A328894 -id:A328894 - cp's OEIS frontend

A343530 Number of steps before being trapped for a knight moving on a square-spiral base-n numbered board when stepping to the closest unvisited square which contains a number that shares no digit with the number of the current square. If two or more such squares are the same distance away the one with the smaller number is chosen.

0, 1, 12, 10, 13, 16, 35, 51, 56, 90, 42, 84, 99, 129, 156, 30, 220, 184, 201, 79, 321, 25, 424, 301, 389, 29, 32, 311, 328, 186, 129, 42, 101, 97, 144, 52, 534, 83, 506, 885, 233, 472, 43, 410, 145, 210, 482, 51, 57, 144, 53, 60, 148, 248, 83, 80, 180, 72, 55

Offset: 2

Scott R. Shannon and Eric Angelini, Apr 19 2021

			The board in base 10 is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12  29
   |   |       |   |   |
  19   6   1---2  11  28
   |   |           |   |
  20   7---8---9--10  27
   |                   |
  21--22--23--24--25--26
.
a(2) = 0 as on a base-2 numbered spiral all surrounding squares contain a 1 digit in their number thus, as the knight starts on the square numbered 1, it has no square to move to which does not contain a 1 digit.
a(3) = 1 as on a base-3 numbered board there are two squares the knight can step to which do not contain a 1 digit -- the squares numbered 200_3 = 18 and 220_3 = 24. The knight steps to 200_3 as it is the lowest numbered square, but after that there are no surrounding unvisited squares the knight can step to which do not contain the digit 0 or 2.
a(4) = 12 as on a base-4 numbered board the knight steps to squares 22_4 = 10, 3_4 = 3, 12_4 = 6, 33_4 = 15, 2_4 = 2, 11_4 = 5, 20_4 = 8, 111_4 = 21, 220_4 = 40, 13_4 = 7, 222_4 = 42, 103_4 = 19. The knight is then trapped as no unvisited square containing only the digit 2 is one knight step away.
See the linked images for other examples.

Rémy Sigrist, Table of n, a(n) for n = 2..3500
Scott R. Shannon, Image of the path for a(4) = 12. In this and other images the starting square is highlighted in white, the visited squares, numbered in base n, in yellow, the final square in red, while the path is colored across the spectrum to show the relative step ordering.
Scott R. Shannon, Image of the path for a(8) = 35.
Scott R. Shannon, Image of the path for a(10) = 56
Scott R. Shannon, Image of the path for a(16) = 156.
Scott R. Shannon, Image of the path for a(24) = 424.
Scott R. Shannon, Image of the path for a(36) = 144.
Rémy Sigrist, PARI program for A343530

Cf. A344325, A343563, A326918, A328894, A326916, A316667.

PARI
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See Links section.
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a(n) = 2015 for any n >= 2979. - Rémy Sigrist, Jun 16 2021

More terms from Rémy Sigrist, Jun 16 2021

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.

Original entry on oeis.org

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

Scott R. Shannon, Apr 02 2020

nonn

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.

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

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).

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

A343563 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the unvisited square containing the spiral number with the smallest digit sum. In case of a tie it chooses the lowest number.

Original entry on oeis.org

1, 10, 3, 30, 11, 4, 13, 2, 5, 20, 23, 6, 21, 40, 105, 202, 103, 100, 141, 250, 315, 190, 251, 140, 61, 14, 31, 12, 15, 32, 55, 130, 91, 180, 301, 234, 127, 52, 25, 50, 121, 222, 119, 220, 117, 80, 51, 124, 231, 126, 53, 26, 9, 22, 41, 106, 203, 104, 201, 102, 143, 252, 321, 480, 323, 400, 403

Offset: 1

Scott R. Shannon, Apr 19 2021

This sequences gives the numbers of the squares visited by a knight moving on a square-spiral numbered board where at each step the knight moves to the unvisited neighbor one knight-leap away which contains the number with the smallest digit sum. If two or more neighbors exist with the same digit sum then from those squares the one with the lowest number is chosen.

The sequence is finite. After 790 steps the square with number 69 is visited, after which all eight neighboring squares have been visited. The largest visit spiral number is a(626) = 6112, while there are four squares with the largest visited digit sum of 19: a(373) = 2683, a(539) = 2737, a(590) = 2944, a(594) = 2728.

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(2) = 10 as the eight unvisited neighbors of the square a(1) = 1 are numbered 10,12,14,16,18,20,22,24, and 10, with a digit sum of 1, has the lowest digit sum of these.
a(4) = 30 as the seven unvisited neighbors of the square a(3) = 3 square are numbered 6,8,28,30,32,34,16, and 30, with a digit sum of 3, has the lowest digit sum of these.
a(9) = 5 as two of the unvisited neighbors of the square a(8) = 2 are 5 and 23, both of which have a digit sum of 5, but 5 is chosen as it is the lower number.

Scott R. Shannon, Image showing the 791 visited squares. The starting square is highlighted in white, the visited squares in yellow, the final square in red, while the path is colored across the spectrum to show the relative step ordering.
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Cf. A007953, A316667, A328894, A326922, A328928.

A333684 The cell number where a knight is trapped when moving on a spirally numbered hexagonal board to the lowest available unvisited cell and starting at cell n.

Original entry on oeis.org

72085, 706243, 270402, 236090, 716518, 716518, 730674, 657313, 236090, 236090, 1318101, 1634797, 482448, 901595, 237177, 1946730, 21429, 726318, 706243, 730674, 793200, 237177, 405933, 3095967, 51035, 159266, 218715, 106443, 101767, 927137, 148315, 786512, 1495770

Offset: 1

Scott R. Shannon, Apr 02 2020

nonn

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.

			The knight starting on cell 1 becomes trapped on cell 72085 after 83965 steps, see A327131.

N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Cf. A333683 (number of steps), A327131, A309918, A328894, A306291.

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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.

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A343563 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the unvisited square containing the spiral number with the smallest digit sum. In case of a tie it chooses the lowest number.

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A333684 The cell number where a knight is trapped when moving on a spirally numbered hexagonal board to the lowest available unvisited cell and starting at cell n.

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