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A341195
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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.
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1
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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MATHEMATICA
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(* 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"]]
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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