|
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 definition f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Magic Cube
A simple magic cube is the lowest of six basic classes of magic cubes. These classes are based on extra features required. The simple magic cube requires only the basic features a cube requires to be magic. Namely, all lines parallel to the faces, and all 4 triagonals sum correctly. i.e. all 1-agonals and all 3-agonals sum to :S = \frac. No planar diagonals (2-agonals) are required to sum correctly, so there are probably no magic squares in the cube. See also * Magic square * Magic cube classes Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics. This new system is more precise in defining magic cubes. But possibly of more importance, it is consistent for all orders and all dimensions of ... References {{reflist External links Aale de Winkel - Magic hypercubes encyclopediaHarvey Heinz - large site on magic squares and cubes John Hendricks site on magic hypercubes Magic squares Recreational mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bimagic 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 |
Nasik Magic Hypercube
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] |
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] |
Magic Cube Classes
Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics. This new system is more precise in defining magic cubes. But possibly of more importance, it is consistent for all orders and all dimensions of magic hypercubes. Minimum requirements for a cube to be magic are: all rows, columns, pillars, and 4 triagonals must sum to the same value. The six classes * Simple: The minimum requirements for a magic cube are: all rows, columns, pillars, and 4 triagonals must sum to the same value. A simple magic cube contains no magic squares or not enough to qualify for the next class. The smallest normal simple magic cube is order 3. Minimum correct summations required = 3''m''2 + 4 * Diagonal: Each of the 3''m'' planar arrays must be a simple magic square. The 6 oblique squares are also simple magic. The smallest normal diagonal magic cube is order 5. These squares were referred to as 'Perfect' by Gardner and others. At the same time he referre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Hypercube
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] |
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 |
Semiperfect Magic Cube
In mathematics, a semiperfect magic cube is a magic cube that is not a perfect magic cube, i.e., a magic cube for which the cross section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ... diagonals do not necessarily sum up to the cube's magic constant. References *. Magic squares {{combin-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Magic Cube
Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * ''Perfect'' (1985 film), a romantic drama * ''Perfect'' (2018 film), a science fiction thriller Literature * ''Perfect'' (Friend novel), a 2004 novel by Natasha Friend * ''Perfect'' (Hopkins novel), a young adult novel by Ellen Hopkins * ''Perfect'' (Joyce novel), a 2013 novel by Rachel Joyce * ''Perfect'' (Shepard novel), a Pretty Little Liars novel by Sara Shepard * ''Perfect'', a young adult science fiction novel by Dyan Sheldon Music * Perfect interval, in music theory * Perfect Records, a record label Artists * Perfect (musician) (born 1980), reggae singer * Perfect (Polish band) * Perfect (American band), an American alternative rock group Albums * ''Perfect'' (Intwine album) (2004) * ''Perfect'' (Half Japanese album) (2016) * ''perfecT'', an album by Sam Shaber * ''Perfect'', an album by True F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetramagic 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 |
Trimagic 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 |
Pandiagonal Magic Cube
In recreational mathematics, a pandiagonal magic cube is a magic cube with the additional property that all broken diagonals (parallel to exactly two of the three coordinate axes) have the same sum as each other. Pandiagonal magic cubes are extensions of diagonal magic cubes (in which only the unbroken diagonals need to have the same sum as the rows of the cube) and generalize pandiagonal magic squares to three dimensions. In a pandiagonal magic cube, all 3''m'' planar arrays must be panmagic squares. The 6 oblique squares are always magic. Several of them may be panmagic squares. A proper pandiagonal magic cube has exactly 9''m''2 lines plus the 4 main triagonals summing correctly (no broken triagonals have the correct sum.) The smallest pandiagonal magic cube has order 7. See also *Magic cube classes Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics. This new system is more precise in defining magic cubes. But possibly of m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
