A306308 Table read by rows: the end square loops for a trapped knight moving on an infinitely large 2-dimensional spirally numbered board starting from any square.

404, 3328, 2666, 1338, 736, 1535, 2168, 406, 2444, 2945, 2245, 605, 684, 2663, 2312, 3323, 935, 910

Offset: 1

Author

Scott R. Shannon, Feb 05 2019

Keywords

• tabf
• nonn
• fini
• full

Comments

Construction: with a knight (a (1,2)-leaper) on an infinite spiral numbered board moving to the lowest numbered unvisited square (see A316884), start the knight on any square and continue moving it until it is trapped. Then start an entirely new sequence starting the knight at the same square at which it was previously trapped. Continue this process until the square at which the knight is trapped has occurred previously, indicating an end square loop. All starting squares for the knight on the infinite board will eventually lead to the knight path falling into one of the 3 end square loops listed here.

As the total number of squares in which the knight can be trapped is finite (see A306291), we expect there to be a finite number of end square loops - in theory, only those values (1518 is all) need to be checked when searching for an end square loop. However, all starting square values up to 302500 have been checked to determine into which of the 3 found loops the knight eventually falls. The 13-member loop with 406 as its lowest number is found to be the dominant loop, with about 89.6% of all initial starting squares going to it. The other 10.4% mostly go to the 3-member loop with 404 as its lowest number, with a decreasingly small remainder going to the 2-member loop with 910 as it lowest number. The attached 3-color image showing the start-value-to-loop mapping shows that the pattern of starting square to end square loops becomes very regular away from the center of the board.

Examples

			The 3 end square loops are:
1: 404, 3328, 2666
2: 1338, 736, 1535, 2168, 406, 2444, 2945, 2245, 605, 684, 2663, 2312, 3323
3: 935, 910
Starting the knight from the square 1 leads to the first 3-member loop after two iterations: the sequence of end squares is 2084, 404, 3328, 2666, 404, ... . Starting from the square 2 leads to the second (13-member) loop after ten iterations: the sequence is 711, 632, 4350, 3727, 3610, 7382, 2411, 4632, 4311, 1338, ... . The third (2-member) loop is not seen until the knight starts from square 284, the sequence being entered after two iterations via 1168, 935, 910, 935, ... .
		

Links

• Scott R. Shannon, Square positions for the 3 loops. The red line connects the 3 points of the first loop, the blue line connects the 13 points of the second loop, and the green line connects the 2 points of the third loop. The white point marks the central square with number 1.
• Scott R. Shannon, Starting square to loop mapping. A plot of the first 302500 starting squares mapped via color to the end square loop into which the corresponding knight path eventually falls: red is the first (3-member) loop, blue the second (13-member) loop, green the third (2-member) loop. The white point marks the central square with number 1 for clarity (it actually falls into the red first loop).
• Scott R. Shannon, The knight's path when starting at square 910. Showing path one of the 2-member loop - the green square is the starting square 910, the red square is the end square 935.
• Scott R. Shannon, The knight's path when starting at square 935. Showing path two of the 2-member loop - the green square is the starting square 935, the red square is the end square 910.
• Scott R. Shannon, Stripped down Java code to produce the loop values.
• N. J. A. Sloane and Brady Haran, The Trapped Knight Numberphile video (2019).

Crossrefs

Cf. A306291, A316884, A316967, A316667, A316328, A317106, A317105, A317416, A317415, A317438, A317437, and A323469, A323470, A323471, A323472.