A341195 Squares visited by knight moves on a diagonally back and forth numbered board in two quadrants and moving to the lowest available unvisited square at every step.

1, 11, 7, 2, 6, 12, 9, 3, 5, 14, 8, 4, 18, 33, 21, 29, 24, 26, 47, 10, 23, 13, 19, 16, 38, 34, 17, 15, 20, 30, 42, 56, 45, 28, 22, 31, 41, 58, 44, 32, 40, 35, 37, 62, 66, 36, 39, 60, 68, 63, 65, 98, 102, 64, 67, 61, 70, 93, 43, 55, 46, 27, 49, 52, 25, 51, 78

Offset: 1

Author

Sander G. Huisman, Feb 06 2021

Keywords

• nonn
• fini
• full

Comments

Board is numbered as follows:

. 17 16 5 4 1 2 9 10 . .

. . 18 15 6 3 8 11 24 . .

. . . 19 14 7 12 23 . . .

. . . . 20 13 22 . . . .

. . . . . 21 . . . . .

. . . . . . . . . . .

This sequence is finite: At step 4408 square 4077 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..4408
• N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (January, 2019).

Crossrefs

Cf. A316588, A316328, A316667, A337170.

Programs

Mathematica

(* Version 12.0 or higher needed *)
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],Style[Disk[Last[output["Coordinates"]]+#,0.2]&/@knightjump,Purple]}]]
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"]["DiagonalZigZagEastQ34",20000];
grid=Association[MapIndexed[#1->#2[[1]]&,grid]];
ShowRoute[fs=FindSequence[{0,0},grid]]
fs
fs["Indices"]
ListPlot[fs["Indices"]]