Editing Magic hypercube (section)

==Pathfinders==
J. R. Hendricks called the directions within a hypercubes "'''pathfinders'''", these directions are simplest denoted in a ternary number system as:
 Pf<sub>p</sub> where: p = <sub>k=0</sub>Σ<sup>n-1</sup> (<sub>k</sub>i + 1) 3<sup>k</sup> &lt;==&gt; &lt;<sub>k</sub>i&gt; ; i ε {-1,0,1}

This gives 3<sup>n</sup> directions. since every direction is traversed both ways one can limit to the upper half [(3<sup>n</sup>-1)/2,..,3<sup>n</sup>-1)] of the full range.

With these pathfinders any line to be summed over (or r-agonal) can be specified:
 [ <sub>j</sub>0 <sub>k</sub>p <sub>l</sub>q ; #j=1 #k=r-1 ; k > j ] &lt; <sub>j</sub>1 <sub>k</sub>θ <sub>l</sub>0 ; θ ε {-1,1} &gt;  ; p,q ε [0,..,m-1]

which specifies all (broken) r-agonals, p and q ranges could be omitted from this description. The main (unbroken) r-agonals are thus given by the slight modification of the above:
 [ <sub>j</sub>0 <sub>k</sub>0 <sub>l</sub>-1 <sub>s</sub>p ; #j=1 #k+#l=r-1 ; k,l > j ] &lt; <sub>j</sub>1 <sub>k</sub>1 <sub>l</sub>-1 <sub>s</sub>0 &gt;