The Magic Encyclopedia ™


Paths through the Hypercube
{note: investigative article (!! currently things being added november 2011 !!)}
(by Aale de Winkel)


This article deals with the various Paths through the Hypercube, and is merely there to define the notations to describe several types of paths, using the notations reached at in the bent-hyperagonal article.
Unless explicitely described the notation implicitely supposes a move to an adjacent position before path continuation
pe: m/2 {<1,1>,<1,-1>} = {m/2<1,1>,<1,0>,m/2<1,-1>}
Further the article presupposes modular space (or wrap around), so paths runs through the hypercubes borders onto the modular equivalent positions
Not too concerned with paths here defined being magic, samples of doubly odd order "panPath(<1,2>)" square (and similar) I recently obtained from <Miguel Angel Amela>, nothing wrong with his notion of "square-" versus "rectangular-" panmagicness as defined in the article he sent, though the latter is a bit foggy in general use as that can be of any size. Stating the paths concidered makes it just more general. The ideas formulated below introduces the parametrized Path(..) qualifier and the panPath(..) variation stating of course the paths start from anywhere.


Paths through the Hypercube
John R. Hendricks's "pathfinders" where defined to run through hyper-r-agonals, below defines a notation for
these and other definable paths through a n-dimensional hypercube, an n-Point can be used to fix the starting
point of a given path of any type, so this article is only concidering the path-form.
1-agonal m {<0r1s>} (r = 0.. m-1; r != s)
main n-agonal m {<1r>} (r = 0.. m-1)
sub n-agnal m {<1r-1s>} (r = 0.. m-1; s = 0.. m-1; r != s)
bent hyper-n-agonal m/2 {<vi; i = 0..n-1>,<wi; i = 0..n-1>} ; with: <v> * <w> = <0>
both n-Vectors elements -1 and 1 only; henche these vectors are orthogonal
knight jump paths m {<vr>} (r = 0.. m-1; vr != 0)
bent knight jump paths m/2 {<vi; i = 0..n-1>,<wi; i = 0..n-1>}
both n-Vectors no 0 elements (vectors need not be orthogonal)
other paths {<vr>i} (r = 0.. m-1;)
(path supposed to be non-circular in less than m steps)
The number of vectors between the curly brackets is not yet defined (probably m), as suggested
above repeated vectors can be indicated by preapending the number of repetitions, this number
is multiplied by the "length-indicator" preappended before the curly brackets. (the mentioned
non-circular condition just avoids double counting the same cell element in the same path).

Stating magicness using paths can become eleborate as shown below on the known situations, the keyword Path in magic qualifiers I reckon introduced the exact definitions of the qualified paths. With panPath to indicate that all positions within the hypercube are concidered. With the usual Hendrick's pathfinder paths concidered "Path" redundantly used possible, but I reckon use is only to draw attention to the special path and things like magic is given by context.


PathMagicness of Hypercubes
Stating magicness of paths means stating starting position n-points and a description of paths with n-vectors
of course the entire set of n-points also defines the paths, using one or the other is a matter of what is convenient
Below some samples of quite familiar situations introducing the (pan)Path(...) qualification keyword properly.
semi magic square
{monagonal[!]}
{ Path([k,0]<0,1>;[0,k]<1,0> ; k=0..m-1) }
{ panPath(<0,1>;<1,0>) }
magic square
{magic}
{ Path([k,0]<0,1>;[0,k]<1,0>;[0,0]<1,1>;[0,m-1]<1,-1> ; k=0..m-1) }
panmagic square
{panmagic}
{ Path([k,0]<0,1>;[0,k]<1,0>;[k,0]<1,1>;[k,m-1]<1,-1> ; k=0..m-1) }
{ panPath(<0,1>;<1,0>;<1,1>;<1,-1>) }
Franklin square
{Franklin}
{ Path([k,0]<0,1>;[0,k]<1,0>;[k,0]<1,1>;[k,m-1]<1,-1> ; k=0..m-1 ;
[k,l]{ m/2{<1,1>}, <1,0>, m/2{<1,-1>}, <1,0> } ; k=0..m-1,l=0,m/2 ;
[l,k]{ m/2{<1,1>}, <0,1>, m/2{<-1,1>}, <0,1> } ; k=0..m-1,l=0,m/2 ) }
(note: needs verification)