A326931 a(n) is the end square spiral number for a knight starting on square n moving on a board with squares numbered with the square of their distance from the 0-square origin and where the knight moves to the smallest numbered unvisited square; the smallest spiral number ordering is used if the distances are equal.

25984, 51159, 8224, 31440, 8224, 31440, 8224, 110081, 131178, 92879, 69289, 59225, 62391, 10042, 66686, 73825, 36212, 123343, 158628, 28616, 74166, 98142, 59386, 50028, 42525, 15828, 7092, 27981, 57726, 27313, 52761, 15586, 47169, 17233, 152620, 73042, 76303, 83957, 59892, 9567

Offset: 1

Author

Scott R. Shannon, Oct 22 2019

Keywords

• nonn

Comments

This is the end square, using its spiral numbered value, for a knight starting on a square with spiral number n for a knight with step rules given in A326922. We use the spiral number to define the start and end square, as opposed to its square distance from the 0-square origin which predominantly determines the knight's path in A326922, as it is a unique value for each square on the board.

The largest end square spiral value for starting squares n from 1 to 200000 is a(72000) = 574108, which has a square distance number of 149725, which was also the largest found value. The largest number of steps before being trapped is for start square 103623, which is trapped after 483425 steps.

The smallest end square spiral value is a(1284) = 1143, which has a square distance number of 298. The smallest number of steps before being trapped is for start square 633, which is trapped after 1127 steps on square 1206. This has a square distance number of 293, the smallest value found.

Examples

			a(1) = 25984. See A326922.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..20000
• Scott R. Shannon, Path for starting square n = 72000. This has the largest found end square spiral number of 574108. In this and other images a green square marks the starting square, a red square the ending square, and blue squares mark the eight blocking squares for the end square. The end square is on the edge at about 12:30 on a clock.
• Scott R. Shannon, Path for starting square n = 103623. This is trapped after 483425 steps, the largest found value. The end square is on the edge at about 6:30 on a clock.
• Scott R. Shannon, Path for starting square n = 1284. This has the smallest found end square spiral number of 1143. Note that the start square acts as one of the eight blocking squares for the end square.
• Scott R. Shannon, Path for starting square n = 633. This is trapped after 1127 steps, the smallest found value.
• N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Crossrefs

Cf. A316667, A309918, A326922, A326924.