Shortest Paths on Cubes.

If we get an Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">L</mi><mi mathvariant="script">S</mi></mrow></math> ht -path, we are done. It follows that no corner move can shorten a 3-face Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">L</mi><mi mathvariant="script">S</mi></mrow></math> ht -path, and so the shortest path must be one of the four 3-face Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">L</mi><mi mathvariant="script">S</mi></mrow></math> ht -paths. This cannot go on forever; if we eventually produce a 3-face path it will have to be an Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">L</mi><mi mathvariant="script">S</mi></mrow></math> ht -path. If we get a pseudopath, then a second corner move will produce a 3-face Graph HT <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">L</mi><mi mathvariant="script">S</mi></mrow></math> ht -path and we are done in this case too. [Extracted from the article]