Pursuant on representations from my extensive fanbase, I have been prevailed upon to implement a modicum of nitpicking clarification concerning the proof of the knight's move distance formula at https://www.dropbox.com/s/nzmzjswtctju23f/knights_path.txt [but it's looking good --- and about bloomin' time!] Also appended is a snippet concerning the number e(x, y) of shortest paths from [0, 0] to [x, y] : this evaluates to a binomial coefficient for points lying on the outer edge of an annulus (2/7 of cases). Does anyone fancy having a go at cracking some of the other cases? WFL On 4/10/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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