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 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 = 3m² + 4

Each of the 3m 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 = 3m² + 6m + 4

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 = 7m². All pan-r-agonals sum correctly for r = 1 and 3.

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 3m 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 = 7m² + 6m

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