A090573 Number of configurations of the 3-dimensional 3 X 3 X 3 sliding cube puzzle that require a minimum of n moves to be reached, starting with the empty space at one of the enclosing cube corners.

1, 3, 9, 30, 102, 324, 1029, 3096, 9960, 30732, 97932, 295924, 929202, 2766958, 8590731, 25271897, 77575820

Offset

0,2

Comments

The 3 X 3 X 3 sliding cube puzzle is a generalization of the 2 X 2 X 2 3-D puzzle A090572. There are various designs of this 3 X 3 X 3 puzzle, but in most of them the central cube is not freely accessible and the empty space cannot be moved to the central location of the cube. The design treated in this sequence assumes full accessibility of the central location.

References

Ji-Ha You, Three dimensional puzzle with blocks - has cube frame with hollow interior and movable blocks filled on one side with magnet. German Patent application DE-4106826 A1, March 4, 1991 - see link

Links

Table of n, a(n) for n=0..16.

Ji-Ha You, Dreidimensionales Spielgeraet. Offenlegungsschrift DE-4106826 A1.

Prog

(Python) # uses alst(), swap() in A089473
def moves3d(p, shape, fixed=None): # n x m x r puzzle
  nxt, (n, m, r) = [], shape
  L, z = n*m, p.find("-") # z: location of blank
  zq, zr = divmod(z, L)
  if zr > n - 1:  nxt.append(swap(p, z, z-n)) # blank up
  if zr < n*m-n:  nxt.append(swap(p, z, z+n)) # blank down
  if zr%n != 0:   nxt.append(swap(p, z, z-1)) # blank left
  if zr%n != n-1: nxt.append(swap(p, z, z+1)) # blank right
  if zq > 0:      nxt.append(swap(p, z, z-L)) # blank in
  if zq < r - 1:  nxt.append(swap(p, z, z+L)) # blank out
  return [m for m in nxt if m.find("-") != fixed]
moves = lambda p, shape: moves3d(p, shape)
start, shape = "-123456789ABCDEFGHIJKLMNOPQ", (3, 3, 3)
print(alst(start, shape, maxd=12)) # Michael S. Branicky, Dec 28 2020

Crossrefs

Cf. A090572 2 X 2 X 2 puzzle, A090574, A090575, A090576 3 X 3 X 3 puzzle with different initial configurations.

Sequence in context: A148946 A096222 A148947 * A228339 A257641 A048119

Adjacent sequences: A090570 A090571 A090572 * A090574 A090575 A090576

Keyword

fini,hard,more,nonn

Author

Hugo Pfoertner, Jan 15 2004

Extensions

a(13)-a(16) from Michael S. Branicky, Dec 28 2020

Status

approved