note: these pages have been superseeded by item in the encyclopedia.
## The HyperCube

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The regular n dimensional HyperCube of order m ^{n}H_{m} is an arangement of numbers

1 .. m^{n} 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)

^{n}S_{m} = [_{i=1}∑^{m^n} i] / m^{n-1} =
[m^{n} (m^{n} + 1) / 2] / m^{n-1} = [m (m^{n} + 1) / 2]

(there are m^{n} 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
Latest Booklet:

Perfect n-dimensional Magic HyperCubes of order 2^{n}

he derives the "treshhold" order 2^{n} for a HyperCube to be (what he calls)

"Perfect" (pan-r-agonal for all r = 0 .. n-1)