Cube (algebra) however, easy to show that most numbers are not perfect cubes because "all" perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only −1, 1 and 0. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3: Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes: The equation has
Magic cube classes 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" + 6"m" ALL 3"m" planar arrays must be pandiagonal magic squares. The 6 oblique squares are always magic (usually simple magic). Several of them