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

Author

Scott R. Shannon and Eric Angelini, Apr 19 2021

Keywords

• nonn
• base

Examples

			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.
		

Links

• 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

Crossrefs

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

Programs

PARI

See Links section.

Formula

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

Extensions

More terms from Rémy Sigrist, Jun 16 2021