cp's OEIS Frontend

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.

This is the end square, using its spiral numbered value, for a knight starting on a square with spiral number n for a knight with step rules given in A326922. We use the spiral number to define the start and end square, as opposed to its square distance from the 0-square origin which predominantly determines the knight's path in A326922, as it is a unique value for each square on the board.

The largest end square spiral value for starting squares n from 1 to 200000 is a(72000) = 574108, which has a square distance number of 149725, which was also the largest found value. The largest number of steps before being trapped is for start square 103623, which is trapped after 483425 steps.

The smallest end square spiral value is a(1284) = 1143, which has a square distance number of 298. The smallest number of steps before being trapped is for start square 633, which is trapped after 1127 steps on square 1206. This has a square distance number of 293, the smallest value found.

For a knight moving on a spirally numbered hexagonal board to the lowest available unvisited cell, see A327131, a(n) gives the cell number where 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 largest cell number where the knight is trapped is for starting starting cell 81479 where the final cell has number 8085793, being reached after 8125572 steps. In the same range the smallest cell number where the night is trapped is for starting cell 1036 where the final cell has number 1267, being reached after 1603 steps. See A333683 for an image of this path.