the existence of an infinite number of line segments does not violate the game theory assumption that the game can be finished in a finite amount of time, provided that there are only finitely many line segments directly "touching" the ground.
On an infinite board, based on the layout of the board the game can continue on forever, assuming there are infinitely many points touching the ground. In the original folklore version of Hackenbush, any player is allowed to cut any edge: as this is an impartial game it is comparatively straightforward to give a complete analysis using the Sprague–Grundy theorem, thus the versions of Hackenbush of interest in combinatorial game theory are more complex partisan games, meaning that the options available to one player would not be the ones available to the other player if it were his turn to move given the same position. This is achieved in one of two ways: Original Hackenbush: All line segments are the same color and may be cut by either player; this means payoffs are symmetric and each player has the same operations based on position on board Blue-Red Hackenbush: Each line segment is colored either red or blue. One player is only allowed to cut blue line segments, while the other player is only allowed to cut red line segments.
Blue-Red-Green Hackenbush: Each line segment is colored red, blue, or green. The rules are the same as for Blue-Red Hackenbush, with the additional stipulation that green line segments can be cut by either player. Blue-Red Hackenbush is a special case of Blue-Red-Green Hackenbush, but it is worth noting separately, as its analysis is much simpler; this is because Blue-Red Hackenbush is a so-called cold game, which means that it can never be an advantage to have the first move. Hackenbush has been used as an example game for demonstrating the definitions and concepts in combinatorial game theory, beginning with its use in the books On Numbers and Games and Winning Ways for your Mathematical Plays by some of the founders of the field. In particular Blue-Red Hackenbush can be used to construct surreal numbers such as nimbers: finite Blue-Red Hackenbush boards can construct dyadic rational numbers, while the values of infinite Blue-Red Hackenbush boards account for real numbers and many more general values that are neither.
Blue-Red-Green Hackenbush allows for the construction of additional games whose values are not real numbers, such as star and all other nimbers. Further analysis of the game can be made using graph theory by considering the board as a collection of vertices and edges and examining the paths to each vertex that lies on the ground. In the impartial version of Hackenbush, it can be thought of using nim heaps by breaking the game up into several cases: vertical and divergent. Played with vertical stacks of line segments referred to as bamboo stalks, the game directly becomes Nim and can be directly analyzed as such. Divergent segments, or trees, add an additional wrinkle to the game and require use of the colon principle stating that when branches come together at a vertex, one may replace the branches by a non-branching stalk of length equal to their nim sum; this principle changes the representation of the game to the more basic version of the bamboo stalks. The last possible set of graphs that can be made are convergent ones known as arbitrarily rooted graphs.
By using the fusion principle, we can state that all vertices on any cycle may be fused together without changing the value of the graph. Therefore, any convergent graph can be interpreted as a simple bamboo stalk graph. By combining all three types of graphs we can add complexity to the game, without changing the nim sum of the game, thereby allowing the game to take the strategies of Nim; the Colon Principle states that when branches come together at a vertex, one may replace the branches by a non-branching stalk of length equal to their nim sum. Consider a fixed but arbitrary graph, G, select an arbitrary vertex, x, in G. Let H1 and H2 be arbitrary trees that have the same Sprague-Grundy value. Consider the two graphs G1 = Gx: H1 and G2 = Gx: H2, where Gx: Hi represents the graph constructed by attaching the tree Hi to the vertex x of the graph G; the colon principle states that the two graphs G2 have the same Sprague-Grundy value. Consider the sum of the two games as in figure 5.4. The claim that G1 and G2 have the same Sprague-Grundy value is equi
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, A contains no greatest element; the set B may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. In other words, A contains every rational number less than the cut, B contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number, in neither set; every real number, rational or not, is equated to only one cut of rationals. Dedekind cuts can be generalized from the rational numbers to any ordered set by defining a Dedekind cut as a partition of a ordered set into two non-empty parts A and B, such that A is closed downwards and B is closed upwards, A contains no greatest element.
See completeness. It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers; every cut of reals is identical to the cut produced by a specific real number. In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps. A similar construction to that used by Dedekind cuts was used in Euclid's Elements to define proportional segments. A Dedekind cut is a partition of the rationals Q into two subsets B such that A is nonempty. A ≠ Q. If x, y ∈ Q, x < y, y ∈ A x ∈ A. If x ∈ A there exists a y ∈ A such that y > x. By relaxing the first two requirements, we formally obtain the extended real number line, it is more symmetrical to use the notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one "half" — say, the lower one — and call any downward closed set A without greatest element a "Dedekind cut".
If the ordered set S is complete for every Dedekind cut of S, the set B must have a minimal element b, hence we must have that A is the interval, B the interval [b, +∞). In this case, we say; the important purpose of the Dedekind cut is to work with number sets. The cut itself can represent a number not in the original collection of numbers; the cut can represent a number b though the numbers contained in the two sets A and B do not include the number b that their cut represents. For example if A and B only contain rational numbers, they can still be cut at √2 by putting every negative rational number in A, along with every non-negative number whose square is less than 2. Though there is no rational value for √2, if the rational numbers are partitioned into A and B this way, the partition itself represents an irrational number. Regard one Dedekind cut as less than another Dedekind cut if A is a proper subset of C. Equivalently, if D is a proper subset of B, the cut is again less than. In this way, set inclusion can be used to represent the ordering of numbers, all other relations can be created from set relations.
The set of all Dedekind cuts is itself a linearly ordered set. Moreover, the set of Dedekind cuts has the least-upper-bound property, i.e. every nonempty subset of it that has any upper bound has a least upper bound. Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set S, which might not have had the least-upper-bound property, within a linearly ordered set that does have this useful property. A typical Dedekind cut of the rational numbers Q is given by the partition with A =, B =; this cut represents the irrational number √2 in Dedekind's construction. To est
Scientific American is an American popular science magazine. Many famous scientists, including Albert Einstein, have contributed articles to it, it is the oldest continuously published monthly magazine in the United States. Scientific American was founded by inventor and publisher Rufus M. Porter in 1845 as a four-page weekly newspaper. Throughout its early years, much emphasis was placed on reports of what was going on at the U. S. Patent Office, it reported on a broad range of inventions including perpetual motion machines, an 1860 device for buoying vessels by Abraham Lincoln, the universal joint which now can be found in nearly every automobile manufactured. Current issues include a "this date in history" section, featuring excerpts from articles published 50, 100, 150 years earlier. Topics include humorous incidents, wrong-headed theories, noteworthy advances in the history of science and technology. Porter sold the publication to Alfred Ely Beach and Orson Desaix Munn a mere ten months after founding it.
Until 1948, it remained owned by Company. Under Munn's grandson, Orson Desaix Munn III, it had evolved into something of a "workbench" publication, similar to the twentieth-century incarnation of Popular Science. In the years after World War II, the magazine fell into decline. In 1948, three partners who were planning on starting a new popular science magazine, to be called The Sciences, purchased the assets of the old Scientific American instead and put its name on the designs they had created for their new magazine, thus the partners—publisher Gerard Piel, editor Dennis Flanagan, general manager Donald H. Miller, Jr.—essentially created a new magazine. Miller retired in 1979, Flanagan and Piel in 1984, when Gerard Piel's son Jonathan became president and editor. In 1986, it was sold to the Holtzbrinck group of Germany. In the fall of 2008, Scientific American was put under the control of Nature Publishing Group, a division of Holtzbrinck. Donald Miller died in December 1998, Gerard Piel in September 2004 and Dennis Flanagan in January 2005.
Mariette DiChristina is the current editor-in-chief, after John Rennie stepped down in June 2009. Scientific American published its first foreign edition in 1890, the Spanish-language La America Cientifica. Publication was suspended in 1905, another 63 years would pass before another foreign-language edition appeared: In 1968, an Italian edition, Le Scienze, was launched, a Japanese edition, Nikkei Science, followed three years later. A new Spanish edition, Investigación y Ciencia was launched in Spain in 1976, followed by a French edition, Pour la Science, in France in 1977, a German edition, Spektrum der Wissenschaft, in Germany in 1978. A Russian edition V Mire Nauki was launched in the Soviet Union in 1983, continues in the present-day Russian Federation. Kexue, a simplified Chinese edition launched in 1979, was the first Western magazine published in the People's Republic of China. Founded in Chongqing, the simplified Chinese magazine was transferred to Beijing in 2001. In 2005, a newer edition, Global Science, was published instead of Kexue, which shut down due to financial problems.
A traditional Chinese edition, known as Scientist, was introduced to Taiwan in 2002. The Hungarian edition Tudomány existed between 1984 and 1992. In 1986, an Arabic edition, Oloom Magazine, was published. In 2002, a Portuguese edition was launched in Brazil. Today, Scientific American publishes 18 foreign-language editions around the globe: Arabic, Brazilian Portuguese, Simplified Chinese, Traditional Chinese, Dutch, German, Hebrew, Japanese, Lithuanian, Romanian and Spanish. From 1902 to 1911, Scientific American supervised the publication of the Encyclopedia Americana, which during some of that period was known as The Americana, it styled itself "The Advocate of Industry and Enterprise" and "Journal of Mechanical and other Improvements". On the front page of the first issue was the engraving of "Improved Rail-Road Cars"; the masthead had a commentary as follows: Scientific American published every Thursday morning at No. 11 Spruce Street, New York, No. 16 State Street, No. 2l Arcade Philadelphia, by Rufus Porter.
Each number will be furnished with from two to five original Engravings, many of them elegant, illustrative of New Inventions, Scientific Principles, Curious Works. Improvements and Inventions; this paper is entitled to the patronage of Mechanics and Manufactures, being the only paper in America, devoted to the interest of those classes. As a family newspaper, it will convey more useful intelligence to children and young people, than five times its cost in school instruction. Another important argument in favor of this paper, is that it will be worth two dollars at the end of the year when the volume is complete, (Old volumes of the New York Mechanic, being now worth double th
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
The United States of America known as the United States or America, is a country composed of 50 states, a federal district, five major self-governing territories, various possessions. At 3.8 million square miles, the United States is the world's third or fourth largest country by total area and is smaller than the entire continent of Europe's 3.9 million square miles. With a population of over 327 million people, the U. S. is the third most populous country. The capital is Washington, D. C. and the largest city by population is New York City. Forty-eight states and the capital's federal district are contiguous in North America between Canada and Mexico; the State of Alaska is in the northwest corner of North America, bordered by Canada to the east and across the Bering Strait from Russia to the west. The State of Hawaii is an archipelago in the mid-Pacific Ocean; the U. S. territories are scattered about the Pacific Ocean and the Caribbean Sea, stretching across nine official time zones. The diverse geography and wildlife of the United States make it one of the world's 17 megadiverse countries.
Paleo-Indians migrated from Siberia to the North American mainland at least 12,000 years ago. European colonization began in the 16th century; the United States emerged from the thirteen British colonies established along the East Coast. Numerous disputes between Great Britain and the colonies following the French and Indian War led to the American Revolution, which began in 1775, the subsequent Declaration of Independence in 1776; the war ended in 1783 with the United States becoming the first country to gain independence from a European power. The current constitution was adopted in 1788, with the first ten amendments, collectively named the Bill of Rights, being ratified in 1791 to guarantee many fundamental civil liberties; the United States embarked on a vigorous expansion across North America throughout the 19th century, acquiring new territories, displacing Native American tribes, admitting new states until it spanned the continent by 1848. During the second half of the 19th century, the Civil War led to the abolition of slavery.
By the end of the century, the United States had extended into the Pacific Ocean, its economy, driven in large part by the Industrial Revolution, began to soar. The Spanish–American War and World War I confirmed the country's status as a global military power; the United States emerged from World War II as a global superpower, the first country to develop nuclear weapons, the only country to use them in warfare, a permanent member of the United Nations Security Council. Sweeping civil rights legislation, notably the Civil Rights Act of 1964, the Voting Rights Act of 1965 and the Fair Housing Act of 1968, outlawed discrimination based on race or color. During the Cold War, the United States and the Soviet Union competed in the Space Race, culminating with the 1969 U. S. Moon landing; the end of the Cold War and the collapse of the Soviet Union in 1991 left the United States as the world's sole superpower. The United States is the world's oldest surviving federation, it is a representative democracy.
The United States is a founding member of the United Nations, World Bank, International Monetary Fund, Organization of American States, other international organizations. The United States is a developed country, with the world's largest economy by nominal GDP and second-largest economy by PPP, accounting for a quarter of global GDP; the U. S. economy is post-industrial, characterized by the dominance of services and knowledge-based activities, although the manufacturing sector remains the second-largest in the world. The United States is the world's largest importer and the second largest exporter of goods, by value. Although its population is only 4.3% of the world total, the U. S. holds 31% of the total wealth in the world, the largest share of global wealth concentrated in a single country. Despite wide income and wealth disparities, the United States continues to rank high in measures of socioeconomic performance, including average wage, human development, per capita GDP, worker productivity.
The United States is the foremost military power in the world, making up a third of global military spending, is a leading political and scientific force internationally. In 1507, the German cartographer Martin Waldseemüller produced a world map on which he named the lands of the Western Hemisphere America in honor of the Italian explorer and cartographer Amerigo Vespucci; the first documentary evidence of the phrase "United States of America" is from a letter dated January 2, 1776, written by Stephen Moylan, Esq. to George Washington's aide-de-camp and Muster-Master General of the Continental Army, Lt. Col. Joseph Reed. Moylan expressed his wish to go "with full and ample powers from the United States of America to Spain" to seek assistance in the revolutionary war effort; the first known publication of the phrase "United States of America" was in an anonymous essay in The Virginia Gazette newspaper in Williamsburg, Virginia, on April 6, 1776. The second draft of the Articles of Confederation, prepared by John Dickinson and completed by June 17, 1776, at the latest, declared "The name of this Confederation shall be the'United States of America'".
The final version of the Articles sent to the states for ratification in late 1777 contains the sentence "The Stile of this Confederacy shall be'The United States of America'". In June 1776, Thomas Jefferson wrote the phrase "UNITED STATES OF AMERICA" in all capitalized letters in the headline of his "original Rough draught" of the Declaration of Independence; this draft of the document did not surface unti
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
Aviezri Siegmund Fraenkel is an Israeli mathematician who has made contributions to combinatorial game theory. Aviezri Siegmund Fraenkel was born in Munich, Germany, to a Jewish family, which moved to Switzerland soon thereafter. In 1939 his family moved once more, to Jerusalem. Fraenkel is married to father of six. One of his grandchildren, Yaacov Naftali Fraenkel, was kidnapped and murdered by Hamas members in June 2014. In 2018, in respect for his life's work, Fraenkel was given the highly-regarded honor to light one of twelve torches—one for each of Twelve Tribes—at the State of Israel's televised opening ceremony on Har Herzl, Jerusalem, in celebration of Israel's 70th Independence Day. Fraenkel received his Ph. D. in 1961 from the University of California, Los Angeles. He was a recipient of the 2005 Euler Medal. Together with Ralph Faudree. On December 5, 2006, he received the "WEIZAC Medal" from the IEEE, as a member of the team that built the WEIZAC, one of the first computers in the world and the first computer built in Israel.
Fraenkel was the founder of the Bar Ilan Responsa Project, serving as its initial director, which received the Israel Prize in 2007. His research delves into computational complexity, as it is important to study the complexity of algorithms which solve games. Biography by Shaula Fraenkel Personal Homepage The official citation from the Israel Prize for the Responsa Project The official Responsa Project CV from the Israel Prize committee