Problem 2.2

Algorithm to solve a Rubik’s Cube

Chose a target edge cubelet, call it X.

Step 1—Place The Edge Cubelets in the UP Layer (ignoring orientation):

• If X is in the UP Layer.
• If X is in the proper positions (orientation may be incorrect), chose a different X.
• Apply F^2, R^2, B^2, or L^2 such that the move places X in the BOT Layer (this is the layer underneath the MID Layer).
• Apply D until X is placed underneath its correct position.
• Apply F^2, R^2, B^2, or L^2 to move X into its proper location
• If not all UP edge cubelets are placed, chose another UP edge cubelet, call it X, and repeat step 1.
• If X is in the MID Layer (this is the layer underneath the UP Layer)
• Apply U until an application of L, R, F or B permutes X while keeping the already placed edge cubelets in the UP layer. Call this process Z.

NOTE: this is possible since not all of the UP edge cubelets have been placed.

• Apply L, R, F or B such that it places X in the BOT Layer
• Apply Z^(-1)
• Apply D until X is placed underneath its correct position.
• Apply F^2, R^2, B^2, or L^2 to move X into its proper location
• If not all UP edge cubelets are placed, chose another UP edge cubelet, call it X, and repeat step 1.
• If X is in the BOT Layer (this is the layer underneath the MID Layer)
• Apply D until X is placed underneath its correct position.
• Apply F^2, R^2, B^2, or L^2 to move X into its proper location
• If not all UP edge cubelets are placed, chose another UP edge cubelet, call it X, and repeat step 1.

Step 2—Orient the UP Edge Cubelets

While there is a UP Edge Cubelet with incorrect orientation

• Chose a UP edge cubelet with incorrect orientation, call it X.
• Reorient the cube such that X is facing you
• Apply FR^(-1)D^(-1)RF^(2)
• Continue if there is a UP Edge Cubelet with incorrect orientation

Step 3—Place UP corner cubelets with position and orientation

Chose a target UP corner cubelet, call it X.

• If X is in the UP Layer
• If X is in the correct position with proper orientation chose another X
• Reorient the cube such that X is facing you and on the left
• Apply LDL^(-1)
• X is now in the BOT Layer
• Apply D until X is underneath its correct position
• Apply D^2
• Reorient the cube such that X is facing you and on the left
• NOTE: X is in the (f d l) position and its correct position is (b r u); i.e. they are “opposite”
• If the permutation (f d l, b r u)… places X correctly
• Apply B^(-1)D^2B
• Repeat step 1 if needed
• If the permutation (f d l, u b r) … places X correctly
• Apply RD^2R^(-1)
• Repeat step 1 if needed
• If the permutation (f d l, r u b) … places X correctly
• Apply RD^2R^(-1)
• NOTE: X is in the correct spot, but wrong orientation
• Apply [B^(-1)D^2B][RD^2R^(-1)]
• Repeat Step 3 while more UP edge cubelets must be placed.

Step 4—Place MID Layer Edge Cubelets (ignoring orientation)

Chose a target edge cubelet, call it X

While there is a misplaced MID edge cubelet

• If X is not in the BOT Layer

· If X is in the MID Layer and correct position chose another X

· If X is in the MID Layer, but in the incorrect positions

o Reorient the cube such that X is in the (f r) position

o Apply L^(-1)D^(-1)(F^2D^2)^3DL

o NOTE: Now X is in the BOT Layer

Now X is in the BOT Layer

• Reorient the cube such that X’s correct position is in the (f r) position
• Apply D until X is in the (d l) position
• Apply L^(-1)D^(-1)(F^2D^2)^3DL
• X is now in the correct position, repeat if needed; otherwise go to Step 5

Step 5—Orient MID Layer Edge Cubelets

Chose a MID layer edge cubelet with incorrect orientation and call it X

While there are incorrectly oriented MID edge cubelets

• Orient the cube such that X is in the (f r) position
• Apply L^(-1)D^(-1)(F^2D^2)^3DL
• Apply D^2
• Apply FD^(-1)F^(-1)D^(-1)FD^2F^(-1)DFDF^(-1)D^2
• Repeat if needed; otherwise go to Step 6

Step 6—Place BOT Layer Corner Cubelets (ignoring orientation)

Apply B until at least one BOT corner cubelet is in its correct position

While all the corner cubelets are not in their correct positions

• If there are three corner cubelets not positioned correctly
• NOTE: There is a pair of corner cubelets (that are in the same face) such that swapping those edge cubelets will correctly place one of them.
• Reorient the cube such that those corner cubelets are in the front face
• Apply R^(-1)D^(-1)RFDF^(-1)R^(-1)DRD^2
• NOTE: Now there are only two misplaced corner cubelets
• Repeat Step 6
• If there are two corner cubelets not positioned
• Chose a corner cubelet that has not been placed, call it X
• Reorient the cube such that X is in the (f d l) position
• If the other unplaced corner cubelet is in the (r b d) position
• Reorient the cube such that X is in the (b l d) position
• Apply R^(-1)D^(-1)RFDF^(-1)R^(-1)DRD^2
• Reorient the cube such that X is in the (f d l) position
• Apply R^(-1)D^(-1)RFDF^(-1)R^(-1)DRD^2
• NOTE: Now there are two unplaced corner cubelets that are in the same face.
• Reorient the cube such that those corner cubelets are in the front face
• Apply R^(-1)D^(-1)RFDF^(-1)R^(-1)DRD^2
• Continue to step 7
• Otherwise the two unplaced corner cubelets are in the same face
• Reorient the cube where that face is the front face
• Apply R^(-1)D^(-1)RFDF^(-1)R^(-1)DRD^2
• Continue to step 7

Step 7—Orient the bottom corner cubelets

NOTE: R^(-1)D^(-1)RD^(-1)R^(-1)D^2RD^2 fixes all corner cubelets and all UP edge cubelets while changing the orientation of the (f r d) , ( r b d) and (l b d) corner cubelets

• If none of the BOT corner cubelets have correct orientation
• Apply R^(-1)D^(-1)RD^(-1)R^(-1)D^2RD^2 until at least one cubelet has proper orientation.
• Reorient the cube such that the properly oriented corner cubelet is in the

(f l d) position.

• Continue applying R^(-1)D^(-1)RD^(-1)R^(-1)D^2RD^2 until all BOT corner cubelets have correct orientation.
• Otherwise reorient the cube such that one of the properly oriented BOT corner cubelets in the (f l d) position and apply R^(-1)D^(-1)RD^(-1)R^(-1)D^2RD^2 until all BOT corner cubelets have correct orientation.

Step 8—Place BOT Layer Edge Cubelets (with correct orientation)

Let Z = L^(-1)RF^(-1)LR^(-1)D^2L^(-1)RF^(-1)LR^(-1)

Y = FLB^(-1)L^(-1)BF^(-1)DBD^(-1)B^(-1)

NOTE: Z = (r d , l d , d b)

Y= (r d , b d , d l)

While the cube is not solved

• Choose a misplaced BOT Edge Cubelet
• Reorient the cube if necessary and apply Z or Y such that X is placed correctly with correct orientation

NOTE: Do not simply place X choose Z or Y such that it places X with correct orientation.

• Repeat if necessary

End Algorithm