A329022 Squares visited by a knight moving on a diagonal spiral numbered board and moving to the lowest available unvisited square at each step.

1, 14, 7, 3, 6, 4, 8, 5, 10, 2, 9, 17, 28, 43, 13, 15, 26, 24, 11, 20, 32, 48, 29, 44, 63, 25, 12, 21, 34, 18, 30, 45, 27, 16, 31, 19, 35, 22, 39, 57, 36, 23, 37, 54, 75, 51, 71, 95, 68, 91, 65, 46, 69, 49, 33, 53, 74, 50, 70, 47, 66, 89, 116, 42, 40, 58, 80, 55, 38, 56, 77, 102, 131, 52, 72, 96, 124

Offset

1,2

Comments

This sequence uses a diagonal spiral of numbers to enumerate the squares on the board. The knight starts on the square with number 1. At each step the knight goes to an unvisited square with the smallest number.

The sequence if finite. After 3722 steps the square with number 3541 is visited, after which all neighboring squares have been visited.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..3723

Scott R. Shannon, Image showing the 3722 steps of the knight's path. The green dot is the starting square and the red dot the final square. Blue dots show the eight occupied squares surrounding the final square.

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

Example

The board is numbered in a spiral moving along the diagonals of the square grid:
.
                       19
                     /   \
                   /       \
                 20    9     18
               /     /   \     \
             /     /       \     \
           21    10    3     8     17
         /     /     /   \     \     \
       /     /     /       \     \     \
     22    11    4     1 --- 2     7     16
       \     \     \             /     /     .
         \     \     \         /     /     .
           23    12    5 --- 6     15    28
             \     \             /     /
               \     \         /     /
                 24    13 -- 14    27
                   \             /
                     \         /
                       25 -- 26
.
    +----+----+----+----+----+----+----+
    | 76 | 53 | 34 | 19 | 32 | 49 | 70 |
    +----+----+----+----+----+----+----+
    | 54 | 35 | 20 |  9 | 18 | 31 | 48 |
    +----+----+----+----+----+----+----+
    | 36 | 21 | 10 |  3 |  8 | 17 | 30 |
    +----+----+----+----+----+----+----+
    | 22 | 11 |  4 |  1 |  2 |  7 | 16 |
    +----+----+----+----+----+----+----+
    | 38 | 23 | 12 |  5 |  6 | 15 | 28 |
    +----+----+----+----+----+----+----+
    | 58 | 39 | 24 | 13 | 14 | 27 | 44 |
    +----+----+----+----+----+----+----+
    | 82 | 59 | 40 | 25 | 26 | 43 | 64 |
    +----+----+----+----+----+----+----+
.

Crossrefs

Cf. A316667.

Cf. A010751(n), A305258(n) for coordinates of point number n+1.

Sequence in context: A353120 A051655 A048932 * A033334 A161914 A162774

Adjacent sequences: A329019 A329020 A329021 * A329023 A329024 A329025

Keyword

nonn,fini,full,walk

Author

Scott R. Shannon, Nov 02 2019

Status

approved