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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 referred to Langman’s 1962 pandiagonal cube also as ‘Perfect’. Christian Boyer and Walter Trump now consider this ''and'' the next two classes to be ''Perfect''. (See ''Alternate Perfect'' below). A. H. Frost referred to all but the simple class as Nasik cubes. The smallest normal diagonal magic cube is order 5. See Diagonal magic cube. Minimum correct summations required = 3''m''2 + 6''m'' + 4 * Pantriagonal: All 4m2 pantriagonals must sum correctly (that is 4 one-segment, 12(''m''-1) two-segment, and 4(''m''-2)(''m''-1) three-segment). There may be some simple AND/OR pandiagonal magic squares, but not enough to satisfy any other classification. The smallest normal pantriagonal magic cube is order 4. See Pantriagonal magic cube. Minimum correct summations required = 7''m''2. All pan-''r''-agonals sum correctly for ''r'' = 1 and 3. * PantriagDiag: A cube of this class was first constructed in late 2004 by Mitsutoshi Nakamura. This cube is a combination Pantriagonal magic cube and Diagonal magic cube. Therefore, all main and broken triagonals sum correctly, and it contains 3''m'' planar simple magic squares. In addition, all 6 oblique squares are pandiagonal magic squares. The only such cube constructed so far is order 8. It is not known what other orders are possible. See Pantriagdiag magic cube. Minimum correct summations required = 7''m''2 + 6''m'' * Pandiagonal: ALL 3''m'' planar arrays must be pandiagonal magic squares. The 6 oblique squares are always magic (usually simple magic). Several of them MAY be pandiagonal magic. Gardner also called this (Langman’s pandiagonal) a ‘perfect’ cube, presumably not realizing it was a higher class then Myer’s cube. See previous note re Boyer and Trump. The smallest normal pandiagonal magic cube is order 7. See Pandiagonal magic cube. Minimum correct summations required = 9''m''2 + 4. All pan-''r''-agonals sum correctly for ''r'' = 1 and 2. * Perfect: ALL 3''m'' planar arrays must be pandiagonal magic squares. In addition, ALL pantriagonals must sum correctly. These two conditions combine to provide a total of 9m pandiagonal magic squares. The smallest normal perfect magic cube is order 8. See Perfect magic cube. Nasik; A. H. Frost (1866) referred to all but the simple magic cube as Nasik! C. Planck (1905) redefined ''Nasik'' to mean magic hypercubes of any order or dimension in which all possible lines summed correctly. i.e. ''Nasik'' is a preferred alternate, and less ambiguous term for the ''perfect'' class. Minimum correct summations required = 13''m''2. All pan-''r''-agonals sum correctly for ''r'' = 1, 2 and 3. Alternate Perfect Note that the above is a relatively new definition of ''perfect''. Until about 1995 there was much confusion about what constituted a ''perfect'' magic cube (see the discussion under diagonal:) . Included below are references and links to discussions of the old definition With the popularity of personal computers it became easier to examine the finer details of magic cubes. Also more and more work was being done with higher dimension magic Hypercubes. For example, John Hendricks constructed the world's first Nasik magic tesseract in 2000. Classed as a perfect magic tesseract by Hendricks definition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「magic cube classes」の詳細全文を読む スポンサード リンク
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