The Magic Encyclopedia ™

Multiplication of magic objects
(by Aale de Winkel)

Combining an hypercube of order m1 and one of order m2 into a hypercube
of order m1*m2 can be viewed upon as a multiplication of its own
The result of the basic multiplication formula is also called a "Composition Magic Hypercube"
The formulae can also be applied with non-magic hypercubes

Hypercube Multiplication
The hypercube multiplication formulae initiated the authors study into the subjects discussed.
The basic formula is proven to be correct, the general formula depends on the used functions
to compensate for off sums. It allows for using blockwise magic hypercubes with multiplication
A few compensating functions are used by this author.
Basic Formula nCm1m2 = nCm1 * nCm2 = {nCm1([nCm2] - 1) m1n)}

The hypercube formulation of this is (analitic numberrange):
nHm1 * nHm2 : n[ki]m1m2 = n[ [[ki \ m2]m1m1n]m2 + [ki % m2]m2]m1m2

The hyperbeam formulation of this is (analitic numberrange):
nB(m..)1 * nB(m..)2 : n[ki](m..)1(m..)2 = n[ [[ki \ mk2] (m..)1k=0n-1mk1](m..)2 + [ki % mk2](m..)2](m..)1(m..)2
(m..) abreviate m0,..,mn-1
Place a copy of nCm1 at each position of nCm2 with each element multiplied by
([nCm2] - 1) m1n where [nCm2] stands for the element on the given position
of the hypercube nCm2
General Formula nCm1m2 = (nCm1;dm1) * (nCm2;dm2) = F { fji { (nCm1;dm1, ([nCm2;dm2]ji - 1) * m1n) }}
Basically the same as in the basic formula, the global function F and the local function fji
denote compensating function for off sums, thus it is possible to multiply with blockwise
Magic hypercubes such as (nC2;1).
fji intents to compensate off sums in (nCm2;dm2) prior to the basic multiplication,
F be used to compensate off sums of this basic multiplication.
The effects of the compensating functions F and fji might be studied prior
to actual application of the basic multiplication, Ralph Strachey's construction for double
odd order squares, or the more general doubling method by Hendricks / Trenkler can be
thus viewed as an application of the general formula above
Although the formulae defined here to produce magic hypercubes, both formulae's can be used to
generate other quality hypercubes, pe basic multiplication of any magic hypercube with a blockwise
magic hypercube will result in a blockwise magic hypercube.