note: these pages have been superseeded by item in the encyclopedia.
The regular n dimensional HyperCube of order m nHm is an arangement of numbers
1 .. mn in an m by ... by m (n m's) hypercube
The magic HyperCube is a regular HyperCube where all pilars and the n-agonals are summing to the same sum:
(the hypercubes magic sum)
nSm = [i=1∑m^n i] / mn-1 =
[mn (mn + 1) / 2] / mn-1 = [m (mn + 1) / 2]
(there are mn pilars in the hypercube, m of which are disjunct in each direction,
the total sum is distributed over these, hence the dividing factor)
These Pages contain a study on the regular (non-magic) HyperCube and the
Magic HyperCube as well.
As such it is a generalisation of things on my regular
Magic Squares and cubes
item, and apply to them as well as to hypercubes of dimension > 3.
In John R. Hendricks's
Perfect n-dimensional Magic HyperCubes of order 2n
he derives the "treshhold" order 2n for a HyperCube to be (what he calls)
"Perfect" (pan-r-agonal for all r = 0 .. n-1)