Knight Tours

I've already made several posts relating mathematics and chess. These are:

• More on the Mathematics of Chess on Thursday, 14th November 2019
• The Mathematics of Chess on Sunday, 6th January 2019
• Hogben Numbers on Wednesday, 28th November 2018
• Chess960 on Monday, 12th February 2018
• The Mathematics of Chess Pairings on Sunday, 11th February 2018

Today I turned 25983 days old and my investigation of the number 25983 revealed an interesting chess connection. Suppose we number the squares on an infinite chess board starting with 0 and then counting in an anticlockwise spiral as shown in Figure 1.

Figure 1

Suppose we devise a Knight Tour such that the Knight starts on square 0 and then moves always to the unvisited square closest to the origin. "Closest to the origin" is meant in the sense of Euclidean distance, and in case of a tie, the square coming earliest on the spiral is chosen.

On the first move, the Knight could move to squares 9, 11, 13, 15, 17, 19, 21 or 23. All these squares are the same distance from the 0 square but 9 is chosen because it comes first in the spiral. From 9, subsequent squares are 2, 5, 8, 3, 6, 1, 4, 7, 10, 13 and so on. 

These numbered squares form OEIS A326924:

A326924

Squares visited by a knight on a spirally numbered board, moving always to the unvisited square closest to the origin.

Figure 2 shows a slightly different depiction of the squares with more squares viewable:

Figure 2

The initial terms are of the sequence are:

0, 9, 2, 5, 8, 3, 6, 1, 4, 7, 10, 13, 28, 31, 14, 11, 26, 23, 44, 19, 22, 43, 40, 17, 34, 37, 18, 15, 32, 29, 52, 25, 46, 21, 76, 47, 50, 27, 12, 33, 16, 39, 20, 45, 24, 51, 48, 77, 114, 73, 70, 105, 38, 35, 60, 93, 30, 53, 84, 49, 78, 115, 74, 41, 68, 103, 36, 61, 94, 57, 54, 85, 124, 81, ...

What's really interesting about this tour is that the knight gets trapped at the 22325th move, where it can't reach any unvisited square. The square on which the Knight is stuck is 25983!

So nothing in this post of deep mathematical significance but it always strikes me as fascinating how numbers can be associated with such diverse phenomena. In this case, it is a Knight's tour of an infinite, spirally numbered chessboard.

For information on other Knight tours visit https://oeis.org/wiki/Knight_tours.