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A344230
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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.
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0
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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, 21, 34, 25, 48, 33, 20, 27, 40, 31, 54, 39, 26, 51, 68, 41, 44, 55, 30, 43, 60, 47, 24, 49, 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
(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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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).
The squares on the board are numbered as follows:
. . . . . . .
. . . . . . .
. . . . . . .
+----+----+----+----+----+----+----+
| 49 | 48 | 47 | 46 | 45 | 44 | 43 | ...
ending +----+----+----+----+----+----+----+
square ---> | 36 | 35 | 34 | 33 | 32 | 31 | 42 | ...
+----+----+----+----+----+----+----+
| 25 | 24 | 23 | 22 | 21 | 30 | 41 | ...
+----+----+----+----+----+----+----+
| 16 | 15 | 14 | 13 | 20 | 29 | 40 | ...
+----+----+----+----+----+----+----+
| 9 | 8 | 7 | 12 | 19 | 28 | 39 | ...
+----+----+----+----+----+----+----+
| 4 | 3 | 6 | 11 | 18 | 27 | 38 | ...
starting +----+----+----+----+----+----+----+
square ---> | 1 | 2 | 5 | 10 | 17 | 26 | 37 | ...
+----+----+----+----+----+----+----+
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LINKS
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MATHEMATICA
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findvalue[{i_, j_}] := If[j > i, (j - 1)^2 + 2 j - i, (i - 1)^2 + j];
possiblemoves[{i_, j_}, prev_List] :=
Block[{moves = {{i + 2, j + 1}, {i + 2, j - 1}, {i + 1,
j + 2}, {i + 1, j - 2}, {i - 1, j + 2}, {i - 1, j - 2}, {i - 2,
j + 1}, {i - 2, j - 1}}, list},
list = DeleteCases[moves, {x_, y_} /; x < 1 || y < 1];
Complement[list, Intersection[list, prev]]];
findnextmove =
Block[{listofmoves = #, nextmove, poss},
pos = possiblemoves[listofmoves[[-1]], listofmoves];
If[Length[pos] > 0,
nextmove = Sort[({findvalue[#], #} &) /@ pos][[1, 2]];
AppendTo[listofmoves, nextmove], listofmoves]] &;
findvalue /@ FixedPoint[findnextmove, {{1, 1}}]
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CROSSREFS
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KEYWORD
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nonn,fini,full,walk
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AUTHOR
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STATUS
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approved
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