A344230 - cp's OEIS frontend

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.

%I A344230 #19 Jul 12 2022 19:52:58
%S A344230 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,
%T A344230 21,34,25,48,33,20,27,40,31,54,39,26,51,68,41,44,55,30,43,60,47,24,49,
%U A344230 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
%N 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.
%C A344230 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).
%C A344230 The squares on the board are numbered as follows:
%C A344230                   .    .    .    .    .    .    .
%C A344230                   .    .    .    .    .    .    .
%C A344230                   .    .    .    .    .    .    .
%C A344230                +----+----+----+----+----+----+----+
%C A344230                | 49 | 48 | 47 | 46 | 45 | 44 | 43 | ...
%C A344230    ending      +----+----+----+----+----+----+----+
%C A344230    square ---> | 36 | 35 | 34 | 33 | 32 | 31 | 42 | ...
%C A344230                +----+----+----+----+----+----+----+
%C A344230                | 25 | 24 | 23 | 22 | 21 | 30 | 41 | ...
%C A344230                +----+----+----+----+----+----+----+
%C A344230                | 16 | 15 | 14 | 13 | 20 | 29 | 40 | ...
%C A344230                +----+----+----+----+----+----+----+
%C A344230                |  9 |  8 |  7 | 12 | 19 | 28 | 39 | ...
%C A344230                +----+----+----+----+----+----+----+
%C A344230                |  4 |  3 |  6 | 11 | 18 | 27 | 38 | ...
%C A344230   starting     +----+----+----+----+----+----+----+
%C A344230    square ---> |  1 |  2 |  5 | 10 | 17 | 26 | 37 | ...
%C A344230                +----+----+----+----+----+----+----+
%H A344230 N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=RGQe8waGJ4w">The Trapped Knight</a>, Numberphile video (2019).
%t A344230 findvalue[{i_, j_}] := If[j > i, (j - 1)^2 + 2 j - i, (i - 1)^2 + j];
%t A344230 possiblemoves[{i_, j_}, prev_List] :=
%t A344230   Block[{moves = {{i + 2, j + 1}, {i + 2, j - 1}, {i + 1,
%t A344230        j + 2}, {i + 1, j - 2}, {i - 1, j + 2}, {i - 1, j - 2}, {i - 2,
%t A344230         j + 1}, {i - 2, j - 1}}, list},
%t A344230    list = DeleteCases[moves, {x_, y_} /; x < 1 || y < 1];
%t A344230    Complement[list, Intersection[list, prev]]];
%t A344230 findnextmove =
%t A344230   Block[{listofmoves = #, nextmove, poss},
%t A344230     pos = possiblemoves[listofmoves[[-1]], listofmoves];
%t A344230     If[Length[pos] > 0,
%t A344230      nextmove = Sort[({findvalue[#], #} &) /@ pos][[1, 2]];
%t A344230      AppendTo[listofmoves, nextmove], listofmoves]] &;
%t A344230 findvalue /@ FixedPoint[findnextmove, {{1, 1}}]
%Y A344230 Cf. A316667.
%K A344230 nonn,fini,full,walk
%O A344230 1,2
%A A344230 _Roman Mecholsky_, May 12 2021

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