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r""" Rubik's cube group functions
.. _sec-rubik:
.. NOTE::
"Rubiks cube" is trademarked. We shall omit the trademark symbol below for simplicity.
NOTATION:
`B` denotes a clockwise quarter turn of the back face, `D` denotes a clockwise quarter turn of the down face, and similarly for `F` (front), `L` (left), `R` (right), and `U` (up). Products of moves are read right to left, so for example, `R \cdot U` means move `U` first and then `R`.
See ``CubeGroup.parse()`` for all possible input notations.
The "Singmaster notation":
- moves: `U, D, R, L, F, B` as in the diagram below,
- corners: `xyz` means the facet is on face `x` (in `R,F,L,U,D,B`) and the clockwise rotation of the corner sends `x-y-z`
- edges: `xy` means the facet is on face `x` and a flip of the edge sends `x-y`.
::
sage: rubik = CubeGroup() sage: rubik.display2d("") +--------------+ | 1 2 3 | | 4 top 5 | | 6 7 8 | +------------+--------------+-------------+------------+ | 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 | | 12 left 13 | 20 front 21 | 28 right 29 | 36 rear 37 | | 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 | +------------+--------------+-------------+------------+ | 41 42 43 | | 44 bottom 45 | | 46 47 48 | +--------------+
AUTHORS:
- David Joyner (2006-10-21): first version
- David Joyner (2007-05): changed faces, added legal and solve
- David Joyner(2007-06): added plotting functions
- David Joyner (2007, 2008): colors corrected, "solve" rewritten (again),typos fixed.
- Robert Miller (2007, 2008): editing, cleaned up display2d
- Robert Bradshaw (2007, 2008): RubiksCube object, 3d plotting.
- David Joyner (2007-09): rewrote docstring for CubeGroup's "solve".
- Robert Bradshaw (2007-09): Versatile parse function for all input types.
- Robert Bradshaw (2007-11): Cleanup.
REFERENCES:
- Cameron, P., Permutation Groups. New York: Cambridge University Press, 1999.
- Wielandt, H., Finite Permutation Groups. New York: Academic Press, 1964.
- Dixon, J. and Mortimer, B., Permutation Groups, Springer-Verlag, Berlin/New York, 1996.
- Joyner,D., Adventures in Group Theory, Johns Hopkins Univ Press, 2002. """
#************************************************************************************** # Copyright (C) 2006 David Joyner <wdjoyner@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #************************************************************************************** from __future__ import print_function from six.moves import range
from sage.groups.perm_gps.permgroup import PermutationGroup, PermutationGroup_generic import random
from sage.structure.sage_object import SageObject from sage.structure.richcmp import richcmp, richcmp_method
from sage.rings.all import RationalField, Integer, RDF #from sage.matrix.all import MatrixSpace from sage.interfaces.all import gap from sage.groups.perm_gps.permgroup_element import PermutationGroupElement from sage.plot.polygon import polygon from sage.plot.text import text pi = RDF.pi()
from sage.plot.plot3d.shapes import Box from sage.plot.plot3d.texture import Texture
####################### predefined colors ##################
named_colors = { 'red': (1,0,0), ## F face 'green': (0,1,0), ## R face 'blue': (0,0,1), ## D face 'yellow': (1,1,0), ## L face 'white': (1,1,1), ## none 'orange': (1,0.6,0.3), ## B face 'purple': (1,0,1), ## none 'lpurple': (1,0.63,1), ## U face 'lightblue': (0,1,1), ## none 'lgrey': (0.75,0.75,0.75), ## sagemath.org color } globals().update(named_colors)
######################################################### #written by Tom Boothby, placed in the public domain
def xproj(x,y,z,r): r""" Return the `x`-projection of `(x,y,z)` rotated by `r`.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import rotation_list, xproj sage: rot = rotation_list(30, 45) sage: xproj(1,2,3,rot) 0.6123724356957945 """
def yproj(x,y,z,r): r""" Return the `y`-projection of `(x,y,z)` rotated by `r`.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import rotation_list, yproj sage: rot = rotation_list(30, 45) sage: yproj(1,2,3,rot) 1.378497416975604 """
def rotation_list(tilt,turn): r""" Return a list `[\sin(\theta), \sin(\phi), \cos(\theta), \cos(\phi)]` of rotations where `\theta` is ``tilt`` and `\phi` is ``turn``.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import rotation_list sage: rotation_list(30, 45) [0.49999999999999994, 0.7071067811865475, 0.8660254037844387, 0.7071067811865476] """
def polygon_plot3d(points, tilt=30, turn=30, **kwargs): r""" Plot a polygon viewed from an angle determined by ``tilt``, ``turn``, and vertices ``points``.
.. WARNING::
The ordering of the points is important to get "correct" and if you add several of these plots together, the one added first is also drawn first (ie, addition of Graphics objects is not commutative).
The following example produced a green-colored square with vertices at the points indicated.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import polygon_plot3d,green sage: P = polygon_plot3d([[1,3,1],[2,3,1],[2,3,2],[1,3,2],[1,3,1]],rgbcolor=green) """
###########################################################
############# lots of "internal" utility plot functions #########
def inv_list(lst): r""" Input a list of ints `1, \ldots, m` (in any order), outputs inverse perm.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import inv_list sage: L = [2,3,1] sage: inv_list(L) [3, 1, 2] """
face_polys = { ### bottom layer L, F, R, B 'ldb': [[-3,0],[-2,0], [-2,1], [-3,1]], #square labeled 14 'ld': [[-2,0],[-1,0], [-1,1], [-2,1]], #square labeled 15 'lfd': [[-1,0],[0,0], [0,1], [-1,1]], #square labeled 16 'fdl': [[0,0],[1,0], [1,1], [0,1]], #square labeled 22 'fd': [[1,0],[2,0], [2,1], [1,1]], #square labeled 23 'frd': [[2,0],[3,0], [3,1], [2,1]], #square labeled 24 'rdf': [[3,0],[4,0], [4,1], [3,1]], #square labeled 30 'rd': [[4,0],[5,0], [5,1], [4,1]], #square labeled 31 'rbd': [[5,0],[6,0], [6,1], [5,1]], #square labeled 32 'bdr': [[6,0],[7,0], [7,1], [6,1]], #square labeled 38 'bd': [[7,0],[8,0], [8,1], [7,1]], #square labeled 39 'bld': [[8,0],[9,0], [9,1], [8,1]], #square labeled 40 ### middle layer L,F,R, B 'lb': [[-3,1],[-2,1], [-2,2], [-3,2]], #square labeled 12 'l_center': [[-2,1],[-1,1], [-1,2], [-2,2]], #center square 'lf': [[-1,1],[0,1], [0,2], [-1,2]], #square labeled 13 'fl': [[0,1],[1,1], [1,2], [0,2]], #square labeled 20 'f_center': [[1,1],[2,1], [2,2], [1,2]], #center square 'fr': [[2,1],[3,1], [3,2], [2,2]], #square labeled 21 'rf': [[3,1],[4,1], [4,2], [3,2]], #square labeled 28 'r_center': [[4,1],[5,1], [5,2], [4,2]], #center square 'rb': [[5,1],[6,1], [6,2], [5,2]], #square labeled 29 'br': [[6,1],[7,1], [7,2], [6,2]], #square labeled 36 'b_center': [[7,1],[8,1], [8,2], [7,2]], #center square 'bl': [[8,1],[9,1], [9,2], [8,2]], #square labeled 37 ## top layer L, F, R, B 'lbu': [[-3,2],[-2,2], [-2,3], [-3,3]], #square labeled 9 'lu': [[-2,2],[-1,2], [-1,3], [-2,3]], #square labeled 10 'luf': [[-1,2],[0,2], [0,3], [-1,3]], #square labeled 11 'flu': [[0,2],[1,2], [1,3], [0,3]], #square labeled 17 'fu': [[1,2],[2,2], [2,3], [1,3]], #square labeled 18 'fur': [[2,2],[3,2], [3,3], [2,3]], #square labeled 19 'ruf': [[3,2],[4,2], [4,3], [3,3]], #square labeled 25 'ru': [[4,2],[5,2], [5,3], [4,3]], #square labeled 26 'rub': [[5,2],[6,2], [6,3], [5,3]], #square labeled 27 'bur': [[6,2],[7,2], [7,3], [6,3]], #square labeled 33 'bu': [[7,2],[8,2], [8,3], [7,3]], #square labeled 34 'bul': [[8,2],[9,2], [9,3], [8,3]], #square labeled 35 # down face 'dlf': [[0,-1],[1,-1], [1,0], [0,0]], #square labeled 41 'df': [[1,-1],[2,-1], [2,0], [1,0]], #square labeled 42 'dfr': [[2,-1],[3,-1], [3,0], [2,0]], #square labeled 43 'dl': [[0,-2],[1,-2], [1,-1], [0,-1]], #square labeled 44 'd_center': [[1,-2],[2,-2], [2,-1], [1,-1]], #center square 'dr': [[2,-2],[3,-2], [3,-1], [2,-1]], #square labeled 45 'dlb': [[0,-3],[1,-3], [1,-2], [0,-2]], #square labeled 46 'db': [[1,-3],[2,-3], [2,-2], [1,-2]], #square labeled 47 'drb': [[2,-3],[3,-3], [3,-2], [2,-2]], #square labeled 48 # up face 'ufl': [[0,3],[1,3], [1,4], [0,4]], #square labeled 6 'uf': [[1,3],[2,3], [2,4], [1,4]], #square labeled 7 'urf': [[2,3],[3,3], [3,4], [2,4]], #square labeled 8 'ul': [[0,4],[1,4], [1,5], [0,5]], #square labeled 4 'u_center': [[1,4],[2,4], [2,5], [1,5]], #center square 'ur': [[2,4],[3,4], [3,5], [2,5]], #square labeled 5 'ulb': [[0,6],[1,6], [1,5], [0,5]], #square labeled 1 'ub': [[1,6],[2,6], [2,5], [1,5]], #square labeled 2 'ubr': [[2,6],[3,6], [3,5], [2,5]], #square labeled 3 }
def create_poly(face, color): """ Create the polygon given by ``face`` with color ``color``.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import create_poly, red sage: create_poly('ur', red) Graphics object consisting of 1 graphics primitive """
####################################################
singmaster_indices = { 1: "ulb", 2: "ub", 3: "ubr", 4: "ul", 5: "ur", 6: "ufl", 7: "uf", 8: "urf", 14: "ldb", 15: "ld", 16: "lfd", 12: "lb", 13: "lf", 9: "lbu", 10: "lu", 11: "luf", 17: "flu", 18: "fu", 19: "fur", 20: "fl", 21: "fr", 22: "fdl", 23: "fd", 24: "frd", 41: "dlf", 42: "df", 43: "dfr", 44: "dl", 45: "dr", 46: "dlb", 47: "db", 48: "drb", 33: "bur", 34: "bu", 35: "bul", 36: "br", 37: "bl", 38: "bdr", 39: "bd", 40: "bld", 25: "ruf", 26: "ru", 27: "rub", 28: "rf", 29: "rb", 30: "rdf", 31: "rd", 32: "rbd", }
def index2singmaster(facet): """ Translate index used (eg, 43) to Singmaster facet notation (eg, fdr).
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import index2singmaster sage: index2singmaster(41) 'dlf' """
def color_of_square(facet, colors=['lpurple', 'yellow', 'red', 'green', 'orange', 'blue']): """ Return the color the facet has in the solved state.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import color_of_square sage: color_of_square(41) 'blue' """
cubie_center_list = { # centers of the cubies on the F,U, R faces 1: [1//2, 1//2, 5//2], # ulb 2: [1//2, 3//2, 5//2], # ub 3: [1//2, 5//2, 5//2], # ubr 4: [3//2, 1//2, 5//2], # ul 5: [3//2, 5//2, 5//2], # ur 6: [5//2, 1//2, 5//2], # ufl 7: [5//2, 3//2, 5//2], # uf 8: [5//2, 5//2, 5//2], # urf 17: [5//2, 1//2, 5//2], # flu 18: [5//2, 3//2, 5//2], # fu 19: [5//2, 5//2, 5//2], # fur 20: [5//2, 1//2, 3//2], # fl 21: [5//2, 5//2, 3//2], # fr 22: [5//2, 1//2, 1//2], # fdl 23: [5//2, 3//2, 1//2], # fd 24: [5//2, 5//2, 1//2], # frd 25: [5//2, 5//2, 5//2], # rfu 26: [3//2, 5//2, 5//2], # ru 27: [1//2, 5//2, 5//2], # rub 28: [5//2, 5//2, 3//2], # rf 29: [1//2, 5//2, 3//2], # rb 30: [5//2, 5//2, 1//2], # rdf 31: [3//2, 5//2, 1//2], # rd 32: [1//2, 5//2, 1//2], # rbd }
def cubie_centers(label): r""" Return the cubie center list element given by ``label``.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import cubie_centers sage: cubie_centers(3) [0, 2, 2] """
def cubie_colors(label,state0): r""" Return the color of the cubie given by ``label`` at ``state0``.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import cubie_colors sage: G = CubeGroup() sage: g = G.parse("R*U") sage: cubie_colors(3, G.facets(g)) [(1, 1, 1), (1, 0.63, 1), (1, 0.6, 0.3)] """ # colors of the cubies on the F,U, R faces
def plot3d_cubie(cnt, clrs): r""" Plot the front, up and right face of a cubie centered at cnt and rgbcolors given by clrs (in the order FUR).
Type ``P.show()`` to view.
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import plot3d_cubie, blue, red, green sage: clrF = blue; clrU = red; clrR = green sage: P = plot3d_cubie([1/2,1/2,1/2],[clrF,clrU,clrR]) """ #ptsD = [[x+0,y+0,0+z],[x+1,y+0,0+z],[x+1,y+1,0+z],[x+0,y+1,0+z],[x+0,y+0,0+z]] #ptsB = [[x+0,y+0,0+z],[x+0,y+1,0+z],[x+0,y+1,1+z],[x+0,y+0,1+z],[x+0,y+0,0+z]] #ptsL = [[x+0,y+0,0+z],[x+1,y+0,0+z],[x+1,y+0,1+z],[x+0,y+0,1+z],[x+0,y+0,0+z]]
####################### end of "internal" utility plot functions #################
class CubeGroup(PermutationGroup_generic): r""" A python class to help compute Rubik's cube group actions.
.. NOTE::
This group is also available via ``groups.permutation.RubiksCube()``.
EXAMPLES:
If G denotes the cube group then it may be regarded as a subgroup of ``SymmetricGroup(48)``, where the 48 facets are labeled as follows.
::
sage: rubik = CubeGroup() sage: rubik.display2d("") +--------------+ | 1 2 3 | | 4 top 5 | | 6 7 8 | +------------+--------------+-------------+------------+ | 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 | | 12 left 13 | 20 front 21 | 28 right 29 | 36 rear 37 | | 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 | +------------+--------------+-------------+------------+ | 41 42 43 | | 44 bottom 45 | | 46 47 48 | +--------------+
::
sage: rubik The Rubik's cube group with generators R,L,F,B,U,D in SymmetricGroup(48).
TESTS::
sage: groups.permutation.RubiksCube() The Rubik's cube group with generators R,L,F,B,U,D in SymmetricGroup(48). """ def __init__(self): """ Initialize ``self``.
EXAMPLES::
sage: rubik = CubeGroup() sage: TestSuite(rubik).run(skip="_test_enumerated_set_contains") # because the group is very large
TESTS:
Check that :trac:`11360` is fixed::
sage: rubik = CubeGroup() sage: rubik.order() 43252003274489856000 """
def gen_names(self): """ Return the names of the generators.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.gen_names() ['B', 'D', 'F', 'L', 'R', 'U'] """
def __repr__(self): """ Return a string representation of ``self``.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik The Rubik's cube group with generators R,L,F,B,U,D in SymmetricGroup(48). """
def B(self): """ Return the generator `B` in Singmaster notation.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.B() (1,14,48,27)(2,12,47,29)(3,9,46,32)(33,35,40,38)(34,37,39,36) """
def D(self): """ Return the generator `D` in Singmaster notation.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.D() (14,22,30,38)(15,23,31,39)(16,24,32,40)(41,43,48,46)(42,45,47,44) """
def F(self): """ Return the generator `F` in Singmaster notation.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.F() (6,25,43,16)(7,28,42,13)(8,30,41,11)(17,19,24,22)(18,21,23,20) """
def L(self): """ Return the generator `L` in Singmaster notation.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.L() (1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12) """
def R(self): """ Return the generator `R` in Singmaster notation.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.R() (3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28) """
def U(self): """ Return the generator `U` in Singmaster notation.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.U() (1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19) """
def parse(self, mv, check=True): r""" This function allows one to create the permutation group element from a variety of formats.
INPUT:
- ``mv`` -- Can one of the following:
- ``list`` - list of facets (as returned by self.facets())
- ``dict`` - list of faces (as returned by ``self.faces()``)
- ``str`` - either cycle notation (passed to GAP) or a product of generators or Singmaster notation
- ``perm_group element`` - returned as an element of ``self``
- ``check`` -- check if the input is valid
EXAMPLES::
sage: C = CubeGroup() sage: C.parse(list(range(1,49))) () sage: g = C.parse("L"); g (1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12) sage: C.parse(str(g)) == g True sage: facets = C.facets(g); facets [17, 2, 3, 20, 5, 22, 7, 8, 11, 13, 16, 10, 15, 9, 12, 14, 41, 18, 19, 44, 21, 46, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 6, 36, 4, 38, 39, 1, 40, 42, 43, 37, 45, 35, 47, 48] sage: C.parse(facets) (1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12) sage: C.parse(facets) == g True sage: faces = C.faces("L"); faces {'back': [[33, 34, 6], [36, 0, 4], [38, 39, 1]], 'down': [[40, 42, 43], [37, 0, 45], [35, 47, 48]], 'front': [[41, 18, 19], [44, 0, 21], [46, 23, 24]], 'left': [[11, 13, 16], [10, 0, 15], [9, 12, 14]], 'right': [[25, 26, 27], [28, 0, 29], [30, 31, 32]], 'up': [[17, 2, 3], [20, 0, 5], [22, 7, 8]]} sage: C.parse(faces) == C.parse("L") True sage: C.parse("L' R2") == C.parse("L^(-1)*R^2") True sage: C.parse("L' R2") (1,40,41,17)(3,43)(4,37,44,20)(5,45)(6,35,46,22)(8,48)(9,14,16,11)(10,12,15,13)(19,38)(21,36)(24,33)(25,32)(26,31)(27,30)(28,29) sage: C.parse("L^4") () sage: C.parse("L^(-1)*R") (1,40,41,17)(3,38,43,19)(4,37,44,20)(5,36,45,21)(6,35,46,22)(8,33,48,24)(9,14,16,11)(10,12,15,13)(25,27,32,30)(26,29,31,28) """ # mv is a perm_group element, return mv # It is a string: may be in cycle notation or Rubik's notation else: else: else:
__call__ = parse
def facets(self, g=None): r""" Return the set of facets on which the group acts. This function is a "constant".
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.facets() == list(range(1,49)) True """ else:
def faces(self, mv): r""" Return the dictionary of faces created by the effect of the move mv, which is a string of the form `X^a*Y^b*...`, where `X, Y, \ldots` are in `\{R,L,F,B,U,D\}` and `a, b, \ldots` are integers. We call this ordering of the faces the "BDFLRU, L2R, T2B ordering".
EXAMPLES::
sage: rubik = CubeGroup()
Here is the dictionary of the solved state::
sage: sorted(rubik.faces("").items()) [('back', [[33, 34, 35], [36, 0, 37], [38, 39, 40]]), ('down', [[41, 42, 43], [44, 0, 45], [46, 47, 48]]), ('front', [[17, 18, 19], [20, 0, 21], [22, 23, 24]]), ('left', [[9, 10, 11], [12, 0, 13], [14, 15, 16]]), ('right', [[25, 26, 27], [28, 0, 29], [30, 31, 32]]), ('up', [[1, 2, 3], [4, 0, 5], [6, 7, 8]])]
Now the dictionary of the state obtained after making the move `R` followed by `L`::
sage: sorted(rubik.faces("R*U").items()) [('back', [[48, 26, 27], [45, 0, 37], [43, 39, 40]]), ('down', [[41, 42, 11], [44, 0, 21], [46, 47, 24]]), ('front', [[9, 10, 8], [20, 0, 7], [22, 23, 6]]), ('left', [[33, 34, 35], [12, 0, 13], [14, 15, 16]]), ('right', [[19, 29, 32], [18, 0, 31], [17, 28, 30]]), ('up', [[3, 5, 38], [2, 0, 36], [1, 4, 25]])] """
def move(self, mv): r""" Return the group element and the reordered list of facets, as moved by the list ``mv`` (read left-to-right)
INPUT:
- ``mv`` -- A string of the form ``Xa*Yb*...``, where ``X``, ``Y``, ... are in ``R``, ``L``, ``F``, ``B``, ``U``, ``D`` and ``a``, ``b``, ... are integers.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.move("")[0] () sage: rubik.move("R")[0] (3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28) sage: rubik.R() (3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28) """
def display2d(self,mv): r""" Print the 2d representation of ``self``.
EXAMPLES::
sage: rubik = CubeGroup() sage: rubik.display2d("R") +--------------+ | 1 2 38 | | 4 top 36 | | 6 7 33 | +------------+--------------+-------------+------------+ | 9 10 11 | 17 18 3 | 27 29 32 | 48 34 35 | | 12 left 13 | 20 front 5 | 26 right 31 | 45 rear 37 | | 14 15 16 | 22 23 8 | 25 28 30 | 43 39 40 | +------------+--------------+-------------+------------+ | 41 42 19 | | 44 bottom 21 | | 46 47 24 | +--------------+ """
def repr2d(self, mv): r""" Displays a 2D map of the Rubik's cube after the move mv has been made. Nothing is returned.
EXAMPLES::
sage: rubik = CubeGroup() sage: print(rubik.repr2d("")) +--------------+ | 1 2 3 | | 4 top 5 | | 6 7 8 | +------------+--------------+-------------+------------+ | 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 | | 12 left 13 | 20 front 21 | 28 right 29 | 36 rear 37 | | 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 | +------------+--------------+-------------+------------+ | 41 42 43 | | 44 bottom 45 | | 46 47 48 | +--------------+
::
sage: print(rubik.repr2d("R")) +--------------+ | 1 2 38 | | 4 top 36 | | 6 7 33 | +------------+--------------+-------------+------------+ | 9 10 11 | 17 18 3 | 27 29 32 | 48 34 35 | | 12 left 13 | 20 front 5 | 26 right 31 | 45 rear 37 | | 14 15 16 | 22 23 8 | 25 28 30 | 43 39 40 | +------------+--------------+-------------+------------+ | 41 42 19 | | 44 bottom 21 | | 46 47 24 | +--------------+
You can see the right face has been rotated but not the left face. """
def plot_cube(self, mv, title=True, colors = [lpurple, yellow, red, green, orange, blue]): r""" Input the move mv, as a string in the Singmaster notation, and output the 2D plot of the cube in that state.
Type ``P.show()`` to display any of the plots below.
EXAMPLES::
sage: rubik = CubeGroup() sage: P = rubik.plot_cube("R^2*U^2*R^2*U^2*R^2*U^2", title = False) sage: # (R^2U^2)^3 permutes 2 pairs of edges (uf,ub)(fr,br) sage: P = rubik.plot_cube("R*L*D^2*B^3*L^2*F^2*R^2*U^3*D*R^3*D^2*F^3*B^3*D^3*F^2*D^3*R^2*U^3*F^2*D^3") sage: # the superflip (in 20f* moves) sage: P = rubik.plot_cube("U^2*F*U^2*L*R^(-1)*F^2*U*F^3*B^3*R*L*U^2*R*D^3*U*L^3*R*D*R^3*L^3*D^2") sage: # "superflip+4 spot" (in 26q* moves) """
def plot3d_cube(self,mv,title=True): r""" Displays `F,U,R` faces of the cube after the given move ``mv``. Mostly included for the purpose of drawing pictures and checking moves.
INPUT:
- ``mv`` -- A string in the Singmaster notation - ``title`` -- (Default: ``True``) Display the title information
The first one below is "superflip+4 spot" (in 26q\* moves) and the second one is the superflip (in 20f\* moves). Type show(P) to view them.
EXAMPLES::
sage: rubik = CubeGroup() sage: P = rubik.plot3d_cube("U^2*F*U^2*L*R^(-1)*F^2*U*F^3*B^3*R*L*U^2*R*D^3*U*L^3*R*D*R^3*L^3*D^2") sage: P = rubik.plot3d_cube("R*L*D^2*B^3*L^2*F^2*R^2*U^3*D*R^3*D^2*F^3*B^3*D^3*F^2*D^3*R^2*U^3*F^2*D^3") """ return P
def legal(self,state,mode="quiet"): r""" Return 1 (true) if the dictionary ``state`` (in the same format as returned by the faces method) represents a legal position (or state) of the Rubik's cube or 0 (false) otherwise.
EXAMPLES::
sage: rubik = CubeGroup() sage: r0 = rubik.faces("") sage: r1 = {'back': [[33, 34, 35], [36, 0, 37], [38, 39, 40]], 'down': [[41, 42, 43], [44, 0, 45], [46, 47, 48]],'front': [[17, 18, 19], [20, 0, 21], [22, 23, 24]],'left': [[9, 10, 11], [12, 0, 13], [14, 15, 16]],'right': [[25, 26, 27], [28, 0, 29], [30, 31, 32]],'up': [[1, 2, 3], [4, 0, 5], [6, 8, 7]]} sage: rubik.legal(r0) 1 sage: rubik.legal(r0,"verbose") (1, ()) sage: rubik.legal(r1) 0 """ else:
def solve(self,state, algorithm='default'): r""" Solves the cube in the ``state``, given as a dictionary as in ``legal``. See the ``solve`` method of the RubiksCube class for more details.
This may use GAP's ``EpimorphismFromFreeGroup`` and ``PreImagesRepresentative`` as explained below, if 'gap' is passed in as the algorithm.
This algorithm
#. constructs the free group on 6 generators then computes a reasonable set of relations which they satisfy
#. computes a homomorphism from the cube group to this free group quotient
#. takes the cube position, regarded as a group element, and maps it over to the free group quotient
#. using those relations and tricks from combinatorial group theory (stabilizer chains), solves the "word problem" for that element.
#. uses python string parsing to rewrite that in cube notation.
The Rubik's cube group has about `4.3 \times 10^{19}` elements, so this process is time-consuming. See http://www.gap-system.org/Doc/Examples/rubik.html for an interesting discussion of some GAP code analyzing the Rubik's cube.
EXAMPLES::
sage: rubik = CubeGroup() sage: state = rubik.faces("R") sage: rubik.solve(state) 'R' sage: state = rubik.faces("R*U") sage: rubik.solve(state, algorithm='gap') # long time 'R*U'
You can also check this another (but similar) way using the ``word_problem`` method (eg, G = rubik.group(); g = G("(3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28)"); g.word_problem([b,d,f,l,r,u]), though the output will be less intuitive). """ except TypeError: return "Illegal or syntactically incorrect state. No solution."
hom = self._gap_().EpimorphismFromFreeGroup() soln = hom.PreImagesRepresentative(gap(str(g))) sol = str(soln) names = self.gen_names() for i in range(6): sol = sol.replace("x%s" % (i+1), names[i]) return sol
########################################################## # 3d object generation ##########################################################
def cubie_faces(): """ This provides a map from the 6 faces of the 27 cubies to the 48 facets of the larger cube.
-1,-1,-1 is left, top, front
EXAMPLES::
sage: from sage.groups.perm_gps.cubegroup import cubie_faces sage: sorted(cubie_faces().items()) [((-1, -1, -1), [6, 17, 11, 0, 0, 0]), ((-1, -1, 0), [4, 0, 10, 0, 0, 0]), ((-1, -1, 1), [1, 0, 9, 0, 35, 0]), ((-1, 0, -1), [0, 20, 13, 0, 0, 0]), ((-1, 0, 0), [0, 0, -5, 0, 0, 0]), ((-1, 0, 1), [0, 0, 12, 0, 37, 0]), ((-1, 1, -1), [0, 22, 16, 41, 0, 0]), ((-1, 1, 0), [0, 0, 15, 44, 0, 0]), ((-1, 1, 1), [0, 0, 14, 46, 40, 0]), ((0, -1, -1), [7, 18, 0, 0, 0, 0]), ((0, -1, 0), [-6, 0, 0, 0, 0, 0]), ((0, -1, 1), [2, 0, 0, 0, 34, 0]), ((0, 0, -1), [0, -4, 0, 0, 0, 0]), ((0, 0, 0), [0, 0, 0, 0, 0, 0]), ((0, 0, 1), [0, 0, 0, 0, -2, 0]), ((0, 1, -1), [0, 23, 0, 42, 0, 0]), ((0, 1, 0), [0, 0, 0, -1, 0, 0]), ((0, 1, 1), [0, 0, 0, 47, 39, 0]), ((1, -1, -1), [8, 19, 0, 0, 0, 25]), ((1, -1, 0), [5, 0, 0, 0, 0, 26]), ((1, -1, 1), [3, 0, 0, 0, 33, 27]), ((1, 0, -1), [0, 21, 0, 0, 0, 28]), ((1, 0, 0), [0, 0, 0, 0, 0, -3]), ((1, 0, 1), [0, 0, 0, 0, 36, 29]), ((1, 1, -1), [0, 24, 0, 43, 0, 30]), ((1, 1, 0), [0, 0, 0, 45, 0, 31]), ((1, 1, 1), [0, 0, 0, 48, 38, 32])] """
cubie_face_list = cubie_faces()
rand_colors = [(RDF.random_element(), RDF.random_element(), RDF.random_element()) for _ in range(56)]
@richcmp_method class RubiksCube(SageObject): r""" The Rubik's cube (in a given state).
EXAMPLES::
sage: C = RubiksCube().move("R U R'") sage: C.show3d()
::
sage: C = RubiksCube("R*L"); C +--------------+ | 17 2 38 | | 20 top 36 | | 22 7 33 | +------------+--------------+-------------+------------+ | 11 13 16 | 41 18 3 | 27 29 32 | 48 34 6 | | 10 left 15 | 44 front 5 | 26 right 31 | 45 rear 4 | | 9 12 14 | 46 23 8 | 25 28 30 | 43 39 1 | +------------+--------------+-------------+------------+ | 40 42 19 | | 37 bottom 21 | | 35 47 24 | +--------------+ sage: C.show() sage: C.solve(algorithm='gap') # long time 'L R' sage: C == RubiksCube("L*R") True """ def __init__(self, state=None, history=[], colors=[lpurple,yellow,red,green,orange,blue]): """ Initialize ``self``.
EXAMPLES::
sage: C = RubiksCube().move("R*U") sage: TestSuite(C).run() """ else: raise ValueError("Not a legal cube.")
def move(self, g): r""" Move the Rubik's cube by ``g``.
EXAMPLES::
sage: RubiksCube().move("R*U") == RubiksCube("R*U") True """
def undo(self): r""" Undo the last move of the Rubik's cube.
EXAMPLES::
sage: C = RubiksCube() sage: D = C.move("R*U") sage: D.undo() == C True """ raise ValueError("no moves to undo")
def _repr_(self): r""" Return a string representation of ``self``.
EXAMPLES::
sage: RubiksCube().move("R*U") +--------------+ | 3 5 38 | | 2 top 36 | | 1 4 25 | +------------+--------------+-------------+------------+ | 33 34 35 | 9 10 8 | 19 29 32 | 48 26 27 | | 12 left 13 | 20 front 7 | 18 right 31 | 45 rear 37 | | 14 15 16 | 22 23 6 | 17 28 30 | 43 39 40 | +------------+--------------+-------------+------------+ | 41 42 11 | | 44 bottom 21 | | 46 47 24 | +--------------+ <BLANKLINE> """
def facets(self): r""" Return the facets of ``self``.
EXAMPLES::
sage: C = RubiksCube("R*U") sage: C.facets() [3, 5, 38, 2, 36, 1, 4, 25, 33, 34, 35, 12, 13, 14, 15, 16, 9, 10, 8, 20, 7, 22, 23, 6, 19, 29, 32, 18, 31, 17, 28, 30, 48, 26, 27, 45, 37, 43, 39, 40, 41, 42, 11, 44, 21, 46, 47, 24] """
def plot(self): r""" Return a plot of ``self``.
EXAMPLES::
sage: C = RubiksCube("R*U") sage: C.plot() Graphics object consisting of 55 graphics primitives """
def show(self): r""" Show a plot of ``self``.
EXAMPLES::
sage: C = RubiksCube("R*U") sage: C.show() """
def cubie(self, size, gap, x,y,z, colors, stickers=True): """ Return the cubie at `(x,y,z)`.
INPUT:
- ``size`` -- The size of the cubie - ``gap`` -- The gap between cubies - ``x,y,z`` -- The position of the cubie - ``colors`` -- The list of colors - ``stickers`` -- (Default ``True``) Boolean to display stickers
EXAMPLES::
sage: C = RubiksCube("R*U") sage: C.cubie(0.15, 0.025, 0,0,0, C.colors*3) Graphics3d Object """ else: return ColorCube(size, [colors[sides[i]+6] for i in range(6)]).translate(-t*x, -t*z, -t*y)
def plot3d(self, stickers=True): r""" Return a 3D plot of ``self``.
EXAMPLES::
sage: C = RubiksCube("R*U") sage: C.plot3d() Graphics3d Object """
def show3d(self): r""" Show a 3D plot of ``self``.
EXAMPLES::
sage: C = RubiksCube("R*U") sage: C.show3d() """
def __richcmp__(self, other, op): """ Comparison.
INPUT:
- ``other`` -- anything
- ``op`` -- comparison operator
EXAMPLES::
sage: C = RubiksCube() sage: D = RubiksCube("R*U") sage: C < D True sage: C > D False sage: C == C True sage: C != D True """ return NotImplemented
def solve(self, algorithm="hybrid", timeout=15): r""" Solve the Rubik's cube.
INPUT:
- ``algorithm`` -- must be one of the following:
- ``hybrid`` - try ``kociemba`` for timeout seconds, then ``dietz`` - ``kociemba`` - Use Dik T. Winter's program (reasonable speed, few moves) - ``dietz`` - Use Eric Dietz's cubex program (fast but lots of moves) - ``optimal`` - Use Michael Reid's optimal program (may take a long time) - ``gap`` - Use GAP word solution (can be slow)
EXAMPLES::
sage: C = RubiksCube("R U F L B D") sage: C.solve() 'R U F L B D'
Dietz's program is much faster, but may give highly non-optimal solutions::
sage: s = C.solve('dietz'); s "U' L' L' U L U' L U D L L D' L' D L' D' L D L' U' L D' L' U L' B' U' L' U B L D L D' U' L' U L B L B' L' U L U' L' F' L' F L' F L F' L' D' L' D D L D' B L B' L B' L B F' L F F B' L F' B D' D' L D B' B' L' D' B U' U' L' B' D' F' F' L D F'" sage: C2 = RubiksCube(s) sage: C == C2 True """
except RuntimeError: solver = sage.interfaces.rubik.CubexSolver() return solver.solve(self.facets())
solver = sage.interfaces.rubik.DikSolver() return solver.solve(self.facets(), timeout=timeout)
elif algorithm == "optimal": # TODO: cache this, startup is expensive solver = sage.interfaces.rubik.OptimalSolver() return solver.solve(self.facets())
elif algorithm == "gap": solver = CubeGroup() return solver.solve(self._state)
else: raise ValueError("Unrecognized algorithm: %s" % algorithm)
def scramble(self, moves=30): """ Scramble the Rubik's cube.
EXAMPLES::
sage: C = RubiksCube() sage: C.scramble() # random +--------------+ | 38 29 35 | | 20 top 42 | | 11 44 30 | +------------+--------------+-------------+------------+ | 48 13 17 | 6 15 24 | 43 23 9 | 1 36 32 | | 4 left 18 | 7 front 37 | 12 right 26 | 5 rear 10 | | 33 31 40 | 14 28 8 | 25 47 16 | 22 2 3 | +------------+--------------+-------------+------------+ | 46 21 19 | | 45 bottom 39 | | 27 34 41 | +--------------+ <BLANKLINE> """
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