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Multimagic Cube
In mathematics, a ''P''-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their ''k''th powers for 1 ≤ ''k'' ≤ ''P''. cubes are called bimagic, cubes are called trimagic, and cubes tetramagic. A cube is said to be semi-perfect if the ''k''th power cubes are perfect for 1 ≤ ''k'' < ''P'', and the ''P''th power cube is . If all ''P'' of the power cubes are perfect, the cube is said to be perfect. The first known example of a bimagic cube was given by in 2000; it is a cube ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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, the so-called 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 ... of the cube, denoted ''M''3(''n''). 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 cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product (mathematics), product of multiplying bases: b^n = \underbrace_.In particular, b^1=b. The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ". The above definition of b^n immediately implies several properties, in particular the multiplication rule:There are three common notations for multiplication: x\times y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variable (mathematics), variables are used; x\cdot y is used for emphasizing that one ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Perfect Magic Cube
In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section (geometry), cross section diagonals sum up to the cube's magic constant. Perfect magic cubes of order one are trivial; cubes of orders two to four can be mathematical proof, proven not to exist, and cubes of orders five and six were first discovered by Walter Trump and Christian Boyer on November 13 and September 1, 2003, respectively. A perfect magic cube of order seven was given by A. H. Frost in 1866, and on March 11, 1875, an article was published in the Cincinnati Commercial newspaper on the discovery of a perfect magic cube of order 8 by Gustavus Frankenstein. Perfect magic cubes of orders nine and eleven have also been constructed. The first perfect cube of order 10 was constructed in 1988 (Li Wen, China). An alternative definition In recent years, an alternative definition for the perfect magic cube was proposed by John ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Semiperfect Magic Cube
In number theory, a semiperfect number or pseudoperfect number is a natural number ''n'' that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... Properties * Every multiple of a semiperfect number is semiperfect. A semiperfect number that is not divisible by any smaller semiperfect number is called ''primitive''. * Every number of the form 2''m''''p'' for a natural number ''m'' and an odd prime number ''p'' such that ''p'' < 2''m''+1 is also semiperfect. ** In particular, every number of the form 2''m''(2''m''+1 − 1) is semiperfect, and indeed perfect if 2''m''+1 − 1 is a . * The sma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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John Hendricks
John Samuel Hendricks (born March 29, 1952)"John Hendricks: An Oral History," The Cable Center, September 2, 2003. is an American businessman and the founder and former chairman of Discovery, Inc. (later Warner Bros. Discovery), a broadcasting and film production company which owned the Discovery Channel, TLC, and Animal Planet networks, among other ventures. On March 20, 2014, after 32 years at the helm, he made public his intention to retire as chairman of Discovery Communications after the annual shareholders' meeting of May 16, 2014. He moved on to found Curiosity Stream, an ad-free, on-demand streaming service. Early life Born in Matewan, West Virginia, Hendricks' father was a home builder and his mother a clerk for city government. In 1958, the Hendricks family moved to Huntsville, Alabama, where Hendricks grew up. His father died when he was 20, and his mother died when he was 30. He attended S. R. Butler High School where he met his first wife, Pattie Miller. Hendric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 . The largest magic constant of normal magic square which is also a: *triangular number is 15 (solve the Diophantine equation where ''y'' is divisible by 4); *square number is 1 (solve the Diophantine equation where ''y'' is even); * generalized pentagonal number is 171535 (solve the Diophantine equation where '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled ''CRC Concise Encyclopedia of Mathematics''. The free online version became only partially accessible to the public. In 1999 Weisstein we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Magic Square
In mathematics, especially History of mathematics, historical and 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, Sagrada Família magic square and the #Parker square, 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). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Multimagic Square
In mathematics, a ''P''-multimagic square (also known as a satanic square) is a magic square that remains magic even if all its numbers are replaced by their ''k''th powers for 1 ≤ ''k'' ≤ ''P''. squares are called bimagic, squares are called trimagic, squares tetramagic, and squares pentamagic. Constants for normal squares If the squares are normal, the constant for the power-squares can be determined as follows: Bimagic series totals for bimagic squares are also linked to the square-pyramidal number sequence is as follows :- Squares 0, 1, 4, 9, 16, 25, 36, 49, .... Sum of Squares 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... )number of units in a square-based pyramid) The bimagic series is the 1st, 4th, 9th in this series (divided by 1, 2, 3, ''n'') etc. so values for the rows and columns in order-1, order-2, order-3 Bimagic squares would be 1, 15, 95, 374, 1105, 2701, 5775, 11180, ... The trimagic series would be related in the same way to the hyper-py ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |