|
Magic Series
A magic series is a set of distinct positive integers which add up to the magic constant of a magic square and a magic cube, thus potentially making up lines in magic tesseracts. So, in an ''n'' × ''n'' magic square using the numbers from 1 to ''n''2, a magic series is a set of ''n'' distinct numbers adding up to ''n''(''n''2 + 1)/2. For ''n'' = 2, there are just two magic series, 1+4 and 2+3. The eight magic series when ''n'' = 3 all appear in the rows, columns and diagonals of a 3 × 3 magic square. Maurice Kraitchik gave the number of magic series up to ''n'' = 7 in ''Mathematical Recreations'' in 1942 . In 2002, Henry Bottomley extended this up to ''n'' = 36 and independently Walter Trump up to ''n'' = 32. In 2005, Trump extended this to ''n'' = 54 (over 2 × 10111) while Bottomley gave an experimental approximation for the numbers of magic series: :\frac \cdot \sqrt \cdot \frac I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is, a magic square which contains the numbers 1, 2, ..., ''n''2 – the magic constant is M = n \cdot \frac. For normal magic squares of orders ''n'' = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). For example, a normal 8 × 8 square will always equate to 260 for each row, column, or diagonal. The normal magic constant of order n is (n^3+n)/2. The largest magic constant of normal magic square which is also a: * triangular number is 15 (solve the Diophantine equation x^2=y^3+16y+16, where y is divisible by 4); *square number is 1 (solve the Diophantine equation x^2=y^3+4y, where y is even); * generalized pentagonal number is 171535 (solve the Dioph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number of integers along one side (''n''), and the constant sum is called the ' magic constant'. If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be 'normal'. Some authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as 'trivial'. Some well-known examples, including the Sagrada Família magic square and the Parker square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant this gives a ''semimagic square (sometimes called orthomagic square). The mathematical study of magic squares typically deals with their construction, classification, and enumeration. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Cube
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an ''n'' × ''n'' × ''n'' pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal to the same number, the so-called magic constant of the cube, denoted ''M''3(''n''). It can be shown that if a magic cube consists of the numbers 1, 2, ..., ''n''3, then it has magic constant :M_3(n) = \frac. If, in addition, the numbers on every cross section diagonal also sum up to the cube's magic number, the cube is called a perfect magic cube; otherwise, it is called a semiperfect magic cube. The number ''n'' is called the order of the magic cube. If the sums of numbers on a magic cube's broken space diagonals also equal the cube's magic number, the cube is called a pandiagonal magic cube. Alternative definition In recent years, an alternative definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Tesseract
In mathematics, a magic hypercube is the ''k''-dimensional generalization of magic squares and magic cubes, that is, an ''n'' × ''n'' × ''n'' × ... × ''n'' array of integers such that the sums of the numbers on each pillar (along any axis) as well as on the main space diagonals are all the same. The common sum is called the magic constant of the hypercube, and is sometimes denoted ''M''''k''(''n''). If a magic hypercube consists of the numbers 1, 2, ..., ''n''''k'', then it has magic number :M_k(n) = \frac. For ''k'' = 4, a magic hypercube may be called a magic tesseract, with sequence of magic numbers given by . The side-length ''n'' of the magic hypercube is called its ''order''. Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks. Marian Trenkler proved the following theorem: A ''p''-dimensional magic hypercube of order ''n'' exists if and only if ''p'' > 1 and ''n'' is different from 2 or ''p'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice Kraitchik
Maurice Borisovich Kraitchik (21 April 1882 – 19 August 1957) was a Belgian mathematician and populariser. His main interests were the theory of numbers and recreational mathematics. He was born to a Jewish family in Minsk. He wrote several books on number theory during 1922–1930 and after the war, and from 1931 to 1939 edited ''Sphinx'', a periodical devoted to recreational mathematics. During World War II, he emigrated to the United States, where he taught a course at the New School for Social Research in New York City on the general topic of "mathematical recreations." Kraïtchik was ''agrégé'' of the Free University of Brussels, engineer at the Société Financière de Transports et d'Entreprises Industrielles (Sofina), and director of the Institut des Hautes Etudes de Belgique. He died in Brussels. Kraïtchik is famous for having inspired the two envelopes problem The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Bottomley
Henry may refer to: People * Henry (given name) * Henry (surname) * Henry Lau, Canadian singer and musician who performs under the mononym Henry Royalty * Portuguese royalty ** King-Cardinal Henry, King of Portugal ** Henry, Count of Portugal, Henry of Burgundy, Count of Portugal (father of Portugal's first king) ** Prince Henry the Navigator, Infante of Portugal ** Infante Henrique, Duke of Coimbra (born 1949), the sixth in line to Portuguese throne * King of Germany ** Henry the Fowler (876–936), first king of Germany * King of Scots (in name, at least) ** Henry Stuart, Lord Darnley (1545/6–1567), consort of Mary, queen of Scots ** Henry Benedict Stuart, the 'Cardinal Duke of York', brother of Bonnie Prince Charlie, who was hailed by Jacobites as Henry IX * Four kings of Castile: **Henry I of Castile **Henry II of Castile ** Henry III of Castile ** Henry IV of Castile * Five kings of France, spelt ''Henri'' in Modern French since the Renaissance to italianize the na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Trump
Walter Trump (born 1952 or 1953 ) is a German mathematician and retired high school teacher. He is known for his work in recreational mathematics. He has made contributions working on both the square packing problem and the magic tile problem. In 1979 he discovered the optimal known packing of 11 equal squares in a larger square, and in 2003, along with Christian Boyer, developed the first known magic cube of order 5. In 2012, Trump ''et al.'' described a model for retention of liquid on random surfaces. In 2014, he and Francis Gaspalou Francis may refer to: People *Pope Francis, the head of the Catholic Church and sovereign of the Vatican City State and Bishop of Rome *Francis (given name), including a list of people and fictional characters *Francis (surname) Places *Rural Mu ... were able to calculate all 8 × 8 bimagic squares. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Gerbicz
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honour, praise, renown" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe it entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including English, German, Dutch, Norwegian, Swedish, Scots, Danish, and Icelandic. It can be use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirk Kinnaes
A dirk is a long bladed thrusting dagger.Chisholm, Hugh (ed.), ''Dagger'', The Encyclopædia Britannica, 11th ed., Vol. VII, New York, NY: Cambridge University Press (1910), p. 729 Historically, it gained its name from the Highland Dirk (Scots Gaelic "Dearg") where it was a personal weapon of officers engaged in naval hand-to-hand combat during the Age of SailO'Brian, Patrick, ''Men-of-War: Life In Nelson's Navy'', New York: W.W. Norton & Co., (1974), p. 35 as well as the personal sidearm of Highlanders. It was also the traditional sidearm of the Highland Clansman and later used by the officers, pipers, and drummers of Scottish Highland regiments around 1725 to 1800 and by Japanese naval officers. Etymology The term is associated with Scotland in the Early Modern Era, being attested from about 1600. The term was spelled ''dork'' or ''dirk'' during the 17th century,Head, T.F. ''The Concise Oxford Dictionary of English Etymology'' Oxford University Press (1996) presumed relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mike Quist
Mike may refer to: Animals * Mike (cat), cat and guardian of the British Museum * Mike the Headless Chicken, chicken that lived for 18 months after his head had been cut off * Mike (chimpanzee), a chimpanzee featured in several books and documentaries Arts * Mike (miniseries), a 2022 Hulu limited series based on the life of American boxer Mike Tyson * Mike (2022 film), a Malayalam film produced by John Abraham * ''Mike'' (album), an album by Mike Mohede * ''Mike'' (1926 film), an American film * MIKE (musician), American rapper, songwriter and record * ''Mike'' (novel), a 1909 novel by P. G. Wodehouse * "Mike" (song), by Elvana Gjata and Ledri Vula featuring John Shahu * Mike (''Twin Peaks''), a character from ''Twin Peaks'' * "Mike", a song by Xiu Xiu from their 2004 album '' Fabulous Muscles'' Businesses * Mike (cellular network), a defunct Canadian cellular network * Mike and Ike, a candies brand Military * MIKE Force, a unit in the Vietnam War * Ivy Mike, the firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
