The Magic Encyclopedia ™

Identification
(by Aale de Winkel)
NOTE: Under construction

(Acknowledgement: Closer analisis I realized that the method "pandiagonal construction" is a minor variation of <John R Hendricks>'s "digit equations". Analitic coordinates used in stead of regular)

Thus far I've not seen a pandiagonal cube that didn't fit the theory I uploaded. Identfying a hypercube is next to qualification one of the major isues in the investigation of hypercubes, since in order to figure out wheter the hypercube is something new or something known one needs to figure out wheter the found hypercube can fit into the known theory. This article atempts to describe the processes involved in identifying an hypercube. Once the hypercube is identified this indentification gives the means to reproduce the hypercube and thus should be concidered to be known by this identification.

Currently this must be seen as preliminairy notes, augmentation of these processes will be described as soon as I can find implementation cq description of the involved process
as always I appreciate contribution to this effort.

Identification
Knight Jump vectors
(odd orders) KJ(P,V₀ .. V_n-1)
The knightjump construction consist of the position of the first number
and a total of n vectors which point from one number onto the nxxt

In the regular number rang P is the postion of the number '1',
V_k k = 0 .. n-1, is the vector between the number m^k to m^k + 1

Latin Prescription vectors
(any orders) lp( a_j ) _=P[0..m-1] (for single component)
The latin prescription construction consist of a series of modular equations
one for each component. Aside from these each component can be subject to
digit changing. (The Hendricks digit and digital equation belong also to this
method thuugh he uses coordinate starting with '1' in stead of '0')

due to other subject I'm currently out of touch with the method, so other then a
trial and error based process I currently can't provide.

pandiagonal construction
(prime orders) LH( a_k )_=P[0..m-1] (for single component)
The pandiagonal construction for prime order hypercubes consist of n parameter
n-Vectors, also digit changing permutation can be applied to each component

when digit changing is applied identification is a bit complicated, thus far
the cubes i've seen where without, and so fairly simple to recognise
take a number at a certain position and write it into m-based numbers
do the same with the numbers next to it along each of the n 1-agonals
each difference in the digits defines a number in the vector.

These qualitative description give the process of reobtaining LH(a_j)_=P[0..m-1]

change hypercube to analitic numberrange, and normalized position (optional)
write the numbers in radix m notation, each digit combine into a latin hypercube (LH)
with each hypercube suppose the poitive difference of a number with the next along a
1-agonal direction is the disired parameter a_j these parameters form LH(a_j)
When LH(a_j) reconstructs the given hypercube the hypercube is found
Positive difference of d1 and d2: dif(d1,d2) = (m + d1 - d2) % m ; thus d2 = (d1 + dif(d1,d2)) % m

note that one might construct the difference hypercube, this should show constant features
(I haven't done this yet, exact feature I'll verify)
(NOTE: thus far I've seen no cube wher I needed to go further then this.)

If LH(a_j) does not reconstruct the given hypercube digit changing might be involved
I suppose with the following one might obtain the parameters and permutation
suppose the parameter of the first 1-agonal and construct the 1 agonal line, this line with the
hypercubes 1-agonal will give the hypercube digit changing permutation P[0..m-1]
Using this permutaton in reverse the other parameters of the latin hypercube can be obtained
The thus found numbers form LH(a_j)_=P[0..m-1]

Note:it might well be possible to use diferent permutation in the various 1-agonal directions
(this generalisation of the construction method needs to be investigated yet)

The description for the cube I singled out since I've done this on a few cubes Harvey uploaded and found those cube identifiable. Personally I consider hypercubes which can be identified by this (and simular) means as known. Historic hypercubes thus identified are of course remarkable, because of their find without the construction method, in these cases the "modern construction" might be included in a description of those hypercubes.

Identification
Knight Jump vectors (odd orders)	KJ(P,V₀ .. V_n-1)
	The knightjump construction consist of the position of the first number and a total of n vectors which point from one number onto the nxxt
	In the regular number rang P is the postion of the number '1', V_k k = 0 .. n-1, is the vector between the number m^k to m^k + 1
Latin Prescription vectors (any orders)	lp( a_j ) _=P[0..m-1] (for single component)
	The latin prescription construction consist of a series of modular equations one for each component. Aside from these each component can be subject to digit changing. (The Hendricks digit and digital equation belong also to this method thuugh he uses coordinate starting with '1' in stead of '0')
	due to other subject I'm currently out of touch with the method, so other then a trial and error based process I currently can't provide.
pandiagonal construction (prime orders)	LH( a_k )_=P[0..m-1] (for single component)
	The pandiagonal construction for prime order hypercubes consist of n parameter n-Vectors, also digit changing permutation can be applied to each component
	when digit changing is applied identification is a bit complicated, thus far the cubes i've seen where without, and so fairly simple to recognise take a number at a certain position and write it into m-based numbers do the same with the numbers next to it along each of the n 1-agonals each difference in the digits defines a number in the vector.
	These qualitative description give the process of reobtaining LH(a_j)_=P[0..m-1] change hypercube to analitic numberrange, and normalized position (optional) write the numbers in radix m notation, each digit combine into a latin hypercube (LH) with each hypercube suppose the poitive difference of a number with the next along a 1-agonal direction is the disired parameter a_j these parameters form LH(a_j) When LH(a_j) reconstructs the given hypercube the hypercube is found Positive difference of d1 and d2: dif(d1,d2) = (m + d1 - d2) % m ; thus d2 = (d1 + dif(d1,d2)) % m note that one might construct the difference hypercube, this should show constant features (I haven't done this yet, exact feature I'll verify) (NOTE: thus far I've seen no cube wher I needed to go further then this.) If LH(a_j) does not reconstruct the given hypercube digit changing might be involved I suppose with the following one might obtain the parameters and permutation suppose the parameter of the first 1-agonal and construct the 1 agonal line, this line with the hypercubes 1-agonal will give the hypercube digit changing permutation P[0..m-1] Using this permutaton in reverse the other parameters of the latin hypercube can be obtained The thus found numbers form LH(a_j)_=P[0..m-1] Note:it might well be possible to use diferent permutation in the various 1-agonal directions (this generalisation of the construction method needs to be investigated yet)