[seqfan] Re: Knight's tour on infinite board

[Hans Havermann]

HP: "Combining all spiral variants with the enumeration of various paths on the 2-d lattice or space filling curves like the knight's tour would be a can of worms."

I've actually thought about it a bit. There appear to be 8 square spiral variants: First step east, north, west, or south, followed by either a clockwise or counterclockwise numbering. This is analogous to the knight's first move which uncoincidentally also numbers 8. Thus we've already set up 64 possible numbering variants of the knight's tour on an infinite board.

HH: "It then jumps 2 south and 1 east and proceeds to circle the 5-by-5 square clockwise four times filling in the now-9-by-9 square and ending in its SE corner."

My idea of flipping the infinite chessboard to reverse the clockwise/counterclockwise direction of the entire knight's tour spiral can (I think) be applied selectively to each two-squares-wide expansion ring that is traveled four times. In my quoted example, if instead of jumping 2 south and 1 east, it jumps 1 south and 2 east, I think it can circle the 5-by-5 square counterclockwise and still end up in the 9-by-9 square's SE corner. If that's correct, then we have at each expansion point a binary choice of whether to go clockwise or counterclockwise. This would suggest an infinite number of possible knight's tours on an infinite chessboard.