A090036 -id:A090036 - cp's OEIS frontend

A089473 Number of configurations of the sliding block 8-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

1, 2, 4, 8, 16, 20, 39, 62, 116, 152, 286, 396, 748, 1024, 1893, 2512, 4485, 5638, 9529, 10878, 16993, 17110, 23952, 20224, 24047, 15578, 14560, 6274, 3910, 760, 221, 2

Offset: 0

Hugo Pfoertner, Nov 19 2003

The sequence was first provided by Alexander Reinefeld in Table 1 of "Complete Solution of the Eight-Puzzle..." (see corresponding link in A087725) with a typo "749" instead of "748" for a(12).

			From the starting configuration
123
456
78-
the two final configurations requiring 31 moves are
647 ... 867
85- and 254
321 ... 3-1

For references and links see A087725.

Hugo Pfoertner, FORTRAN program to solve small sliding block puzzles.
Hugo Pfoertner, Table of solution lengths of the 8-puzzle
I. Kapustik, Cesta.
J. Knuuttila, S. Broman and P. Ranta, n-Puzzle, in Finnish (2002).
A. Reinefeld, Complete Solution of the Eight-Puzzle and the Benefit of Node Ordering in IDA*. Proc. 13th Int. Joint Conf. on Artificial Intelligence (1993), Chambery Savoi, France, pp. 248-253.
Takaken, n-Puzzle Page.
Takaken, No. 33 (8 puzzles).

Cf. A087725 = maximum number of moves for n X n puzzle, A089474 = 8-puzzle starting with blank square at center, A089483 = 8-puzzle starting with blank square at mid-side, A089484 = solution lengths for 15-puzzle, A090031 - A090036 = other sliding block puzzles.

# alst(), swap(), moves() useful for other sliding puzzle problems
def swap(p, z, nz):
  lp = list(p)
  lp[z], lp[nz] = lp[nz], "-"
  return "".join(lp)
def moves(p, shape): # moves for n x m sliding puzzle
  nxt, (n, m), z = [], shape, p.find("-") # z: blank location
  if z > n - 1:  nxt.append(swap(p, z, z-n)) # blank up
  if z < n*m-n:  nxt.append(swap(p, z, z+n)) # blank down
  if z%n != 0:   nxt.append(swap(p, z, z-1)) # blank left
  if z%n != n-1: nxt.append(swap(p, z, z+1)) # blank right
  return nxt
def alst(start, shape, v=False, maxd=float('inf')):
  alst, d, expanded, frontier = [], 0, set(), {start}
  alst.append(len(frontier))
  if v: print(len(frontier), end=", ")
  while len(frontier) > 0 and d < maxd:
    reach1 = set(m for p in frontier for m in moves(p, shape) if m not in expanded)
    expanded |= frontier # expanded = frontier # ALTERNATE using less memory
    if len(reach1):
      alst.append(len(reach1))
      if v: print(len(reach1), end=", ")
    frontier = reach1
    d += 1
  return alst
print(alst("-12345678", (3, 3))) # Michael S. Branicky, Dec 28 2020

A090033 Triangle T(j,k) read by rows, where T(j,k) is the number of single tile moves in the longest optimal solution of the j X k generalization of the sliding block 15-puzzle, starting with the empty square in a corner.

Original entry on oeis.org

0, 1, 6, 2, 21, 31, 3, 36, 53, 80, 4, 55, 84

Offset: 1

Hugo Pfoertner, Nov 23 2003

T(k,j) = T(j,k).

T(2,2), T(2,3), T(4,2), T(4,3) from Karlemo and Östergård, T(3,3) from Reinefeld, T(4,4) from Bruengger et al.

			The triangle begins
  0
  1   6
  2  21  31
  3  36  53  80
  4  55  84  ...
.
a(6)=T(3,3)=31 because the A090163(3,3)=2 longest optimal solution paths of the 3 X 3 (9-) sliding block puzzle have length 31 (see A089473).

For references and links see A087725(n)=T(n,n).

Cf. A087725, A089473, A089484, A090034, A090035, A090036, A090166, A090163 corresponding number of different configurations with largest distance.

Cf. A151944 same as this sequence, but written as full array.

# alst(), moves(), swap() in A089473
def T(j, k):  # chr(45) is '-'
    start, shape = "".join(chr(45+i) for i in range(j*k)), (j, k)
    return len(alst(start, shape))-1
for j in range(1, 5):
    for k in range(1, j+1):
        print(T(j,k), end=", ") # Michael S. Branicky, Aug 02 2021

T(5,3) copied from A151944 by Hugo Pfoertner, Aug 02 2021

A090034 Number of configurations of the 3 X 2 variant of Sam Loyd's sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 12, 12, 16, 23, 25, 28, 39, 44, 40, 29, 21, 18, 12, 6, 1

Offset: 0

Hugo Pfoertner, Nov 23 2003

Data from Karlemo and Östergård. See corresponding link in A087725.

			Starting with
123
45-
the most distant configuration corresponding to a(21)=1 is
45-
123 (i.e., it takes longest just to swap the two rows).

See A087725.

Hugo Pfoertner, Solutions of small n*2 sliding block puzzles.
Takaken, n-Puzzle Page.
Takaken, No. 32 (5 puzzles).

Cf. A087725, A089473, A090033, A090035, A090036, A090163, A090167.

# alst(), moves(), swap() in A089473
print(alst("-12345", (3, 2))) # Michael S. Branicky, Dec 30 2020

A090035 Number of configurations of the 4 X 2 variant of Sam Loyd's sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

Original entry on oeis.org

1, 2, 3, 6, 10, 14, 19, 28, 42, 61, 85, 119, 161, 215, 293, 396, 506, 632, 788, 985, 1194, 1414, 1664, 1884, 1999, 1958, 1770, 1463, 1076, 667, 361, 190, 88, 39, 19, 7, 1

Offset: 0

Hugo Pfoertner, Nov 26 2003

Data from Karlemo, Östergård. See corresponding link in A087725.

			Starting from
1234
567-
the most distant configuration corresponding to a(36)=1 is
-721
4365

See A087725.

Hugo Pfoertner, Solutions of small n*2 sliding block puzzles.
Takaken, n-Puzzle Page.
Takaken, No. 42 (7 puzzles).

Cf. A087725, A089473. Index of last sequence term: A090033. Other nonsquare sliding block puzzles: A090034, A090036, A090166, A090167.

# alst(), moves(), swap() in A089473
print(alst("-1234567", (4, 2))) # Michael S. Branicky, Dec 30 2020

A090167 Number of configurations of the 6 X 2 variant of the so-called "Sam Loyd" sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 36, 60, 95, 155, 258, 426, 688, 1106, 1723, 2615, 3901, 5885, 8851, 13205, 19508, 28593, 41179, 58899, 83582, 118109, 165136, 228596, 312542, 423797, 568233, 755727, 994641, 1296097, 1667002, 2119476, 2660415, 3300586, 4038877

Offset: 0

Hugo Pfoertner, Nov 27 2003

Herman Jamke, Table of n, a(n) for n = 0..80 (complete sequence)
Hugo Pfoertner, Solutions of small n*2 sliding block puzzles.
Takaken, 11-Puzzle Page.
Takaken, No. 62 (11 puzzles).

Cf. A087725, A089473. Index of last sequence term: A090033. Other nonsquare sliding block puzzles: A090034, A090035, A090036, A090166.

# alst(), moves(), swap() in A089473
alst("-123456789AB", (6, 2), v=True) # Michael S. Branicky, Dec 31 2020

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 22 2006

A090166 Number of configurations of the 4 X 3 variant of Sam Loyd's sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

Original entry on oeis.org

1, 2, 4, 9, 20, 37, 63, 122, 232, 431, 781, 1392, 2494, 4442, 7854, 13899, 24215, 41802, 71167, 119888, 198363, 323206, 515778, 811000, 1248011, 1885279, 2782396, 4009722, 5621354, 7647872, 10065800, 12760413, 15570786, 18171606, 20299876, 21587248, 21841159, 20906905, 18899357, 16058335, 12772603, 9515217, 6583181, 4242753, 2503873, 1350268, 643245, 270303, 92311, 27116, 5390, 1115, 86, 18

Offset: 0

Hugo Pfoertner, Nov 27 2003, Jul 07 2007

Data from Karlemo and Östergård.

Filip R. W. Karlemo and Patric R. J. Östergård, On Sliding Block Puzzles, J. Combin. Math. Combin. Comp. 34 (2000), 97-107.
Takaken, 11-Puzzle Page.
Takaken, No. 43 (11 puzzles).

Cf. A087725. Index of last sequence term: A090033. Other nonsquare sliding block puzzles: A090034, A090035, A090036, A090167.

# alst(), moves(), swap() in A089473
alst("-123456789AB", (4, 3), v=True) # Michael S. Branicky, Dec 31 2020

A346736 Number of configurations of the 7 X 2 variant of the sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 37, 67, 117, 198, 329, 557, 942, 1575, 2597, 4241, 6724, 10535, 16396, 25515, 39362, 60532, 92089, 138969, 207274, 307725, 453000, 664240, 964874, 1392975, 1992353, 2832063, 3988528, 5586275, 7756511, 10698721, 14621717, 19840724, 26676629

Offset: 0

Ben Whitmore, Jul 31 2021

This sequence was originally computed by Richard Korf, but the full sequence was not included in his paper. It was later re-computed by Tomas Rokicki.

			Starting from the solved configuration
  1  2  3  4  5  6  7
  8  9 10 11 12 13
the unique configuration requiring 108 moves is
  7  6 12  4  3  9  1
    13  5 11 10  2  8

Ben Whitmore, Table of n, a(n) for n = 0..108
Richard Korf, Linear-time Disk-Based Implicit Graph Search, Journal of the ACM 55 (2008), No. 6.
Tomas Rokicki, Twenty-Four puzzle, some observations, 2011.

Cf. A090033, A090034, A090035, A090036, A090167, A089473, A089484.

# alst(), moves(), swap() in A089473
print(alst("-123456789abcd", (7, 2), v=True)) # Michael S. Branicky, Jul 31 2021

A346737 Number of configurations of the 5 X 3 variant of the sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

Original entry on oeis.org

1, 2, 4, 9, 21, 42, 89, 164, 349, 644, 1349, 2473, 5109, 9110, 18489, 32321, 64962, 112445, 223153, 378761, 740095, 1231589, 2364342, 3847629, 7246578, 11506172, 21233764, 32854049, 59293970, 89146163, 157015152, 228894783, 392648931, 553489877, 922382155

Offset: 0

Ben Whitmore, Jul 31 2021

This sequence was originally computed by Richard Korf, but the full sequence was not included in his paper. It was later re-computed by Tomas Rokicki.

			Starting from the solved configuration
   1  2  3  4  5
   6  7  8  9 10
  11 12 13 14
the unique configuration requiring 84 moves is
   5  4  3  2  1
  10  9  8  7  6
     14 13 12 11

Ben Whitmore, Table of n, a(n) for n = 0..84
Richard Korf, Linear-time Disk-Based Implicit Graph Search, Journal of the ACM 55 (2008), No. 6.
Tomas Rokicki, Twenty-Four puzzle, some observations, 2011.

Cf. A090033, A090034, A090035, A090036, A090167, A346736, A090166, A089473, A089484.

# alst(), moves(), swap() in A089473
print(alst("-123456789abcde", (5, 3), v=True)) # Michael S. Branicky, Jul 31 2021

A355560 Number of configurations of the 8 X 2 variant of the sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 37, 68, 125, 227, 394, 672, 1151, 1983, 3373, 5703, 9508, 15640, 25293, 40732, 65032, 103390, 162830, 255543, 397013, 613104, 938477, 1431068, 2162964, 3255845, 4860428, 7223861, 10649867, 15628073, 22747718, 32963838, 47397514, 67825949, 96317070

Offset: 0

Ben Whitmore, Jul 06 2022

This sequence was computed by Richard Korf in "Linear-time Disk-Based Implicit Graph Search" (see links), but was not included in the paper.

			Starting from the solved configuration
   1  2  3  4  5  6  7  8
   9 10 11 12 13 14 15
the unique configuration requiring 140 moves is
      8  6  5  4  3 10  1
  15  7 14 13 12 11  2  9

Ben Whitmore, Table of n, a(n) for n = 0..140
Richard Korf, Linear-time Disk-Based Implicit Graph Search, Journal of the ACM 55 (2008), No. 6.

Cf. A090033, A090034, A090035, A090036, A090167, A346736.

# alst(), moves(), swap() in A089473
print(alst("-123456789abcdef", (8, 2), v=True)) # Michael S. Branicky, Jul 06 2022

cp's OEIS Frontend

A089473 Number of configurations of the sliding block 8-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

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A090033 Triangle T(j,k) read by rows, where T(j,k) is the number of single tile moves in the longest optimal solution of the j X k generalization of the sliding block 15-puzzle, starting with the empty square in a corner.

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A090034 Number of configurations of the 3 X 2 variant of Sam Loyd's sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

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A090035 Number of configurations of the 4 X 2 variant of Sam Loyd's sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

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A090167 Number of configurations of the 6 X 2 variant of the so-called "Sam Loyd" sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

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A090166 Number of configurations of the 4 X 3 variant of Sam Loyd's sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

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A346736 Number of configurations of the 7 X 2 variant of the sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

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A346737 Number of configurations of the 5 X 3 variant of the sliding block 15-puzzle that require a minimum of n moves to be reached, starting with the empty square in one of the corners.

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