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

Author

Sander G. Huisman, Jan 28 2021

Keywords

• nonn
• fini
• full
• look

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

Crossrefs

Cf. A316588, A316328, A316667.

Programs

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"]]