A333683 The number of steps for a knight to be trapped when moving on a spirally numbered hexagonal board to the lowest available unvisited cell and starting at cell n.

83965, 738091, 277614, 252431, 731818, 731818, 765367, 622644, 252431, 252431, 1409949, 1720441, 512861, 925161, 251386, 1967478, 24228, 759058, 738091, 765367, 813609, 251386, 427289, 3220511, 48709, 151878, 231983, 121515, 113147, 894298, 158680, 815439, 1452850, 231479

Offset: 1

Author

Scott R. Shannon, Apr 02 2020

Keywords

• nonn

Comments

For a knight moving on a spirally numbered hexagonal board to the lowest available unvisited cell, see A327131, a(n) gives the number of steps before the knight is trapped when the knight starts on the cell numbered n.

See A327131 for the allowed knight moves, a diagram of the hexagonal board, and an illustration of the knight's path for n = 1.

For the first 100000 terms the longest path before the knight is trapped is for starting starting cell 81479 where it is trapped after 8125572 steps, the final cell being 8085793. In the same range the shortest path before being trapped is for starting cell 1036 where it is trapped after 1603 steps, the final cell being 1267. See the image in the links. This is likely the shortest path to being trapped for all starting cells.

Examples

			The knight starting on cell 1 becomes trapped after 83965 steps, see A327131.
		

Links

• Scott R. Shannon, The knight's path for starting cell n = 1036. The start cell 1036 is marked with a green dot, the final cell 1267 with a red dot, and the twelve surrounding blocking cell with blue dots.
• N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Crossrefs

Cf. A333684 (trapped cell number), A327131, A309918, A328894, A306291.