The Two Path Claim:If {$T$} is triangular, then there are just two path lengths. Proof: If {$T$} has a greatest member, say {$c$}, then the path starting on the {$a\times c$} face ends at the starting point, moving on the {$b\times c$} face (Just Right). Thus all eight paths starting on either the {$a\times c$} or the {$b\times c$} face have the same length. If instead, {$T=\langle a,b,b\rangle$}, {$a<b$}, there are four paths of length {$b$} and eight of length {$2a+b$}.  |