|
Diagonal Magic Cube
The class of diagonal magic cubes is the second of the six magic cube classes (when ranked by the number of lines summing correctly), coming after the simple magic cubes. In a diagonal magic cube of order ''m'',Traditionally, ''n'' has been used to indicate the order of the magic hypercube. However, in recent years, due to the increasing emphasis on higher dimension hypercubes, there is a trend to use ''m'' to indicate order and ''n'' to indicate dimension. all 6''m'' of the diagonals in the ''m'' planes parallel to the top, front, and sides of the cube must sum correctly. This means that the cube contains 3''m'' simple magic squares of order ''m''. Because the cube contains so many magic squares, it was considered for many years to be "perfect" (although other types of cubes were also sometimes called a "perfect magic cube"). It is now known that there are three higher classes of cubes. The (proper) diagonal magic cube has a total of 3m2 + 6''m'' + 4 correctly summing lines and ... [...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] |
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] |
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. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pandiagonal Magic Square
A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant. A pandiagonal magic square remains pandiagonally magic not only under rotation or reflection, but also if a row or column is moved from one side of the square to the opposite side. As such, an n \times n pandiagonal magic square can be regarded as having 8n^2 orientations. 3×3 pandiagonal magic squares It can be shown that non-trivial pandiagonal magic squares of order 3 do not exist. Suppose the square :\begin \hline \!\!\!\; a_ \!\!\! & \!\! a_\!\!\!\!\; & \!\! a_ \!\!\\ \hline \!\!\!\; a_ \!\!\! & \!\! a_\!\!\!\!\; & \!\! a_ \!\!\\ \hline \!\!\!\; a_ \!\!\! & \!\! a_\!\!\!\!\; & \!\! a_ \!\!\\ \hline \end is pandiagonally magic with magic constant . Adding sums and results in . Subtracting ... [...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] |
