Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
Cosette is a fictional character in the novel Les Misérables by Victor Hugo and in the many adaptations of the story for stage and television. Her birth name, Euphrasie, is only mentioned briefly; as the orphaned child of an unmarried mother deserted by her father, Hugo never gives her a surname. In the course of the novel, she is mistakenly identified as Lark, or Mademoiselle Lanoire, she is the daughter of Fantine, who leaves her to be looked after by the Thénardiers, who exploit and victimise her. Rescued by Jean Valjean, who raises Cosette as if she were his own, she grows up in a convent school, she falls in love with a young lawyer. Valjean's struggle to protect her while disguising his past drives much of the plot until he recognizes "that this child had a right to know life before renouncing it"—and must yield to her romantic attachment to Marius. Euphrasie, nicknamed Cosette by her mother, illegitimate daughter of Fantine and Félix Tholomyès, a rich student, is born in Paris circa 1815.
Tholomyès abandons Fantine, who leaves three-year-old Cosette with the Thénardiers at their inn in Montfermeil, paying them to care for her child while she works in the city of Montreuil-sur-Mer. Unbeknownst to Fantine, the Thénardiers abuse Cosette, they force her to perform heavy labor in the inn. Under the Thénardiers' care she is described as "thin and pale", wears rags for clothing, has chilblains on her hands as well as bruised and reddened skin, she is forced to go barefoot in winter. The narrator says that "fear was spread all over her". While Fantine is in the hospital, the mayor of Montreuil-sur-Mer—Jean Valjean masquerading as "Madeleine" to avoid revealing that he is a paroled convict—vows to retrieve Cosette for her. Although Fantine dies before seeing her daughter, Valjean becomes determined to look after the young girl; when he arrives in Montfermeil on Christmas Eve, Valjean finds Cosette fetching a pail of water and accompanies her back to the inn, where he witnesses her mistreatment by the Thénardiers as well as the unkindnesses of their daughters Éponine and Azelma.
Valjean leaves the inn and returns with an expensive new doll. At first reluctant, Cosette joyfully accepts it; the next morning, Christmas Day, Valjean informs the Thénardiers that he has come to take Cosette with him. Madame Thénardier agrees but Thénardier pretends affection for Cosette and acts reluctant. Valjean pays 1500 francs to settle Fantine's leaves with Cosette. Thénardier tries to swindle more money out of Valjean by saying that he has changed his mind and wants Cosette back, informing Valjean that Cosette's mother had entrusted her to his care and he cannot release her without a note from her mother. Valjean, agreeing with him, hands him a letter signed by Fantine. Thénardier attempts to order Valjean to return Cosette or to pay a thousand crowns, but Valjean ignores him and leaves with Cosette. Valjean takes Cosette to Paris; when Inspector Javert discovers Valjean's whereabouts, he and Cosette must flee. Valjean climbs over a wall and they find themselves in a garden attached to a convent.
The gardener Fauchelevent recognizes Valjean as the man who rescued him years earlier and agrees to shelter them. Valjean says he is Cosette's grandfather, they live peacefully in the convent for many years as Valjean works with Fauchelevent and Cosette attends the convent school. Over time, she appears to have no recollection of her childhood before arriving at the convent; as Cosette matures and healthy, with chestnut hair, beautiful eyes, rosy cheeks, pale skin, a radiant smile, Valjean realizes it would be unfair to allow her to become a nun without having experienced the outside world which a cloistered nun must renounce. In a chapter, Cosette remembers her childhood—praying for the mother she never knew. In the same chapter, she asks Valjean about her mother; when Cosette has a dream about her mother as an angel, she remarks that her mother must have been a saint. Valjean replies, "through martyrdom". Marius sees Cosette for the first time in the Luxembourg Garden, she is fourteen years old, fresh out of the convent, so he pays little attention to her.
After a few months, Marius notices her and sees that she has grown to be an beautiful young woman. Soon Cosette and Marius exchange glances and fall in love. Valjean notices; when he learns that Marius has followed them home and inquired about them, he moves to a more obscure address with Cosette. Marius spots Cosette again during a charitable visit she and Valjean make to the Thénardiers at Gorbeau House, directly next door to Marius, he asks Éponine to find her address for him, she reluctantly agrees. After many weeks, Éponine takes Marius to Cosette's new address. Marius watches Cosette for a few nights before approaching her; when Cosette and Marius meet again in the garden, they confess their mutual love, share their first kiss, introduce themselves. They continue to meet in secret. Éponine prevents Patron-Minette and Brujon from robbing Valjean and Cosette's house. The same night, Cosette informs Marius; this news devastates them both. Marius attempts to obtain money and permission to marry from his grandfather to circumvent this issue.
Their discussion dissolves int
Hans Julius Zassenhaus was a German mathematician, known for work in many parts of abstract algebra, as a pioneer of computer algebra. He was born in Koblenz in 1912, his father was a advocate for Reverence for Life as expressed by Albert Schweitzer. Hans had two brothers and Wilfred, sister Hiltgunt, who wrote an autobiography in 1974. According to her, their father lost his position as school principal due to his philosophy, she wrote: Hans, my eldest brother, studied mathematics. My brothers Guenther and Wilfred were in medical school.... Only students who participated in Nazi activities would get scholarships; that left us out. Together we made an all-out effort.... Soon our house became a beehive. Day in and day out for the next four years a small army of children of all ages would arrive to be tutored. At the University of Hamburg Zassenhaus came under the influence of Emil Artin; as he wrote later: His introductory course in analysis that I attended at the age of 17 converted me from a theoretical physicist to a mathematician.
When just 21, Zassenhaus was studying composition series in group theory. He proved his butterfly lemma that provides a refinement of two normal chains to isomorphic central chains. Inspired by Artin, Zassenhaus wrote a textbook Lehrbuch der Gruppentheorie, translated as Theory of Groups, his thesis was on doubly transitive permutation groups with Frobenius groups as stabilizers. These groups are now called Zassenhaus groups, they have had a deep impact on the classification of finite simple groups. He obtained his doctorate in June 1934, took the teachers’ exam the next May, he became a scientific assistant at University of Rostock. In 1936 he became assistant to Artin back in Hamburg, but Artin departed for the USA the following year. Zassenhaus gave his Habilitation in 1938. According to his sister Hiltgunt, Hans was "called up as a research scientist at a weather station" for his part in the German war effort. Zassenhaus married Lieselotte Lohmann in 1942; the couple raised three children: Michael and Peter.
In 1943 Zassenhaus became extraordinary professor. He became Managing Director of the Hamburg Mathematical Seminar. After the war, as a fellow of the British Council, Zassenhaus visited University of Glasgow in 1948. There he was given an honorary Master of Arts degree; the following year he joined the faculty of McGill University where the endowments of Peter Redpath financed a professorship. He was at McGill for a decade with leaves of absence to Institute for Advanced Study and California Institute of Technology. There he was using computers to advance number theory. In 1959 Zassenhaus began teaching at University of Notre Dame and became director of its computing center in 1964. Zassenhaus was a Mershon visiting professor at Ohio State University in the fall of 1963. In 1965 he came to Ohio State permanently; the mathematics department was led by Arnold Ross. Nonetheless, he continued to take leaves of absence for visits to Göttingen, Heidelberg, UCLA, Warwick, CIT, U Montreal, Saarbrücken, he served as editor in chief of the Journal of Number Theory from its first issue in 1967.
He won a Lester R. Ford Award in 1968. Hans Zassenhaus died in Columbus, Ohio on November 21, 1991, his doctoral students include Joachim Lambek. Hans Julius Zassenhaus, Lehrbuch der Gruppentheorie, 2nd edition,The theory of groups. A famous group theory book based on a course by Emil Artin given at the University of Hamburg during winter semester 1933 and summer semester 1934. Zassenhaus showed that there are just seven near-fields that are not division rings or Dickson near-fields in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11, pp 187–220. In 1977 Academic Press published Number Theory and Algebra, a collection of papers dedicated to Henry B. Mann, Arnold E. Ross, Olga Taussky-Todd, edited by Zassenhaus, it included "A Theorem on Cyclic Algebras" by Zassenhaus. Cambridge University Press published Algorithmic Algebraic Number Theory written by Zassenhaus and M. Pohst in 1989. A second edition appeared in 1993. Pohst, M.. 1st paperback edition. ISBN 978-0-521-59669-5. Cantor, David G..
The paper that introduced the Cantor–Zassenhaus algorithm for factoring polynomials. Zassenhaus dual expansion Schur–Zassenhaus theorem M. Pohst "Hans Zassenhaus", Journal of Number Theory 47:1–19. O'Connor, John J.. "Hans Zassenhaus", MacTutor History of Mathematics archive, University of St Andrews. Biography from the Ohio State University
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
Partition of a set
In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets. Every equivalence relation on a set defines a partition of this set, every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid in type theory and proof theory. A partition of a set X is a set of nonempty subsets of X such that every element x in X is in one of these subsets. Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: The family P does not contain the empty set; the union of the sets in P is equal to X. The sets in P are said to cover X; the intersection of any two distinct sets in P is empty. The elements of P are said to be pairwise disjoint; the sets in P are called the parts or cells of the partition. The rank of P is | X | − | P |; the empty set ∅ has one partition, namely ∅. For any nonempty set X, P = is a partition of X, called the trivial partition.
Every singleton set has one partition, namely. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, namely; the set has these five partitions:, sometimes written 1|2|3. Or 12|3. Or 13|2. Or 1|23. Or 123; the following are not partitions of: is not a partition because one of its elements is the empty set. Is not a partition because the element 2 is contained in more than one block. Is not a partition of because none of its blocks contains 3. For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y when x and y are in the same part in P, thus the notions of equivalence relation and partition are equivalent. The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing one element from each part of the partition; this implies that given an equivalence relation on a set one can select a canonical representative element from every equivalence class.
A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that α ≤ ρ; this finer-than relation on the set of partitions of X is a partial order. Each set of elements has a least upper bound and a greatest lower bound, so that it forms a lattice, more it is a geometric lattice; the partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left. Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a finite set corresponds to a matroid in which the base set of the matroid consists of the atoms of the lattice, the partitions with n − 2 singleton sets and one two-element set; these atomic partitions correspond one-for-one with the edges of a complete graph. The matroid closure of a set of atomic partitions is the finest common coarsening of them all.
In this way, the lattice of partitions corresponds to the lattice of flats of the graphic matroid of the complete graph. Another example illustrates the refining of partitions from the perspective of equivalence relations. If D is the set of cards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalence classes: the sets and; the 2-part partition corresponding to ~C has a refinement that yields the same-suit-as relation ~S, which has the four equivalence classes, and. A partition of the set N = with corresponding equivalence relation ~ is noncrossing if it has the following property: If four elements a, b, c and d of N having a < b < c < d satisfy a ~ c and b ~ d a ~ b ~ c ~ d. The name comes from the following equivalent definition: Imagine the elements 1, 2... N of N drawn as the n vertices of a regular n-gon. A partition can be visualized by drawing each block as a polygon; the partition is noncrossing if and only if these polygons do not intersect.
The lattice of noncrossing partitions of a finite set has taken on importance because of its role in free probability theory. These form a subset of the lattice of
In geometry, parallel lines are lines in a plane which do not meet. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same three-dimensional space. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism; the parallel symbol is ∥. For example, A B ∥ C D indicates that line AB is parallel to line CD. In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation "equal and parallel to". Given parallel straight lines l and m in Euclidean space, the following properties are equivalent: Every point on line m is located at the same distance from line l.
Line m is in the same plane as line l but does not intersect l. When lines m and l are both intersected by a third straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, so, are "more complicated" than the second. Thus, the second property is the one chosen as the defining property of parallel lines in Euclidean geometry; the other properties are consequences of Euclid's Parallel Postulate. Another property that involves measurement is that lines parallel to each other have the same gradient; the definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. Alternative definitions were discussed by other Greeks as part of an attempt to prove the parallel postulate. Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein.
Simplicius mentions Posidonius' definition as well as its modification by the philosopher Aganis. At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools; the traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines; these reform texts were not without their critics and one of them, Charles Dodgson, wrote a play and His Modern Rivals, in which these texts are lambasted. One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction. According to Wilhelm Killing the idea may be traced back to Leibniz. Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, the difference of their directions is the angle between them."
Wilson In definition 15 he introduces parallel lines in this way. Wilson Augustus De Morgan reviewed this text and declared it a failure on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson devotes a large section of his play to denouncing Wilson's treatment of parallels. Wilson edited this concept out of the third and higher editions of his text. Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better; the main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line; this must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles all transversals must do so.
Again, a new axiom is needed to justify this statement. The three properties above lead to three different methods of construction of parallel lines; because parallel lines in a Euclidean plane are equidistant there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines, y = m x + b 1 y = m x + b 2
Rubik's Cube is a 3-D combination puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Called the Magic Cube, the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 via businessman Tibor Laczi and Seven Towns founder Tom Kremer, won the German Game of the Year special award for Best Puzzle that year. As of January 2009, 350 million cubes had been sold worldwide making it the world's top-selling puzzle game, it is considered to be the world's best-selling toy. On the original classic Rubik's Cube, each of the six faces was covered by nine stickers, each of one of six solid colours: white, blue, orange and yellow; the current version of the cube has been updated to coloured plastic panels instead, which prevents peeling and fading. In sold models, white is opposite yellow, blue is opposite green, orange is opposite red, the red and blue are arranged in that order in a clockwise arrangement. On early cubes, the position of the colours varied from cube to cube.
An internal pivot mechanism enables each face to turn thus mixing up the colours. For the puzzle to be solved, each face must be returned to have only one colour. Similar puzzles have now been produced with various numbers of sides and stickers, not all of them by Rubik. Although the Rubik's Cube reached its height of mainstream popularity in the 1980s, it is still known and used. Many speedcubers continue to practice similar puzzles. Since 2003, the World Cube Association, the Rubik's Cube's international governing body, has organised competitions worldwide and recognises world records. In March 1970, Larry D. Nichols invented a 2×2×2 "Puzzle with Pieces Rotatable in Groups" and filed a Canadian patent application for it. Nichols's cube was held together by magnets. Nichols was granted U. S. Patent 3,655,201 on 11 April 1972, two years before Rubik invented his Cube. On 9 April 1970, Frank Fox applied to patent an "amusement device", a type of sliding puzzle on a spherical surface with "at least two 3×3 arrays" intended to be used for the game of noughts and crosses.
He received his UK patent on 16 January 1974. In the mid-1970s, Ernő Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest. Although it is reported that the Cube was built as a teaching tool to help his students understand 3D objects, his actual purpose was solving the structural problem of moving the parts independently without the entire mechanism falling apart, he did not realise that he had created a puzzle until the first time he scrambled his new Cube and tried to restore it. Rubik applied for a patent in Hungary for his "Magic Cube" on 30 January 1975, HU170062 was granted that year; the first test batches of the Magic Cube were produced in late 1977 and released in Budapest toy shops. Magic Cube was held together with interlocking plastic pieces that prevented the puzzle being pulled apart, unlike the magnets in Nichols's design. With Ernő Rubik's permission, businessman Tibor Laczi took a Cube to Germany's Nuremberg Toy Fair in February 1979 in an attempt to popularise it.
It was noticed by Seven Towns founder Tom Kremer and they signed a deal with Ideal Toys in September 1979 to release the Magic Cube worldwide. Ideal wanted at least a recognisable name to trademark; the puzzle made its international debut at the toy fairs of London, Paris and New York in January and February 1980. After its international debut, the progress of the Cube towards the toy shop shelves of the West was halted so that it could be manufactured to Western safety and packaging specifications. A lighter Cube was produced, Ideal decided to rename it. "The Gordian Knot" and "Inca Gold" were considered, but the company decided on "Rubik's Cube", the first batch was exported from Hungary in May 1980. After the first batches of Rubik's Cubes were released in May 1980, initial sales were modest, but Ideal began a television advertising campaign in the middle of the year which it supplemented with newspaper adverts. At the end of 1980 Rubik's Cube won a German Game of the Year special award, won similar awards for best toy in the UK, the US.
By 1981 Rubik's Cube had become a craze, it is estimated that in the period from 1980 to 1983 around 200 million Rubik's Cubes were sold worldwide. In March 1981 a speedcubing championship organised by the Guinness Book of World Records was held in Munich, a Rubik's Cube was depicted on the front cover of Scientific American that same month. In June 1981 The Washington Post reported that the Rubik's Cube is "a puzzle that's moving like fast food right now... this year's Hoola Hoop or Bongo Board", by September 1981 New Scientist noted that the cube had "captivated the attention of children of ages from 7 to 70 all over the world this summer."As most people could only solve one or two sides, numerous books were published including David Singmaster's Notes on Rubik's "Magic Cube" and Patrick Bossert's You Can Do the Cube. At one stage in 1981 three of the top ten best selling books in the US were books on solving the Rubik's Cube, the best-selling book of 1981 was James G. Nourse's The Simple Solution to Rubik's Cube which sold over 6 million copies.
In 1981 the Museum of Modern Art in New York exhibited a Rubik's Cube, at the 1982 World's Fair in Knoxville, Tennessee a six-foot Cube was put on display. ABC Television developed a cartoon show called Rubik, the Amazing Cube. In June 1982 the First Rubik's Cube World Championship