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%I #28 Jan 26 2020 01:06:51
%S 1,10,3,6,9,4,7,2,5,8,11,14,29,32,15,12,27,24,45,20,23,44,41,18,35,38,
%T 19,16,33,30,53,26,47,22,43,70,21,40,17,34,13,28,25,46,75,42,69,104,
%U 37,62,95,58,55,86,51,48,77,114,73,108,151,68,103,64,67,36
%N Squares visited by a knight moving on a square-spiral numbered board where the knight moves to an unvisited square with the lowest spiral number and with seven or fewer visited neighbors. It only moves to squares with eight visited neighbors when no other square is available.
%C This is a variation of A316667. The same knight move rules apply, but the knight will not move to a square which will result in it being trapped (the square will have eight visited surrounding neighbors) unless no other squares are available. If the only squares available will all result in the knight being trapped it will choose the one with the lowest board spiral number.
%C The sequence is finite. After 23014 steps the square with spiral number 25809 is reached after which all surrounding squares have been visited. This is the third largest possible path using the given knight-leap rules for the eight possible values of visited neighbor count. A329520 gives the other path lengths.
%C The sequences matches the values of A316667 for the first 2015 terms, but on the 2015th step the knight sees that square 2084 will result in it being trapped and thus chooses square 2668 instead. Along its path the knight encounters sixteen squares where it would be trapped if it had chosen the smallest numbered available square. These occurs after steps 2015, 2983, 3116, 3372, 7485, 8775, 9726, 10971, 11845, 11918, 12140, 18477, 18706, 19921, 22223, 23014. The corresponding board numbers which were rejected are given by the first fifteen values of A323714. On step 23014 there is only one square available which is it forced to move to, resulting in it being trapped on square 25809, the sixteenth entry of A323714.
%H Scott R. Shannon, <a href="/A329518/b329518.txt">Table of n, a(n) for n = 1..23015</a>
%H Scott R. Shannon, <a href="/A329518/a329518.png">Image showing the 23104 steps of the knight's path</a>. The start square is shown in green and the final square in red. All squares where the knight would have been trapped if it had chosen the lowest numbered available square are shown in yellow, along with the number of steps completed. The square it rejected at that point is shown in gray, and eight purple squares are shown around these rejected squares to show the square would have resulted in trapping. The second last square is also surrounded by eight pink squares showing that the only available square at that point was the final square which it was thus forced to move to. The final square, on the edge at about the 7 o'clock position, is surrounded by its eight blocking squares in blue.
%H N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=RGQe8waGJ4w">The Trapped Knight</a>, Numberphile video (2019).
%e See A316667 for the spiral board numbering.
%Y Cf. A316667, A323714, A329519, A329520.
%K nonn,fini,full,walk
%O 1,2
%A _Scott R. Shannon_, Nov 18 2019
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