Matthew Burke

Problems 2.1 and 2.2

An Algorithm to Solve the Rubik’s Cube

 

Notation:

·        F is a 90 degree clockwise turn of the front face. B, R, L, U, D are similar moves for the Back, Right, Left, Up, and Down faces respectively.

·        F-1 is a 90 degree counterclockwise turn of the front face.

·        [FR] = FRF-1R-1

·        Let fuo denote the cubelet that should be in the front upper position and let fu simply denote the front upper position.  Ignore facelet orientation unless specified.

·        Reorientations of the cube are often written as permutations of the faces in cycle notation.  For example: (F,L,B,R) means fix the up and down faces and send the front face to the left, the left face to the back, the back face to the right, and the right face to the front.

 

I. Solving the UP layer:

A.     Positioning edge cubelets:

i.         Find fuo and move it to df.

a.      If fuo in UP layer then it is in one of the following positions: uf, ur, ub, ul.

1.      If it is in uf, use F2 to move it to the df position.

2.      If it is in ur, use R2D-1 to move it to the df position.

3.      If it is in ub, use B2D2 to move it to the df position.

4.      If it is in ul, use L2D to move it to the df position.

 

b.      If fuo in Middle layer then it is in one of the following positions: fr, fl, bl, br.

1.      If it is in fr, use F to move it to the df position.

2.      If it is in fl, use F-1 to move it to the df position.

3.      If it is in bl, use U2BU-2D2 to move it to the df position.

4.      If it is in br, use U2B-1U-2D2 to move it to the df position.

 

c.      If fuo in Down layer then it is in one of the following positions: df, dr, db, dl.

1.      If it is in df, continue to (ii).

2.      If it is in dr, use D-1 to move to df position.

3.      If it is in db, use D2 to move to df position.

4.      If it is in dl, use D to move to df position.

 

ii.       Properly position fuo with proper orientation.

a.      If the Up facelet of fuo is facing down, use F2 to properly position the cubelet.           

b.      If the Up facelet of fuo is facing front, use R-1DRF-1 to properly position the cubelet.

 

iii.      Reorient: (F,L,B,R), and repeat A. until all of the Up Edge cubelets are in position.  Note they will be properly oriented.

 

B.     Positioning UP corner cubelets.

i.         Find furo.

a.      If furo in UP layer then it is in one of the following positions: urb, ulb, ulf, urf.

1.      If furo is in urb, use UR-1D-1RU-1 to move to the dfl position.

2.      If furo is in ulb, use U2R-1D-1RU-2 to move to the dfl position.

3.      If furo is in ulf, use U-1R-1D-1RU to move to the dfl position.

4.      If furo is in urf, use R-1D-1R to move to the dfl position.

 

b.      If furo in Down layer then it is in one of the following positions: drb, dlb, drf, dlf.

1.      If furo is in drb, use D2 to move to the dfl position.

2.      If furo is in dlb, use D to move to the dfl position.

3.      If furo is in drf, use D-1 to move to the dfl position.

4.      If furo is in dlf, continue to the next step.

 

ii.       Position furo in proper position with proper orientation.

a.      If the Up facelet of furo is facing left, use R-1DR to properly position the cubelet.           

b.      If the Up facelet of furo is facing front, use D2FD-1F-1 to properly position the cubelet.

c.      If the Up facelet of furo is facing down, use (R-1DR) (FDF-1) D2 (R-1DR) to properly position the cubelet.

 

iii.      Reorient: (F,L,B,R), and repeat B. until all of the Up corner cubelets are in position.  Note they will be properly oriented.

 

II. Position Down corner cubelets.

  1. Reorient: (U,D) (F,B).
  2. One of three cases will occur:

i.         It is possible using an element of <U> to put only one corner cubelet in its proper position. Note that if two corner cublets are properly positioned and share a face, we are in case (i.).

a.      Properly orient cube.

1.      if the properly positioned corner cubelet is in the ful position, Reorient: (F,L,B,R).

2.      if the properly positioned corner cubelet is in the fur position, Reorient: (R,L)(F,B).

3.      if the properly positioned corner cubelet is in the bur position, Reorient: (F,R,B,L).

4.      if the properly positioned corner cubelet is in the bul position, then continue to next step.

 

b.   Then one or two applications of [FR] U [RF] U-1 (R-1F(U2R2)3F-1R) should properly position the other three corner cubelets.

             

ii.       It is possible to properly position only two corner cubelets on a diagonal.

a.      Properly orient the cube.

1.      If the properly positioned corner cubelets are in the ful and bur positions, then continue to (b.).

2.      If not, Reorient: (F,L,B,R).

 

b.      Next do [FR] U [RF] U-1 (R-1F(U2R2)3F-1R).  Now we are in case (i.) from above. So complete case (i.) to properly position the corner cubelets.

 

iii.  All corner cubelets are in their proper position! Carry on.

 

III. Orient remaining corner cubelets. Reorient: (U,F,D,B). Note that after reorientation the current Front face was originally the Down face.

A.     Either none, 1, 2, or all will be properly oriented.

i.         If none of the  Front corner cubelets are properly oriented:

a.      Find the facelets that should be Front on the fulo and fdlo cubelets.

1.      If both or neither are facing left, then one or two applications of [FR]2 R-1 [RD]2 R will properly orient the fulo and fdlo cubelets.

2.      If not Reorient: (L,U,R,D); then one or two applications of [FR]2 R-1 [RD]2 R will properly orient the fulo and fdlo cubelets.

 

b.      Next Reorient: (R,L) (U,D).

c.      Repeat (a.) then all corner cubelets should be properly oriented in their proper positions.

 

ii.       If one front corner cubelet is properly oriented.

a.      The properly oriented cubelet is in one of the following positions: dfr, dfl, ful, fur.

1.      If in the dfr position, Reorient: (R,U,L,D).

2.      If in the dfl position, Reorient: (R,L) (U,D).

3.      If in the ful position, Reorient: (R,D,L,U).

4.      If in the fru position continue to next step.

 

b.      Then one or two applications of [FR]2 R-1 [RD]2 R will properly orient the fulo cubelet.

c.      Reorient: (U,R,D,L).

d.      Then one or two applications of [FR]2 R-1 [RD]2 R will properly orient the fulo and fdlo cubelets.

 

iii.      If two Front corner cubelets are properly oriented. They either share a side or are on the diagonal.

a.      If they share a side:

1.      Fix the front face and reorient so that the shared side is the right side.

2.      One or two applications of [FR]2 R-1 [RD]2 R will properly orient the fulo and fdlo cubelets.

 

b.      If the two properly oriented cubelets are on a diagonal:

1.      Fix the front face and reorient so that the properly oriented cubelets are in the fur and dfl positions.

2.      Then one or two applications of [FR]2 R-1 [RD]2 R will properly orient the fulo cubelet.

3.      Reorient: (U,R,D,L).

4.      Then one or two applications of [FR]2 R-1 [RD]2 R will properly orient the fulo and fdlo cubelets.

 

IV. Position Middle edge cubelets.  Reorient: (F,D,B,U). Note that after reorientation the Up and Down faces are the original Up and Down faces.

A.     Find fro.

i.         If in Middle layer, fro is in one of the following positions: br, bl, fl, fr.

a.      If fro is in br, use (R2D2)3 to properly position fro.

b.      If fro is in bl, use (R2D2)3 (B2D2)3 (R2D2)3 to properly position fro.

c.      If fro is in fl, use (F2D2)3 to properly position fro.

d.      If fro is in fr, continue to (iii.).

 

ii.       If fro is in Down layer, it is in one of the following positions: df, dr, db, dl.

a.      If fro is in df, use D-1L-1D-1(F2D2)3 DLD to properly position fro.

b.      If fro is in dr, use D2L-1D-1(F2D2)3 DLD2 to properly position fro.

c.      If fro is in db, use DL-1D-1(F2D2)3 DLD-1 to properly position fro.

d.      If fro is in dl, use L-1D-1(F2D2)3 DL to properly position fro.

 

iii.      Reorient: (F,L,B,R).  Repeat (A.) until all of the Middle edge cubelets are properly positioned.

 

V.     Orient Middle edge cubelets.

A.     Either none, one, two, three or all four Middle edge cubelets are properly oriented.

i.         If none are properly oriented, Reorient: (U,L,D,R).

a.      Then do (F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1). Reorient: (F,B) (U,D).

b.      Then do (F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1) to properly orient the remaining middle edge cubelets.

c.      Reorient: (U,L,D,R).

ii.       If one of the middle edge cubelets is properly oriented Reorient the cube.  Fix the Up face and the reorient so that the properly oriented middle edge cubelet is in the bl position.

a.      Reorient: (U,L,D,R).

b.      Do (F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1).

c.      Reorient: (U,L,B) (F,D,R).

d.      Do U-1 [(F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1)] U.

e.      Reorient (F,U,B,D)

iii.      If two of the middle edge cubelets are properly oriented, either the two oriented cubelets share a face or they do not.

a.      If they share a common face Reorient by fixing the Up face, and send the common face to the Back.

1.      Reorient: (U,L,D,R).

2.      Do (F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1) to properly orient the other middle edge cubelets.

3.      Reorient: (U,L,D,R)

 

b.      If they do not share a common face, fix the Up face and reorient so that one of the properly oriented middle edge cubelets is in the fr position.

1.      Reorient: (U,L,D,R).

2.      Do (F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1).

3.      Reorient: (U,F,D,B).

4.      Then do (F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1) again, and now all of the middle edge cubelets should be properly oriented.

5.      Reorient: (U,L,D,R)

 

iv.     If three middle edge cubelets are properly oriented, then fix the Up face and reorient so that the non-oriented middle edge cubelet is in the fl position.

a.      Reorient: (U,B,D,F).

b.      Then do U-1[(F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1)]U to properly orient the last middle edge cubelet.

c.      Reorient: (U,B,D,F).

v.       If all four are properly oriented Reorient: (U,D) (R,L).

 

VI.  Position Up (originally Down) edge cubelets.

A.     Either none, one or all will be in proper position.

i.         If none are in proper position, do (U2R2)3 (B-1UB) (U2R2)3 (B-1U-1B), then one of the edge cubelets will be properly positioned.  Proceed to (ii.).

 

ii.       If one is properly positioned, fix the Up face and reorient so that the properly positioned cubelet is in the ul position.

a.      Then one or two applications of (U2R2)3 (B-1UB) (U2R2)3 (B-1U-1B) will properly position all of the edge cubelets.

 

iii.      If all are properly positioned carry on.

 

VII.   Orient Up (originally Down) edge cubelets. Reorient: (U,F,D,B).

A.     Either none, two or all of the edge cubelets will be properly oriented.

i.         If none:

a.      Do (F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1).

b.      Then do F [(F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1)] F-1 (then the Rubik’s Cube should be solved!).

 

ii.       If two of the edge cubelets are properly oriented, then the two oriented cubelets will either be opposite each other or adjacent to each other.

a.      If the two properly oriented edge cubelets are opposite each other, fix the Front face and reorient so that the non-oriented cubelets are in the fu and fd positions.

1.      Do (F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1) (then the Rubik’s Cube should be solved!).

 

b.      If the two properly oriented edge cubelets are adjacent to each other, fix the Front face and reorient so that they non-oriented cubelets are in the fu and fr positions.

1.      Do R-1D-1 [(F2R2)3 (RD-1RFL-1F) (F2R2)3 (F-1LF-1R-1DR-1)] DR (then the Rubik’s Cube should be solved!).

 

iii.   If all of the edge cubelets are properly oriented then proceed to (VIII.).

 

VIII. Now celebrate and be pleased with yourself!  On the off chance it didn’t work, try again after a little break.