Taking a fresh look at that infernal knight's move distance formula, I had to admit that version 6 of my Lemma 4 proof was confusingly cryptic about folding neighbours of the target point back into the first octant; and anyway this tortuous (and arguably dubious) manoeuvre may be avoided altogether, by weakening the constraints on Lemma 3. Warren D Smith <warren.wds@gmail.com>, Fri, Apr 11, 2014 at 11:28 PM --- << As I remarked to Lunnon in email, I still do not accept that he has proved the N-distance formula. Despite him seeming to think he has. >> Whether I have identified the issue raised above (with customary daisy-cutter precision) remains unclear. Regardless, relevant amendments, correction of gruesome typos, a modicum of signposting, flag-waving, grand-standing (er, maybe time to take a break) and further tidying are provided for the benefit anyone who might still be following, by version 7 at https://www.dropbox.com/s/nzmzjswtctju23f/knights_path.txt If only this would stabilise for a while it might even manage to get properly typeset. Also my plots of symmetric 4-valent graphs derived from a 6x6 toric board were inadvertantly trashed; subtly enhanced replacements are at https://www.dropbox.com/s/8b0stotr8343mq2/semi_6x6.pdf https://www.dropbox.com/s/gyispnh717ayz86/semi0_6x6.pdf I have in mind an animation showing one graph morphing symmetrically into the other; but cannot presently visualise a suitable permutation. Resort to the computer seems tedious overkill ... Fred Lunnon