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On a doubly-infinite chessboard, number all the cells in a counterclockwise spiral starting at a central cell labeled 0. Start with a knight at cell 0, and thereafter always move the knight to the smallest unvisited cell. Sequence gives succession of squares visited.

Sequence ends if knight is unable to move.

Inspired by A316588 and, like that sequence, has only finitely many terms; see A316667 for details.

See A326924 for a variant where the knight prefers squares closest to the origin, and gets trapped only after 22325 moves. - M. F. Hasler, Oct 21 2019

See A323809 for an infinite extension of this sequence, obtained by allowing the knight to go back in case it was trapped. See A328908 for a variant of length > 10^6, using the taxicab distance, and A328909 for a variant using the sup norm. - M. F. Hasler, Nov 04 2019

Differs from A326924 (where only the 2-norm is considered) from a(34) = 42 on, and from A316328 (which considers only the number of the square) from a(48) = 38 on.

When the knight lands on square number a(25108) = 21040 of coordinates (73, -57), there is no unvisited square within reach. The sequence then stops, or can be extended by specifying that the knight has to go back on its path until an unvisited square comes within reach, as in A323809.

The least unvisited square at move 25108 is square number 17822 at (67,67). It is however close to the border of the visited region and the knight will visit it in the infinite extension of the sequence shortly after, at move n = 25358. Is there a square that will never be visited in that infinite extension? (Cf. comments in A323809.) - M. F. Hasler, Nov 04 2019

"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.

Differs from the original A316328 from a(34) = 76 on. See there for more information and other related sequences.

The knight gets trapped at the 22325th move at position (x,y) = (81, -18), from which it can't reach any unvisited square.

Sequence A326922 gives the distance^2 of the square number a(n) visited at move n. - M. F. Hasler, Oct 22 2019

From M. F. Hasler, Nov 04 2019: (Start)

When a(22325) = 25983 at (81, -18) is reached, at distance sqrt(6885) from the origin, the last unvisited square has number 13924, at (-59, 59), distance sqrt(6962) from the origin. This suggests that in an infinite extension (knight moves one step back if no unvisited square is available, cf. A323809) the knight might eventually visit every square. Can this be disproved by a counterexample of a square which will never be visited in the infinite extension? (In A328908 such a counterexample exists even before the knight gets stuck.)

The ratio a(n)/n oscillates between 0.5 and less than 1.7 for all n > 3000, even < 1.5 for all n > 14000, cf. graph of the sequence. What is the superior and inferior limit of this ratio, assuming the infinite extension beyond n = 22325?

(End)

This sequence uses the same board numbering as A326413, and like that sequence, if the next step encounters two or more squares with the same square number, it then chooses the square which is the closest to the original 0-squared origin. But if two or more squares are found which also have the same distance to the origin, then the square which was first drawn in the spiral numbering is chosen, i.e., the smallest standard spiral numbered square as in A316667.

The sequence is finite. After 1069 steps a square with the number 9 (standard spiral number = 473) is visited, after which all neighboring squares have been visited.

Sequence A326916(n) gives the number of the square visited at step n, i.e., its rank in the spiral, starting with 0, as illustrated, e.g., in A326413. The digit on that square, i.e., a(n) can be obtained through A007376, cf. formula. - M. F. Hasler, Oct 21 2019

Take the standard counterclockwise square spiral starting at 0, as in A304586, but only write one digit at a time in the cells of the spiral: 0,1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,...

Place a chess knight at cell 0. Move it to the lowest-numbered cell it can attack, and if there is a tie, move it to the cell closest (in Euclidean distance) to the start, and if there is still a tie, move to the left(*).

No cell can be visited more than once.

Inspired by the Trapped Knight video and A316667.

Just as for A316667, the sequence is finite. After a while, the knight has no unvisited squares it can reach, and the sequence ends with a(1217) = 4.

(*)Moving to the left means choose the point with the lowest x-coordinate. This leads to an unambiguous choice of tied squares only for the 'move left' case.

This is an infinite extension of A316667 with which it agrees for the first 2016 terms. - N. J. A. Sloane, Jan 28 2019

This is an infinite extension of A316328, with which it coincides for the first 2016 terms. - N. J. A. Sloane, Jan 29 2019

From M. F. Hasler, Nov 04 2019: (Start)

At move 99999, the least yet unvisited square has number 66048, which is near the border of the visited region. This suggests that the knight will eventually visit every square. Can this be proved or disproved through a counterexample?

More formally, let us call "isolated" a set of unvisited squares which is connected through knight moves, but which cannot be extended to a larger such set by adding a further square. Can there exist at some moment a finite isolated set which the knight cannot reach? (Without the last condition, the square a(2016) would clearly satisfy the condition just before the knight reaches it.)

Such subsets have a good chance of preserving this property forever. It should be possible to prove that an isolated subset sufficiently far (2 knight moves?) from any other unvisited square (or from the infinite connected subset of unvisited squares) remains so forever. (This condition of distance could replace the time-dependent condition "reachable by the knight".)

If such (forever) isolated sets do exist, with what frequency will they occur? Could they have a nonzero asymptotic density? Will this (if so, how) depend on the way the knight measures "lowest available" (cf. variants where the taxicab or Euclidean or sup norm distance from the origin is used)? (End)

A variant of Angelini's "Kneil's Knumberphile Knight", inspired by Sloane's "The Trapped Knight", cf. A316328 and links:

Consider an infinite chess board with squares numbered along the infinite square spiral starting with 0 at the origin (as in A174344, A274923 and A296030). The squares are filled with successive digits of the integers: 0, 1, 2, ..., 9, 1, 0, 1, 1, ... (= A007376 starting with 0). The knight moves at each step to the yet unvisited square with the lowest digit on it, and in case of a tie, the one closest to the origin, first by Euclidean distance, then by appearance on the spiral (i.e., number of the square). This sequence lists the number of the square visited in the n-th move, if the knight starts at the origin, viz a(0) = 0.

It turns out that following these rules, the knight gets trapped at the 1070th move, when he can't reach any unvisited square.

See A326918 for the sequence of visited digits, given as A007376(a(n)).

Many squares, e.g., 2: (1,1), 4: (-1,1), 5: (-1,0), 6: (-1,-1), 7: (0,-1), 8: (1,-1), 9: (2,-1), ..., will never be visited, even in the infinite extension of the sequence where the knight can move back if it gets trapped, in order to resume with a new unvisited square, as in A323809. - M. F. Hasler, Nov 08 2019

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 moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the number with the fewest divisors. If two or more adjacent squares exist with the same fewest number of divisors then the square with the lowest spiral number is chosen. Note that if the king simply moves to the smallest available number, as the knight does in A316667, the sequence will be infinite as the king will just follow the square spiral path.

The sequence is finite. After 411 steps the square with number 760 is visited, after which all adjacent neighboring squares have been visited.

Due to the king's preference for squares with the fewest divisors it will move to a prime numbered square when possible, and the lowest prime if two or more unvisited primes are in adjacent squares. Of the 411 visited squares 134 contain prime numbers while 277 contain composites. The largest visited square is a(365) = 3061.

This sequence 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 moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the number with the most divisors. If two or more adjacent squares exist with the same highest number of divisors then the square with the highest spiral number is chosen. Given both of these rules tend to force the king to squares with larger numbers, and thus move away from the central 1 starting square, it is remarkable that the king is eventually trapped. Note that if the king simply moves to the highest available number the sequence will be infinite as the king will step along the southeast diagonal from square 1 forever.

The sequence is finite. After 1113 steps the square with number 855481 is visited, after which all adjacent neighboring squares have been visited.

Due to the king's preference for squares with the most divisors it will avoid prime numbers unless no other choice exists. Of the 1113 visited squares only once does it visit a square with a prime number, at a(308) = 108223. This is due to a(307) = 106913 having square 108223 as its sole neighboring unvisited square. This is the only time in the sequence where only one unvisited adjacent neighbor is available.

As even numbers >= 6 will always contain 4 or more divisors the king will tend to visit more even numbers than odd numbers; in the 1113 visited squares 929 contain an even number while only 184 contain an odd number.

As the even numbers are diagonally adjacent in the square spiral the king's path will be dominated by diagonal steps, often taking many diagonal steps in succession - see the attached link image. In fact after the first downward step to 8 the next 110 steps are along the southeast diagonal, stepping to successively larger even numbers. This sequence is finally broken on the 112th step when the square with number 50624, with 28 divisors, is the next square in the southeast direction. However the square with number 50622, with 32 divisors, is in the southwest direction so is the next square chosen. It is not until the 166th step, to the square with number 108230, that the path takes a step to a lower number than the one it is currently on.

The largest visited square is a(1050) = 942676. The visited square with the maximum number of divisors is a(680) = 388080, which has 180 divisors. The lowest unvisited square is 2.