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Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the spiral number with the most divisors. In case of a tie it chooses the square with the lowest spiral number.
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6
1, 6, 18, 40, 70, 108, 72, 42, 20, 21, 44, 45, 75, 114, 160, 216, 280, 350, 351, 352, 432, 520, 616, 720, 832, 952, 1080, 1216, 1360, 1512, 1672, 1840, 2016, 2200, 2392, 2592, 2800, 3016, 3240, 3472, 3710, 3956, 4212, 4476, 4746, 5024, 5310, 5022, 4743, 4472, 4473, 4209, 4208, 3952, 3705
COMMENTS
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 most divisors. If two or more adjacent squares exist with the same highest number of divisors then the square with the lowest spiral number is chosen. Note that if the king simply moves to the highest available number the sequence will be infinite as the king will step along the south-east diagonal from square 1 forever.
The sequence is finite. After 1784 steps the square with number 1478 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 1784 visited squares only 27 contain prime numbers while 1757 contain composites. 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 1784 visited squares 1289 contain an even number while 495 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 numerous diagonal steps is succession - see the attached link image.
The largest visited square is a(390) = 17664. The lowest unvisited square is 2.
LINKS
Scott R. Shannon, Image showing the 1784 steps of the king's path. A green dot marks the starting 1 square and a red dot the final square with number 1478. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dots marks the smallest unvisited square with number 2.
EXAMPLE
The board is numbered with the square spiral:
.
17--16--15--14--13 .
| | .
18 5---4---3 12 29
| | | | |
19 6 1---2 11 28
| | | |
20 7---8---9--10 27
| |
21--22--23--24--25--26
.
a(1) = 1, the starting square for the king.
a(2) = 6. The eight unvisited squares around a(1) the king can move to are numbered 2,3,4,5,6,7,8,9. Of these 6 and 8 both have the maximum four divisors, and of those 6 is the smallest.
a(3) = 18. The seven unvisited squares around a(2) = 6 the king can move to are numbered 4,5,18,19,20,7,8. Of these 18 and 20 have the maximum six divisors, and of those 18 is the smallest.
a(603) = 821. This is the first prime number visited; a(602) = 939 has square 821 as the sole unvisited adjacent neighbor.
Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the largest spiral number.
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5
1, 7, 23, 47, 79, 49, 25, 9, 11, 29, 53, 87, 127, 177, 233, 299, 373, 454, 543, 641, 746, 859, 979, 1109, 1247, 1393, 1249, 1111, 983, 863, 751, 647, 753, 866, 865, 985, 1115, 1253, 1399, 1553, 1714, 1883, 2059, 2243, 2437, 2638, 2846, 3063, 3287, 3061, 2843, 2633, 2841, 3057, 3281, 3513, 3755
COMMENTS
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 largest spiral number is chosen. Note that if the king simply moves to the largest available number the sequence will be infinite as the king will step along the south-east diagonal from square 1 forever.
The sequence is finite. After 21276 steps the square with spiral number 281747427 is visited, after which all adjacent neighboring squares have been visited. The end square is extremely far from the starting square, approximately 8860 units away, as the king is drawn generally outward due to its preference for the largest numbered square when the divisor counts are tied - see the link image. This end square spiral number is currently the largest for any square spiral single-visit trapped knight or trapped king path in the OEIS.
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 21276 visited squares 4363 contain prime numbers while 16913 contain composites. The largest visited square is a(21208) = 282486458.
LINKS
Scott R. Shannon, Image showing the 21276 steps of the king's path. A green dot, far lowest left, marks the starting 1 square and a red dot, far upper right, the final square with number 281747427. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dots marks the smallest unvisited square with number 2. This is a high resolution image and may need to be downloaded to be viewed correctly.
EXAMPLE
The board is numbered with the square spiral:
.
17--16--15--14--13 .
| | .
18 5---4---3 12 29
| | | | |
19 6 1---2 11 28
| | | |
20 7---8---9--10 27
| |
21--22--23--24--25--26
.
a(1) = 1, the starting square for the king.
a(2) = 7. The eight unvisited squares around a(1) the king can move to are numbered 2,3,4,5,6,7,8,9. Of these 2,3,5,7 have the minimum two divisors, and of those 7 is the largest.
a(3) = 23. The seven unvisited squares around a(2) the king can move to are numbered 6,8,19,20,21,22,23. Of these 19 and 23 have the minimum two divisors, and of those 23 is the largest.
Squares visited by a chess king on a square-spiral numbered board and stepping to the lowest unvisited adjacent square, where each step is not in the same direction as the previous step.
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4
1, 2, 3, 4, 6, 5, 15, 14, 12, 11, 9, 8, 22, 7, 19, 18, 16, 17, 35, 34, 60, 32, 13, 29, 28, 10, 25, 24, 46, 23, 45, 21, 20, 40, 39, 67, 37, 36, 38, 66, 64, 63, 97, 61, 62, 96, 95, 59, 33
COMMENTS
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, which is not in the same direction as its previous step, moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the lowest spiral number.
The sequence is finite. After 48 steps the square with spiral number 33 is reached after which all eight adjacent squares have been visited.
If the king simply moved to the lowest numbered unvisited adjacent square the walk would be infinite as the king would just follow the path of the square spiral. By not allowing consecutive moves in the same direction forces the king off this minimal numbered path. The first time this happens is a(5) = 6 as from a(4) = 4 the lowest numbered adjacent square is 5 but that would require a step directly to the left, the same as the previous step from a(3) = 3 to a(4).
LINKS
Scott R. Shannon, Image showing the 48 steps of the king's path. The green dot is the first square with number 1 and the red dot the last square with number 33. The red dot is surrounded by blue dots to show the eight occupied squares. The yellow dots marks the smallest unvisited square with number 26.
EXAMPLE
The board is numbered with the square spiral:
.
17--16--15--14--13 .
| | .
18 5---4---3 12 29
| | | | |
19 6 1---2 11 28
| | | |
20 7---8---9--10 27
| |
21--22--23--24--25--26
.
a(1) = 1, the starting square of the king.
a(2) = 2. The eight adjacent unvisited squares around a(1) are numbered 2,3,4,5,6,7,8,9. Of these 2 is the lowest.
a(5) = 6. The five adjacent unvisited squares around a(4) = 4 are numbered 5,6,14,15,16. Of these 5 is the lowest but that would require a step directly left from 4, which is the same step as a(3) = 3 to a(4) = 4, so is not allowed. The next lowest available square is 6.
Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the lowest prime number. In no adjacent prime numbered squares are present it chooses the lowest spiral number.
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3
1, 2, 3, 11, 29, 13, 31, 59, 32, 14, 4, 5, 17, 37, 67, 103, 149, 104, 66, 38, 18, 19, 7, 23, 47, 79, 48, 24, 8, 6, 20, 41, 71, 43, 73, 109, 72, 42, 21, 22, 44, 45, 46, 76, 75, 113, 74, 112, 110, 111, 157, 211, 271, 209, 269, 337, 267, 205, 151, 107, 69, 39, 40
COMMENTS
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 the adjacent unvisited square containing the lowest prime number. If no adjacent unvisited square contains a prime number then the square with the lowest spiral number is chosen. Note that if the king simply moves to the lowest unvisited number the sequence will be infinite as the king will just follow the square spiral path.
The sequence is finite. After 719 steps the square with number 437 is visited, after which all adjacent neighboring squares have been visited.
Of the 719 visited squares 165 contain prime numbers while 554 contain composites. As the odd numbers are diagonally adjacent in the square spiral the king's path will contain many diagonal steps, often taking numerous diagonal steps is succession - see the attached link image.
The largest visited square is a(709) = 1367. The lowest unvisited square is 33.
The 719 steps until self-trapping occurs are significantly larger than the expected average of 210 moves to self-trapping for a random walk of the king on an infinite chessboard. See the link to the probability density graphs in A323562. - Hugo Pfoertner, Jul 19 2020
LINKS
Scott R. Shannon, Image showing the 719 steps of the path. A green dot marks the starting 1 square and a red dot the final square with number 437. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dots marks the smallest unvisited square with number 33.
EXAMPLE
The board is numbered with the square spiral:
.
17--16--15--14--13 .
| | .
18 5---4---3 12 29
| | | | |
19 6 1---2 11 28
| | | |
20 7---8---9--10 27
| |
21--22--23--24--25--26
.
a(1) = 1, the starting square for the king.
a(2) = 2. The four unvisited squares around a(1) the king can move which contain prime numbers are 2,3,5,7. Of those 2 is the lowest.
a(4) = 11. The two unvisited squares around a(3) = 3 the king can move to which contain prime numbers are 11 and 13. Of those 11 is the lowest.
a(9) = 32. There are no unvisited squares around a(8) = 59 which contain prime numbers. The seven other unvisited squares are numbered 32,33,58,60,93,94,95. Of those 32 is the lowest.
Squares visited by a fairy chess wazir moving on a square-spiral numbered board where the wazir moves to the unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the lowest spiral number.
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1
1, 2, 3, 4, 5, 6, 7, 8, 23, 22, 21, 20, 19, 18, 17, 38, 37, 64, 65, 66, 67, 68, 39, 40, 41, 42, 43, 74, 73, 110, 109, 154, 155, 208, 269, 268, 337, 338, 339, 340, 271, 272, 211, 274, 275, 346, 347, 426, 427, 514, 515, 428, 349, 278, 277, 214, 159, 158, 157, 212, 213, 276
COMMENTS
A fairy chess wazir can move one step in each of the four orthogonal grid directions, i.e., the same directions as a chess rook but only one square. In this sequence the wazir moves to the closest unvisited neighboring square which contains the number with the fewest divisors, and in case of a tie the square with the lowest spiral number. Note that if the wazir simply moves to the lowest available number the sequence will be infinite as the wazir will just follow the square spiral path.
The sequence is finite. After 61 steps the square with number 276 is visited, after which all four neighboring squares have been visited.
Due to the wazir'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 neighboring squares. Of the 61 visited squares, 21 contain prime numbers, while 40 contain composites. The largest visited square is a(51) = 515.
LINKS
Scott R. Shannon, Image showing the 61 steps of the wazir's path. A green dot marks the starting 1 square and a red dot the final square with number 276. The red dot is surrounded by four blue dots to show the unavailable neighboring squares. A yellow dot marks the smallest unvisited square with number 9.
EXAMPLE
The board is numbered with the square spiral:
.
17--16--15--14--13 .
| | .
18 5---4---3 12 29
| | | | |
19 6 1---2 11 28
| | | |
20 7---8---9--10 27
| |
21--22--23--24--25--26
.
a(1) = 1, the starting square for the wazir.
a(2) = 2. The four unvisited squares around a(1) to which the wazir can move are numbered 2,4,6,8. Of these, 2 has only two divisors, so it is the square chosen.
a(9) = 23. The two unvisited squares around a(8) = 8 to which the wazir can move are numbered 9 and 23. Of these, 23 has only two divisors, so it is the square chosen.
Triangle read by columns where T(n,k) is the number of points in Z^n such that |x1| + ... + |xn| = k, |x1|, ..., |xn| > 0.
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0
2, 2, 4, 2, 8, 8, 2, 12, 24, 16, 2, 16, 48, 64, 32, 2, 20, 80, 160, 160, 64, 2, 24, 120, 320, 480, 384, 128, 2, 28, 168, 560, 1120, 1344, 896, 256, 2, 32, 224, 896, 2240, 3584, 3584, 2048, 512, 2, 36, 288, 1344, 4032, 8064, 10752, 9216, 4608, 1024
COMMENTS
T(n,k) is the number of points on the n-dimensional cross polytope with facets at distance k from the origin which have no coordinate equal to 0.
T(n,n) = 2^n. The (n-1)-dimensional simplex at distance n from the origin in Z^n has exactly 1 point with no zero coordinates, at (1,1,...,1). There are 2^n (n-1)-dimensional simplexes at distance n from the origin as part of the cross polytope in Z^n. (The lower dimensional facets do not count as they have at least one 0 coordinate.)
T(2*n,3*n) = T(2*n+1,3*n), and this is A036909.
FORMULA
T(n,k) = binomial(k-1,n-1)*2^n.
EXAMPLE
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-----------------------------------------------------------------------
1 | 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 | 4 8 12 16 20 24 28 32 36 40 44 48 52 56
3 | 8 24 48 80 120 168 224 288 360 440 528 624 728
4 | 16 64 160 320 560 896 1344 1920 2640 3520 4576 5824
5 | 32 160 480 1120 2240 4032 6720 10560 15840 22880 32032
6 | 64 384 1344 3584 8064 16128 29568 50688 82368 128128
7 | 128 896 3584 10752 26880 59136 118272 219648 384384
8 | 256 2048 9216 30720 84480 202752 439296 878592
9 | 512 4608 23040 84480 253440 658944 1537536
10 | 1024 10240 56320 225280 732160 2050048
11 | 2048 22528 135168 585728 2050048
12 | 4096 49152 319488 1490944
13 | 8192 106496 745472
14 | 16384 229376
15 | 32768
The cross polytope in Z^3 (the octahedron) with points at distance 3 from the origin has 8 triangle facets, each with edge length 4. There is one point in the center of each triangle with coordinates (+-1,+-1,+-1).
MATHEMATICA
T[n_, k_]:=Binomial[k-1, n-1]*2^n; Table[T[n, k], {k, 10}, {n, k}]//Flatten
PROG
(Python)
from math import comb
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