[math-fun] Atropa Belladonna redux
Eventually I did get my head properly around Gareth's combinatorial solution to my Atropa Belladonna (Deadly Knightshade) Q3. Whereupon several strange things happened. Against all my expectations, his argument did generalise relatively straightforwardly to counting the number of shortest knight paths to any point on an infinite chessboard. Even more bizarrely, it did not depend on knowing the shortest distance beforehand: instead that comes almost for free as a consequence, along with the knight circle circumference (number of points at given distance). With the dust finally settled, the outcome is concise, rigorous combinatorial constructions of solutions to those three problems, very nearly also settling which points have maximal paths counts for given distance. Tantalisingly, the last stage of this last, tougher nut --- establishing dominance order among 7 axial and diagonal sequences --- still seems to require wheeling in heavy equipment, in the shape of computer-assisted `real algebraic geometry'. Waiting for its to-do list to empty promises to last for ever; so the draft has finally been uploaded to https://www.dropbox.com/s/1t51ivltm6l32gq/knight_path.pdf --- comments are solicited as usual. Fred Lunnon
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Fred Lunnon