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=0∏n-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.