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At Most Three Paths:

For {$T\in\mathcal {B}^+$}, there are at most three different path lengths (from corner to corner). If {$T$} is not an axle, there are at most two different path lengths.

Proof:

Let {$T=\langle a,b,c\rangle$}. Since there are eight corners on a box and each corner borders on three faces, there are exactly twelve paths. If one path starts on one face and ends on the same face, there will be at least three more paths with the same shape. If a path begins on one face and ends on another, there will be seven more with the same shape. A box has faces of at most three different dimensions. Thus, a box may have:

1. three different paths, each with four specimens,
2. two different paths, one with four, one with eight specimens, or
3. one path with twelve specimens (a cube).

Now suppose that {$T$} is not an axle.

The Two Path Claim: If {$T$} is triangular, then there are just two path lengths.

If {$T$} is not triangular, {$\Delta^k(T)$} is triangular for some {$k$} and by {$\Delta$} and Destinations, paths which are congruent on one box have corresponding congruent paths on any {$\Delta$} pre-image. Thus {$T$} has at most two different path lengths.