On 4/10/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This has got legs. Does it stand up?
Perhaps, but it staggers rather than runs. And now we shall lower it into quick-setting concrete and discreetly withdraw. For we already knew (Lemma 3 of WFL's earlier Dropbox post) that For f(x, y) > 4 in the first octant x >= y >= 0 , f(x, y) = f(x-2, y-1) + 1 from which it is an easy induction to establish WDS's distance constraint f(x, y) = 1 + min( f(x-2, y-1), ..., f(x+2, y+1) ) for [x, y] <> [0, 0] then a further trivial induction to deduce the hallowed d(x, y) = f(x, y) QED (tries to ignore derisive groans emanating from the assembled company). No messy geometry; no modulo 24 or whopping initial tabulation; why on earth did that take so long? This tantalising little beauty demonstrates a notable capacity to generate red herrings! The screed at https://www.dropbox.com/s/nzmzjswtctju23f/knights_path.txt has been updated (and substantially shortened). Fred Lunnon