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%I A336038 #40 Jan 20 2024 03:22:38
%S A336038 1,2,3,4,6,5,15,14,12,11,9,8,22,7,19,18,16,17,35,34,60,32,13,29,28,10,
%T A336038 25,24,46,23,45,21,20,40,39,67,37,36,38,66,64,63,97,61,62,96,95,59,33
%N A336038 Squares visited by a chess king on a square-spiral numbered board and stepping to the lowest unvisited adjacent square, where each step is not in the same direction as the previous step.
%C A336038 This sequences gives the numbers of the squares visited by a chess king moving on a square-spiral numbered board where the king starts on the 1 numbered square and at each step, which is not in the same direction as its previous step, moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the lowest spiral number.
%C A336038 The sequence is finite. After 48 steps the square with spiral number 33 is reached after which all eight adjacent squares have been visited.
%C A336038 If the king simply moved to the lowest numbered unvisited adjacent square the walk would be infinite as the king would just follow the path of the square spiral. By not allowing consecutive moves in the same direction forces the king off this minimal numbered path. The first time this happens is a(5) = 6 as from a(4) = 4 the lowest numbered adjacent square is 5 but that would require a step directly to the left, the same as the previous step from a(3) = 3 to a(4).
%H A336038 Scott R. Shannon, <a href="/A336038/a336038.png">Image showing the 48 steps of the king's path</a>. The green dot is the first square with number 1 and the red dot the last square with number 33. The red dot is surrounded by blue dots to show the eight occupied squares. The yellow dots marks the smallest unvisited square with number 26.
%H A336038 N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=RGQe8waGJ4w">The Trapped Knight</a>, Numberphile video (2019).
%e A336038 The board is numbered with the square spiral:
%e A336038 .
%e A336038   17--16--15--14--13   .
%e A336038    |               |   .
%e A336038   18   5---4---3  12   29
%e A336038    |   |       |   |   |
%e A336038   19   6   1---2  11   28
%e A336038    |   |           |   |
%e A336038   20   7---8---9--10   27
%e A336038    |                   |
%e A336038   21--22--23--24--25--26
%e A336038 .
%e A336038 a(1) = 1, the starting square of the king.
%e A336038 a(2) = 2. The eight adjacent unvisited squares around a(1) are numbered 2,3,4,5,6,7,8,9. Of these 2 is the lowest.
%e A336038 a(5) = 6. The five adjacent unvisited squares around a(4) = 4 are numbered 5,6,14,15,16. Of these 5 is the lowest but that would require a step directly left from 4, which is the same step as a(3) = 3 to a(4) = 4, so is not allowed. The next lowest available square is 6.
%Y A336038 Cf. A336208, A335816, A333713, A333714, A316667.
%K A336038 nonn,fini,full,walk
%O A336038 1,2
%A A336038 _Scott R. Shannon_, Jul 12 2020

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