Rubik’s cube group functions¶

Note

“Rubik’s 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), U (up) Products of moves are read right to left, so for example, R*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.

class sage.groups.perm_gps.cubegroup.CubeGroup¶

Bases: sage.groups.perm_gps.permgroup.PermutationGroup_generic

A python class to help compute Rubik’s cube group actions.

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 PermutationGroup of all legal moves of the Rubik's cube.
sage: print rubik
The Rubik's cube group with generators R,L,F,B,U,D in SymmetricGroup(48).

B()¶

D()¶

F()¶

L()¶

R()¶

U()¶

display2d(mv)¶

faces(mv)¶

Returns the dictionary of faces created by the effect of the move mv, which is a string of the form $X^a*Y^b*...$ , where $X$ , $Y$ , ... are in $\{R,L,F,B,U,D\}$ and $a,b, ...$ are integers. We call this ordering of the faces the “BDFLRU, L2R, T2B ordering”.

EXAMPLES:

sage: rubik = CubeGroup()

Now type rubik.faces("") for the dictionary of the solved state and rubik.faces("R*L") for the dictionary of the state obtained after making the move R followed by L.

facets(g=None)¶

Returns the set of facets on which the group acts. This function is a “constant”.

EXAMPLES:

sage: rubik = CubeGroup()
sage: rubik.facets()==range(1,49)
True

gen_names()¶

gens()¶

group()¶

legal(state, mode='quiet')¶

Returns 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. Returns 0 (false) otherwise.

EXAMPLES:

sage: rubik = CubeGroup()
sage: G = rubik.group()
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

move(mv)¶

Returns the group element and the reordered list of facets, as moved by the list mv (read left-to-right)

INPUT: mv is 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)

parse(mv)¶

This function allows one to create the permutation group element from a variety of formats.

INPUT: 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.group()

EXAMPLES:

sage: C = CubeGroup()
sage: C.parse(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
{'right': [[25, 26, 27], [28, 0, 29], [30, 31, 32]], 'up': [[17, 2, 3], [20, 0, 5], [22, 7, 8]], '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]]}
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)

plot3d_cube(mv, title=True)¶

Displays F,U,R faces of the cube after the given move mv, where mv is a string in the Singmaster notation. Mostly included for the purpose of drawing pictures and checking moves.

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")

plot_cube(mv, title=True, colors=[, (1, 0.63, 1), (1, 1, 0), (1, 0, 0), (0, 1, 0), (1, 0.59999999999999998, 0.29999999999999999), (0, 0, 1)])¶

Input the move mv, as a string in the Singmaster notation, and output the 2-d 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)

repr2d(mv)¶

Displays a 2d map of the Rubik’s cube after the move mv has been made. Nothing is returned.

EXAMPLES:

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.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 |
             +--------------+

You can see the right face has been rotated but not the left face.

solve(state, algorithm='default')¶

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).

class sage.groups.perm_gps.cubegroup.RubiksCube(state=None, history=[], colors=[, (1, 0.63, 1), (1, 1, 0), (1, 0, 0), (0, 1, 0), (1, 0.59999999999999998, 0.29999999999999999), (0, 0, 1)])¶

Bases: sage.structure.sage_object.SageObject

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

cubie(size, gap, x, y, z, colors, stickers=True)¶

facets()¶

move(g)¶

plot()¶

plot3d(stickers=True)¶

sage: C = RubiksCube().move("R*U")
sage: C.plot3d()
sage: C.plot()

scramble(moves=30)¶

show()¶

show3d()¶

solve(algorithm='hybrid', timeout=15)¶

Algorithm must be one of : 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)

EXAMPLE:

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

undo()¶

sage.groups.perm_gps.cubegroup.color_of_square(facet, colors=[, 'lpurple', 'yellow', 'red', 'green', 'orange', 'blue'])¶

Returns the color the facet has in the solved state.

EXAMPLES:

sage: from sage.groups.perm_gps.cubegroup import *
sage: color_of_square(41)
'blue'

sage.groups.perm_gps.cubegroup.create_poly(face, color)¶

sage.groups.perm_gps.cubegroup.cubie_centers(label)¶

sage.groups.perm_gps.cubegroup.cubie_colors(label, state0)¶

sage.groups.perm_gps.cubegroup.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

sage.groups.perm_gps.cubegroup.index2singmaster(facet)¶

Translates index used (eg, 43) to Singmaster facet notation (eg, fdr).

EXAMPLES:

sage: from sage.groups.perm_gps.cubegroup import *
sage: index2singmaster(41)        
'dlf'

sage.groups.perm_gps.cubegroup.inv_list(lst)¶

Input a list of ints 1, ..., m (in any order), outputs inverse perm.

EXAMPLES:

sage: from sage.groups.perm_gps.cubegroup import *
sage: L = [2,3,1]
sage: inv_list(L)
[3, 1, 2]

sage.groups.perm_gps.cubegroup.plot3d_cubie(cnt, clrs)¶

Plots 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 *
sage: clrF = blue; clrU = red; clrR = green
sage: P = plot3d_cubie([1/2,1/2,1/2],[clrF,clrU,clrR])

sage.groups.perm_gps.cubegroup.polygon_plot3d(points, tilt=30, turn=30, **kwargs)¶

Plots 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 *
sage: P = polygon_plot3d([[1,3,1],[2,3,1],[2,3,2],[1,3,2],[1,3,1]],rgbcolor=green)

sage.groups.perm_gps.cubegroup.rotation_list(tilt, turn)¶

sage.groups.perm_gps.cubegroup.xproj(x, y, z, r)¶

sage.groups.perm_gps.cubegroup.yproj(x, y, z, r)¶

Rubik’s cube group functions¶

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