A337170 Squares visited by knight moves on a diagonally back and forth numbered board and moving to the lowest available unvisited square at every step.

1, 8, 4, 2, 13, 3, 6, 9, 11, 19, 5, 7, 26, 16, 14, 31, 15, 18, 12, 21, 24, 10, 23, 33, 39, 20, 22, 34, 25, 17, 28, 47, 43, 29, 27, 32, 40, 35, 37, 53, 57, 36, 58, 52, 38, 55, 80, 76, 56, 54, 59, 51, 42, 30, 45, 48, 62, 70, 44, 46, 64, 49, 41, 72, 60, 50, 63

Offset

1,2

Comments

Board is numbered as follows:

1 3 4 10 11 .

2 5 9 12 . .

6 8 13 19 . .

7 14 18 . . .

15 17 . . . .

16 . . . . .

This sequence is finite: At step 343 square 276 is visited, after which there are no unvisited squares within one knight move.

Links

Sander G. Huisman, Table of n, a(n) for n = 1..343

Scott R. Shannon, Image of the path. The green and red square are the first and last visited squares.

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

Mathematica

ClearAll[ShowRoute, MakeMove, FindSequence]
knightjump=Select[Tuples[Range[-2, 2], 2], Norm[#]==Sqrt[5]&];
ShowRoute[output_Association]:=Module[{colors}, colors=(ColorData["Rainbow"]/@Subdivide[Length[output["Coordinates"]]-1.0]);
Graphics[{Line[output["Coordinates"], VertexColors->colors], Disk[Last@output["Coordinates"], 0.2]}]]
MakeMove[spiral_Association, visited_List]:=Module[{poss, hj}, poss=Table[Last[Last[visited]]+hj, {hj, knightjump}];
poss=DeleteMissing[{spiral[#], #}&/@poss, 1, \[Infinity]];
poss=Select[poss, FreeQ[visited[[All, 2]], Last[#]]&];
If[Length[poss]>0, First[TakeSmallestBy[poss, First, 1]], Missing[]]]
FindSequence[start_:{0, 0}, grid_]:=Module[{positions, j, next}, positions={{grid[start], start}};
PrintTemporary[Dynamic[j]];
Do[next=MakeMove[grid, positions];
If[next=!=Missing[], AppendTo[positions, next], Break[]; ], {j, \[Infinity]}];
<|"Coordinates"->positions[[All, 2]], "Indices"->positions[[All, 1]]|>]
grid=ResourceFunction["LatticePointsArrangement"]["DiagonalZigZagEastQ4", 10000];
grid=Association[MapIndexed[#1->#2[[1]]&, grid]];
ShowRoute[fs=FindSequence[{0, 0}, grid]]
fs
fs["Indices"]
ListPlot[fs["Indices"]]

Crossrefs

Cf. A316588, A316328, A316667.

Sequence in context: A089729 A173670 A382004 * A050135 A109595 A256782

Adjacent sequences: A337167 A337168 A337169 * A337171 A337172 A337173

Keyword

nonn,fini,full,look

Author

Sander G. Huisman, Jan 28 2021

Status

approved