Rubik’s cube group functions#
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 │
└──────────────┘
>>> from sage.all import *
>>> rubik = CubeGroup()
>>> 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[source]#
Bases:
PermutationGroup_genericA 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 │ └──────────────┘
>>> from sage.all import * >>> rubik = CubeGroup() >>> 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).
>>> from sage.all import * >>> rubik The Rubik's cube group with generators R,L,F,B,U,D in SymmetricGroup(48).
- B()[source]#
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)
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.B() (1,14,48,27)(2,12,47,29)(3,9,46,32)(33,35,40,38)(34,37,39,36)
- D()[source]#
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)
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.D() (14,22,30,38)(15,23,31,39)(16,24,32,40)(41,43,48,46)(42,45,47,44)
- F()[source]#
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)
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.F() (6,25,43,16)(7,28,42,13)(8,30,41,11)(17,19,24,22)(18,21,23,20)
- L()[source]#
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)
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.L() (1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12)
- R()[source]#
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)
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.R() (3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28)
- U()[source]#
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)
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.U() (1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19)
- display2d(mv)[source]#
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 │ └──────────────┘
>>> from sage.all import * >>> rubik = CubeGroup() >>> 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 │ └──────────────┘
- faces(mv)[source]#
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()
>>> from sage.all import * >>> 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]])]
>>> from sage.all import * >>> 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]])]
>>> from sage.all import * >>> 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]])]
- facets(g=None)[source]#
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
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.facets() == list(range(Integer(1),Integer(49))) True
- gen_names()[source]#
Return the names of the generators.
EXAMPLES:
sage: rubik = CubeGroup() sage: rubik.gen_names() ['B', 'D', 'F', 'L', 'R', 'U']
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.gen_names() ['B', 'D', 'F', 'L', 'R', 'U']
- legal(state, mode='quiet')[source]#
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
>>> from sage.all import * >>> rubik = CubeGroup() >>> r0 = rubik.faces("") >>> r1 = {'back': [[Integer(33), Integer(34), Integer(35)], [Integer(36), Integer(0), Integer(37)], [Integer(38), Integer(39), Integer(40)]], 'down': [[Integer(41), Integer(42), Integer(43)], [Integer(44), Integer(0), Integer(45)], [Integer(46), Integer(47), Integer(48)]],'front': [[Integer(17), Integer(18), Integer(19)], [Integer(20), Integer(0), Integer(21)], [Integer(22), Integer(23), Integer(24)]],'left': [[Integer(9), Integer(10), Integer(11)], [Integer(12), Integer(0), Integer(13)], [Integer(14), Integer(15), Integer(16)]],'right': [[Integer(25), Integer(26), Integer(27)], [Integer(28), Integer(0), Integer(29)], [Integer(30), Integer(31), Integer(32)]],'up': [[Integer(1), Integer(2), Integer(3)], [Integer(4), Integer(0), Integer(5)], [Integer(6), Integer(8), Integer(7)]]} >>> rubik.legal(r0) 1 >>> rubik.legal(r0,"verbose") (1, ()) >>> rubik.legal(r1) 0
- move(mv)[source]#
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 formXa*Yb*..., whereX,Y, … are inR,L,F,B,U,Danda,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)
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik.move("")[Integer(0)] () >>> rubik.move("R")[Integer(0)] (3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28) >>> rubik.R() (3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28)
- parse(mv, check=True)[source]#
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 byself.faces())str– either cycle notation (passed to GAP) or a product of generators or Singmaster notationperm_group element– returned as an element ofself
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)
>>> from sage.all import * >>> C = CubeGroup() >>> C.parse(list(range(Integer(1),Integer(49)))) () >>> 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) >>> C.parse(str(g)) == g True >>> 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] >>> C.parse(facets) (1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12) >>> C.parse(facets) == g True >>> 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]]} >>> C.parse(faces) == C.parse("L") True >>> C.parse("L' R2") == C.parse("L^(-1)*R^2") True >>> 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) >>> C.parse("L^4") () >>> 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)[source]#
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 notationtitle– (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") # needs sage.plot 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") # needs sage.plot
>>> from sage.all import * >>> rubik = CubeGroup() >>> 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") # needs sage.plot >>> 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") # needs sage.plot
- plot_cube(mv, title=True, colors=[(1, 0.63, 1), (1, 1, 0), (1, 0, 0), (0, 1, 0), (1, 0.6, 0.3), (0, 0, 1)])[source]#
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) # needs sage.plot 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") # needs sage.plot 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") # needs sage.plot sage: # "superflip+4 spot" (in 26q* moves)
>>> from sage.all import * >>> rubik = CubeGroup() >>> P = rubik.plot_cube("R^2*U^2*R^2*U^2*R^2*U^2", title=False) # needs sage.plot >>> # (R^2U^2)^3 permutes 2 pairs of edges (uf,ub)(fr,br) >>> 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") # needs sage.plot >>> # the superflip (in 20f* moves) >>> 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") # needs sage.plot >>> # "superflip+4 spot" (in 26q* moves)
- repr2d(mv)[source]#
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 │ └──────────────┘
>>> from sage.all import * >>> rubik = CubeGroup() >>> 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 │ └──────────────┘
>>> from sage.all import * >>> 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.
- solve(state, algorithm='default')[source]#
Solve the cube in the
state, given as a dictionary as inlegal. See thesolvemethod of the RubiksCube class for more details.This may use GAP’s
EpimorphismFromFreeGroupandPreImagesRepresentativeas 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 https://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'
>>> from sage.all import * >>> rubik = CubeGroup() >>> state = rubik.faces("R") >>> rubik.solve(state) 'R' >>> state = rubik.faces("R*U") >>> rubik.solve(state, algorithm='gap') # long time 'R*U'
You can also check this another (but similar) way using the
word_problemmethod (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, 0.6, 0.3), (0, 0, 1)])[source]#
Bases:
SageObjectThe Rubik’s cube (in a given state).
EXAMPLES:
sage: C = RubiksCube().move("R U R'") sage: C.show3d() # needs sage.plot
>>> from sage.all import * >>> C = RubiksCube().move("R U R'") >>> C.show3d() # needs sage.plot
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() # needs sage.plot sage: C.solve(algorithm='gap') # long time 'L*R' sage: C == RubiksCube("L*R") True
>>> from sage.all import * >>> 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 │ └──────────────┘ >>> C.show() # needs sage.plot >>> C.solve(algorithm='gap') # long time 'L*R' >>> C == RubiksCube("L*R") True
- cubie(size, gap, x, y, z, colors, stickers=True)[source]#
Return the cubie at \((x,y,z)\).
INPUT:
size– The size of the cubiegap– The gap between cubiesx,y,z– The position of the cubiecolors– The list of colorsstickers– (DefaultTrue) Boolean to display stickers
EXAMPLES:
sage: C = RubiksCube("R*U") sage: C.cubie(0.15, 0.025, 0,0,0, C.colors*3) # needs sage.plot Graphics3d Object
>>> from sage.all import * >>> C = RubiksCube("R*U") >>> C.cubie(RealNumber('0.15'), RealNumber('0.025'), Integer(0),Integer(0),Integer(0), C.colors*Integer(3)) # needs sage.plot Graphics3d Object
- facets()[source]#
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]
>>> from sage.all import * >>> C = RubiksCube("R*U") >>> 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]
- move(g)[source]#
Move the Rubik’s cube by
g.EXAMPLES:
sage: RubiksCube().move("R*U") == RubiksCube("R*U") True
>>> from sage.all import * >>> RubiksCube().move("R*U") == RubiksCube("R*U") True
- plot()[source]#
Return a plot of
self.EXAMPLES:
sage: C = RubiksCube("R*U") sage: C.plot() # needs sage.plot Graphics object consisting of 55 graphics primitives
>>> from sage.all import * >>> C = RubiksCube("R*U") >>> C.plot() # needs sage.plot Graphics object consisting of 55 graphics primitives
- plot3d(stickers=True)[source]#
Return a 3D plot of
self.EXAMPLES:
sage: C = RubiksCube("R*U") sage: C.plot3d() # needs sage.plot Graphics3d Object
>>> from sage.all import * >>> C = RubiksCube("R*U") >>> C.plot3d() # needs sage.plot Graphics3d Object
- scramble(moves=30)[source]#
Scramble the Rubik’s cube.
EXAMPLES:
sage: C = RubiksCube() sage: C.scramble() # random ┌──────────────┐ │ 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 │ └──────────────┘
>>> from sage.all import * >>> C = RubiksCube() >>> C.scramble() # random ┌──────────────┐ │ 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 │ └──────────────┘
- show()[source]#
Show a plot of
self.EXAMPLES:
sage: C = RubiksCube("R*U") sage: C.show() # needs sage.plot
>>> from sage.all import * >>> C = RubiksCube("R*U") >>> C.show() # needs sage.plot
- show3d()[source]#
Show a 3D plot of
self.EXAMPLES:
sage: C = RubiksCube("R*U") sage: C.show3d() # needs sage.plot
>>> from sage.all import * >>> C = RubiksCube("R*U") >>> C.show3d() # needs sage.plot
- solve(algorithm='hybrid', timeout=15)[source]#
Solve the Rubik’s cube.
INPUT:
algorithm– must be one of the following:hybrid– trykociembafor timeout seconds, thendietzkociemba– 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)
Any choice other than
gaprequires the optional packagerubiks. Otherwise, thegapalgorithm is used.EXAMPLES:
sage: C = RubiksCube("R U F L B D") sage: C.solve() # optional - rubiks 'R U F L B D'
>>> from sage.all import * >>> C = RubiksCube("R U F L B D") >>> C.solve() # optional - rubiks '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 # optional - rubiks "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) # optional - rubiks sage: C == C2 # optional - rubiks True
>>> from sage.all import * >>> s = C.solve('dietz'); s # optional - rubiks "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'" >>> C2 = RubiksCube(s) # optional - rubiks >>> C == C2 # optional - rubiks True
- sage.groups.perm_gps.cubegroup.color_of_square(facet, colors=['lpurple', 'yellow', 'red', 'green', 'orange', 'blue'])[source]#
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'
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import color_of_square >>> color_of_square(Integer(41)) 'blue'
- sage.groups.perm_gps.cubegroup.create_poly(face, color)[source]#
Create the polygon given by
facewith colorcolor.EXAMPLES:
sage: from sage.groups.perm_gps.cubegroup import create_poly, red sage: create_poly('ur', red) # needs sage.plot Graphics object consisting of 1 graphics primitive
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import create_poly, red >>> create_poly('ur', red) # needs sage.plot Graphics object consisting of 1 graphics primitive
- sage.groups.perm_gps.cubegroup.cubie_centers(label)[source]#
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]
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import cubie_centers >>> cubie_centers(Integer(3)) [0, 2, 2]
- sage.groups.perm_gps.cubegroup.cubie_colors(label, state0)[source]#
Return the color of the cubie given by
labelatstate0.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)]
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import cubie_colors >>> G = CubeGroup() >>> g = G.parse("R*U") >>> cubie_colors(Integer(3), G.facets(g)) [(1, 1, 1), (1, 0.63, 1), (1, 0.6, 0.3)]
- sage.groups.perm_gps.cubegroup.cubie_faces()[source]#
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])]
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import cubie_faces >>> 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])]
- sage.groups.perm_gps.cubegroup.index2singmaster(facet)[source]#
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'
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import index2singmaster >>> index2singmaster(Integer(41)) 'dlf'
- sage.groups.perm_gps.cubegroup.inv_list(lst)[source]#
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]
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import inv_list >>> L = [Integer(2),Integer(3),Integer(1)] >>> inv_list(L) [3, 1, 2]
- sage.groups.perm_gps.cubegroup.plot3d_cubie(cnt, clrs)[source]#
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]) # needs sage.plot
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import plot3d_cubie, blue, red, green >>> clrF = blue; clrU = red; clrR = green >>> P = plot3d_cubie([Integer(1)/Integer(2),Integer(1)/Integer(2),Integer(1)/Integer(2)],[clrF,clrU,clrR]) # needs sage.plot
- sage.groups.perm_gps.cubegroup.polygon_plot3d(points, tilt=30, turn=30, **kwargs)[source]#
Plot a polygon viewed from an angle determined by
tilt,turn, and verticespoints.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]], # needs sage.plot ....: rgbcolor=green)
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import polygon_plot3d,green >>> P = polygon_plot3d([[Integer(1),Integer(3),Integer(1)],[Integer(2),Integer(3),Integer(1)],[Integer(2),Integer(3),Integer(2)],[Integer(1),Integer(3),Integer(2)],[Integer(1),Integer(3),Integer(1)]], # needs sage.plot ... rgbcolor=green)
- sage.groups.perm_gps.cubegroup.rotation_list(tilt, turn)[source]#
Return a list \([\sin(\theta), \sin(\phi), \cos(\theta), \cos(\phi)]\) of rotations where \(\theta\) is
tiltand \(\phi\) isturn.EXAMPLES:
sage: from sage.groups.perm_gps.cubegroup import rotation_list sage: rotation_list(30, 45) [0.49999999999999994, 0.7071067811865475, 0.8660254037844387, 0.7071067811865476]
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import rotation_list >>> rotation_list(Integer(30), Integer(45)) [0.49999999999999994, 0.7071067811865475, 0.8660254037844387, 0.7071067811865476]
- sage.groups.perm_gps.cubegroup.xproj(x, y, z, r)[source]#
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
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import rotation_list, xproj >>> rot = rotation_list(Integer(30), Integer(45)) >>> xproj(Integer(1),Integer(2),Integer(3),rot) 0.6123724356957945
- sage.groups.perm_gps.cubegroup.yproj(x, y, z, r)[source]#
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
>>> from sage.all import * >>> from sage.groups.perm_gps.cubegroup import rotation_list, yproj >>> rot = rotation_list(Integer(30), Integer(45)) >>> yproj(Integer(1),Integer(2),Integer(3),rot) 1.378497416975604