A344230 Squares visited by a knight (chess piece) moving to the lowest-numbered unvisited square at each step on a semi-infinite chessboard numbered by starting in the lower left and filling in squares in a counterclockwise way moving to the bottom leftmost unnumbered square when the edge of the board is encountered.

1, 6, 9, 2, 7, 4, 5, 8, 11, 14, 3, 10, 19, 22, 15, 12, 17, 28, 13, 18, 29, 32, 23, 16, 35, 46, 21, 34, 25, 48, 33, 20, 27, 40, 31, 54, 39, 26, 51, 68, 41, 44, 55, 30, 43, 60, 47, 24, 49, 62, 45, 42, 53, 38, 65, 52, 37, 66, 85, 70, 57, 76, 61, 80, 97, 116, 75, 56, 59, 74, 71, 58, 73, 88, 69, 84, 101, 124, 83, 50, 67, 82, 103, 86, 107, 72, 87, 104, 123, 148, 105, 128, 89, 92, 109, 112, 93, 90, 111, 130, 91, 108, 127, 152, 131, 134, 113, 94, 77, 98, 63, 36

Offset

1,2

Comments

The sequence is finite and ends at the 111th move, which takes the knight to the square numbered 36 (the leftmost square on the 6th row).

The squares on the board are numbered as follows:

. . . . . . .

. . . . . . .

. . . . . . .

+----+----+----+----+----+----+----+

| 49 | 48 | 47 | 46 | 45 | 44 | 43 | ...

ending +----+----+----+----+----+----+----+

square ---> | 36 | 35 | 34 | 33 | 32 | 31 | 42 | ...

+----+----+----+----+----+----+----+

| 25 | 24 | 23 | 22 | 21 | 30 | 41 | ...

+----+----+----+----+----+----+----+

| 16 | 15 | 14 | 13 | 20 | 29 | 40 | ...

+----+----+----+----+----+----+----+

| 9 | 8 | 7 | 12 | 19 | 28 | 39 | ...

+----+----+----+----+----+----+----+

| 4 | 3 | 6 | 11 | 18 | 27 | 38 | ...

starting +----+----+----+----+----+----+----+

square ---> | 1 | 2 | 5 | 10 | 17 | 26 | 37 | ...

+----+----+----+----+----+----+----+

Links

Table of n, a(n) for n=1..112.

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

Mathematica

findvalue[{i_, j_}] := If[j > i, (j - 1)^2 + 2 j - i, (i - 1)^2 + j];
possiblemoves[{i_, j_}, prev_List] :=
  Block[{moves = {{i + 2, j + 1}, {i + 2, j - 1}, {i + 1,
       j + 2}, {i + 1, j - 2}, {i - 1, j + 2}, {i - 1, j - 2}, {i - 2,
        j + 1}, {i - 2, j - 1}}, list},
   list = DeleteCases[moves, {x_, y_} /; x < 1 || y < 1];
   Complement[list, Intersection[list, prev]]];
findnextmove =
  Block[{listofmoves = #, nextmove, poss},
    pos = possiblemoves[listofmoves[[-1]], listofmoves];
    If[Length[pos] > 0,
     nextmove = Sort[({findvalue[#], #} &) /@ pos][[1, 2]];
     AppendTo[listofmoves, nextmove], listofmoves]] &;
findvalue /@ FixedPoint[findnextmove, {{1, 1}}]

Crossrefs

Cf. A316667.

Sequence in context: A198676 A198616 A215668 * A010502 A254292 A188618

Adjacent sequences: A344227 A344228 A344229 * A344231 A344232 A344233

Keyword

nonn,fini,full,walk

Author

Roman Mecholsky, May 12 2021

Status

approved