A338288 Squares visited by the white knight when a white knight and a black knight are moving on a spirally numbered board, always to the lowest available unvisited square; white moves first.

1, 10, 3, 6, 15, 2, 5, 8, 11, 24, 27, 48, 23, 20, 39, 16, 19, 22, 41, 44, 71, 74, 45, 42, 69, 38, 63, 66, 99, 36, 61, 94, 31, 54, 85, 124, 51, 80, 83, 120, 123, 168, 81, 118, 77, 114, 73, 108, 151, 68, 103, 64, 67, 102, 143, 146, 195, 100, 141, 96, 137, 60, 93, 90

Offset

1,2

Comments

Board is numbered with the square spiral:

17--16--15--14--13 .

| | .

18 5---4---3 12 .

| | | | .

19 6 1---2 11 .

| | | .

20 7---8---9--10 .

| .

21--22--23--24--25--26

Both knights start on square 1, white moves to the lowest unvisited square (10), black then moves to the lowest unvisited square (12) and so on...

This sequence is finite, on the white knight's 3999th step, square 3606 is visited, after which there are no unvisited squares within one knight move.

The sequences generated by 4 knights and 8 knights also produce new sequences not yet in the OEIS.

Note that the black knight is trapped after the 1879th step when it visits square 4242; the sequence gives all the squares visited by the white knight up to and beyond that step until it is also trapped. - Scott R. Shannon, Apr 27 2026

Links

Scott R. Shannon, Table of n, a(n) for n = 1..4000

Scott R. Shannon, Image of the white knight's path. The final square is shown in red while the 8 visited squares around the final square are highlighted in dark blue.

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

Crossrefs

Cf. A338289, A338290.

Sequence in context: A335214 A394387 A394363 * A330008 A335844 A110409

Adjacent sequences: A338285 A338286 A338287 * A338289 A338290 A338291

Keyword

nonn,fini,full

Author

Andrew Smith, Oct 20 2020

Extensions

a(27) corrected by Scott R. Shannon, Apr 26 2026

Status

approved