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 
^{n}C_{m1m2} = ^{n}C_{m1} * ^{n}C_{m2} =
{^{n}C_{m1}([^{n}C_{m2}]  1) m1^{n})} The hypercube formulation of this is (analitic numberrange): ^{n}H_{m1} * ^{n}H_{m2} : ^{n}[_{k}i]_{m1m2} = ^{n}[ [[_{k}i \ m_{2}]_{m1}m_{1}^{n}]_{m2} + [_{k}i % m_{2}]_{m2}]_{m1}_{m2} The hyperbeam formulation of this is (analitic numberrange): ^{n}B_{(m..)1} * ^{n}B_{(m..)2} : ^{n}[_{k}i]_{(m..)1(m..)2} = ^{n}[ [[_{k}i \ m_{k2}]_{ (m..)1}_{k=0}∏^{n1}m_{k1}]_{(m..)2} + [_{k}i % m_{k2}]_{(m..)2}]_{(m..)1}_{(m..)2} (m..) abreviate m_{0},..,m_{n1} 

Place a copy of ^{n}C_{m1} at each position of ^{n}C_{m2}
with each element multiplied by ([^{n}C_{m2}]  1) m1^{n} where [^{n}C_{m2}] stands for the element on the given position of the hypercube ^{n}C_{m2} 

General Formula  ^{n}C_{m1m2} = (^{n}C_{m1};d_{m1}) * (^{n}C_{m2};d_{m2}) = F { f_{j}i { (^{n}C_{m1};d_{m1}, ([^{n}C_{m2};d_{m2}]_{j}i  1) * m1^{n}) }}  
Basically the same as in the basic formula, the global function F and the local function f_{j}i denote compensating function for off sums, thus it is possible to multiply with blockwise Magic hypercubes such as (^{n}C_{2};1). f_{j}i intents to compensate off sums in (^{n}C_{m2};d_{m2}) prior to the basic multiplication, F be used to compensate off sums of this basic multiplication. 

The effects of the compensating functions F and f_{j}i 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. 