The Magic Encyclopedia ™

The Hypercube Doubling Method
{note: investigative article}
(by Aale de Winkel)

in the spring of 1999 <John R Hendricks> came up with a general method for doubling any hypercube. Independently <Marián Trenkler> came up with the same method. This Author merely added a little twist to apply it to the order 2 hypercube as well.

Note: the procedure can be viewed upon as left multiplication of the order m hypercube by the order 2 hypercube. The below formulated can be viewed as samples(?) of the there needed compensations for off sums.

I raised the investigative flag on this article since writing it I noticed some elements which needs some carefull study the below described might have some generalisation. Further the question I raised wheter the augmentation hypercube could be made pan-n-aganal itself.

The Hypercube Doubling Theorem
Given any semi magic latin order m hypercube ⁿL_m distributed over the order 2 hypercube ⁿT₂
which after changing digits turns into an "augmentation" magic latin hypercube of order 2m
ⁿA_2m = ⁿT₂(ⁿL_m) = mⁿ (D(F_i,j(ⁿL_m)) - 1) (i,j = (0,1))
To this augmentation hypercube any order n hypercube can be added to each hyperquadrant such that
ⁿH_2m = ⁿA_2m + G_i,j(ⁿH_m) is a regular magic hypercube

Distribution functions
F_i,j() and G_i,j() The distribution functions where in the original method mirroring in the central
horizontal and vertical. In order to obtain an order 4 form the order 2 I found
G_i,j() need also to include a center mirroring (NEED TO VERIFY)

latin hypercube
ⁿL_m Regularly the used hypercube is a zero n-agonal order m hypercube
(need checking, possibly more general possibilities!?)

Digit changing
D_i,j() Changing digits in the distributed latin hypercube is needed to compensate the
off-sums in the obtained order 2m square. Off sums (+c) in one hyperquadrant
need to be compensated into an adjecent hyperquadrant by a factor (-c) thus
forming the hyperquadrants compenstion patterns (+--+) or (-++-) for squares
and (+--+-++-) or (-++-+--+) for cubes. The encyclopedias database lists the
data for the order 2 and 3 square and cube, and a derivation method to obtain
data for other orders and dimensions as well.

Augmentation hypercube
ⁿA_2m Multiplying the above obtained square by mⁿ an augmentation hypercube
is obtained to which any order m hypercube can be added, a little care is needed
in the choise of the distribution function G_i,j() to avoid doubly appearing numbers

The Hypercube Doubling Theorem
Given any semi magic latin order m hypercube ⁿL_m distributed over the order 2 hypercube ⁿT₂ which after changing digits turns into an "augmentation" magic latin hypercube of order 2m ⁿA_2m = ⁿT₂(ⁿL_m) = mⁿ (D(F_i,j(ⁿL_m)) - 1) (i,j = (0,1)) To this augmentation hypercube any order n hypercube can be added to each hyperquadrant such that ⁿH_2m = ⁿA_2m + G_i,j(ⁿH_m) is a regular magic hypercube
Distribution functions F_i,j() and G_i,j()	The distribution functions where in the original method mirroring in the central horizontal and vertical. In order to obtain an order 4 form the order 2 I found G_i,j() need also to include a center mirroring (NEED TO VERIFY)
latin hypercube ⁿL_m	Regularly the used hypercube is a zero n-agonal order m hypercube (need checking, possibly more general possibilities!?)
Digit changing D_i,j()	Changing digits in the distributed latin hypercube is needed to compensate the off-sums in the obtained order 2m square. Off sums (+c) in one hyperquadrant need to be compensated into an adjecent hyperquadrant by a factor (-c) thus forming the hyperquadrants compenstion patterns (+--+) or (-++-) for squares and (+--+-++-) or (-++-+--+) for cubes. The encyclopedias database lists the data for the order 2 and 3 square and cube, and a derivation method to obtain data for other orders and dimensions as well.
Augmentation hypercube ⁿA_2m	Multiplying the above obtained square by mⁿ an augmentation hypercube is obtained to which any order m hypercube can be added, a little care is needed in the choise of the distribution function G_i,j() to avoid doubly appearing numbers