The HyperCube

The regular n dimensional HyperCube of order m ⁿH_m is an arangement of numbers
1 .. mⁿ in an m by ... by m (n m's) hypercube

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The magic HyperCube is a regular HyperCube where all pilars and the n-agonals are summing to the same sum: (the hypercubes magic sum)

ⁿS_m = [_i=1∑^{m^n} i] / m^n-1 = [mⁿ (mⁿ + 1) / 2] / m^n-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)

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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.

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In John R. Hendricks's Latest Booklet:
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)