The nettle has finally been grasped --- as intuition has been prompting from the start, if only I would listen --- and the resulting geometrical proof of the formula for knight's-move distance posted at https://www.dropbox.com/s/nzmzjswtctju23f/knights_path.txt Maybe not as short as initially hoped, but neither as long and tedious as recently feared. Third time lucky? And since it was easy to do --- not to mention more fun than writing up that dratted proof --- version 3 also tabulates the number of shortest paths from [0, 0] to [x, y] . Which in turn raises the question of finding the maximum number possible of shortest paths to a given [x, y] at given distance d . Unsurprisingly, the relevant points appear to hug the axis, along path [x, y] = [2d-3, d+1 mod 2] for d > 4 , these being at smallest Euclidean distance from the origin. The number of paths appears to approach 2^d d^2 for large d ; the counts for 0 <= d <= 48 are tabulated below, pending addition to OEIS. Fred Lunnon [ 1, 1, 2, 12, 54, 100, 330, 1050, 3024, 8736, 23220, 62700, 158004, 406692, 986986, 2452450, 5788640, 14002560, 32357052, 76640148, 174174520, 405623400, 909582212, 2089064516, 4633556448, 10519464000, 23120533800, 51977741400, 113365499940, 252725219460, 547593359850, 1211884139250, 2610998927040, 5741708459520, 12309472580460, 26917328938500, 57457069777800, 125016198060600, 265832233972140, 575824335603660, 1220234181784800, 2632570331352000, 5561593101431640, 11955215864166120, 25186887938801160, 53963277227145000, 113403569798134980, 242237549346710580, 507901544029952064 ];