Pantriagonal magic cube
A pantriagonal magic cube is a magic cube where all 4m2 pantriagonals sum correctly. There are 4 one-segment, 12(m − 1) two-segment, and 4(m − 2)(m − 1) three-segment pantriagonals. This class of magic cubes may contain some simple magic squares and/or pandiagonal magic squares, but not enough to satisfy any other classifications.
The constant for magic cubes is S = m(m3 + 1)/2.
A proper pantriagonal magic cube has 7m2 lines summing correctly. It contains no magic squares.
Order 4 is the smallest pantriagonal magic cube possible. A pantriagonal magic cube is the 3-dimensional equivalent of the pandiagonal magic square. Only, instead of the ability to move a line from one edge to the opposite edge of the square with it remaining magic, you can move a plane from one edge to the other.
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