# Rubik's Cube/Moving Edge Cubies

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+---------------------------------------------------------------------------------------------+ | HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE | | (Preliminary April 20, 2007 version) | | by Mr. Ray Calvin Baker | | FREE Educational Material | | | | Chapter Eight - - - - - - - - - Moving Edge Cubies | | | | Goal Three is to locate each of the twelve edge cubies at the proper edge of the Cube. This | | is going to take some different moves that leave the locations and orientations of the | | corner cubies unchanged. We will be using some two-layer turns that move some of the | | centers of several faces as well. Consequently, we will need some new landmarks. | | | | This is not as difficult as it sounds, because, at this stage, all of the corner cubies | | should be properly located and oriented. Here is what is called the "down-slice" move. | | (It's called a "down-slice" because the three cubies on the FRONT move DOWN.) | | | | _ * _ (1) _ * _ (7) _ * _ (7) | | _ * _ T _ * _ / _ * _ T _ * _ / _ * _ T _ * _ / | | _ * _ ? _ * _ 2 _ * _ _ * _ ? _ * _ 8 _ * _ _ * _ ? _ * _ 8 _ * _ | | * _ T _ * _ T _ * _ T _ * * _ T _ * _ k _ * _ k _ * * _ T _ * _ k _ * _ T _ * | | | * _ 3 _ * _ ? _ * | | * _ 1 _ * _ ? _ * | | * _ 1 _ * _ ? _ * | | | | F | * _ T _ * | R | | F | * _ k _ * | R | | F | * _ T _ * | R | | | * _ | 4 | * | ? | _ * * _ | 2 | * | ? | _ * * _ | 2 | * | ? | _ * | | | * _ | F | R | _ * | | * _ | t | R | _ * | | * _ | F | R | _ * | | | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | | * _ | F | * | R | _ * 2R^ * _ | t | * | R | _ * Rv * _ | t | * | R | _ * | | | * _ | ? | ? | _ * | --> | * _ | ? | ? | _ * | --> | * _ | ? | ? | _ * | | | | F | * _ | _ * | R | | F | * _ | _ * | R | | F | * _ | _ * | R | | | * _ | 5 | * | ? | _ * * _ | 3 | * | ? | _ * * _ | 3 | * | ? | _ * | | * _ | F | R | _ * * _ | t | R | _ * * _ | F | R | _ * | | / * _ | _ * / * _ | _ * / * _ | _ * | | (6) * (4) * (4) * | | | | DIAGRAM 8-1. The Down-Slice Move | | | | Notice the new landmarks in this process -- the LEFT and RIGHT sides, and the BOTTOM, | | FRONT, BACK, and TOP edges of the LEFT layer have all stayed neatly arranged. The change to | | the BOTTOM, FRONT, BACK, and TOP was only temporary. (I used "t" and "k" to indicate those | | faces that started on TOP or on the BACK; notice how they ended up exactly where they | | started.) | | | | Next, I will show you how a simple move (T2) and the "up-slice" can be used together to | | create a very useful three way shuffle of edge cubies. | | | | _ * _ (7) _ * _ (2) | | _ * _ T _ * _ / _ * _ T _ * _ / | | _ * _ ? _ * _ 8 _ * _ _ * _ ? _ * _ 1 _ * _ | | * _ T _ * _ k _ * _ T _ * * _ T _ * _ k _ * _ T _ * | | | * _ 1 _ * _ ? _ * | | * _ 8 _ * _ ? _ * | | | | F | * _ T _ * | R | | k | * _ T _ * | l | | | * _ | 2 | * | ? | _ * * _ | 7 | * | ? | _ * | | | * _ | F | R | _ * | | * _ | k | l | _ * | | | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | | * _ | t | * | R | _ * T2 * _ | t | * | R | _ * | | | * _ | ? | ? | _ * | --> | * _ | ? | ? | _ * | | | | F | * _ | _ * | R | | F | * _ | _ * | R | | | * _ | 3 | * | ? | _ * * _ | 3 | * | ? | _ * | | * _ | F | R | _ * * _ | F | R | _ * | | / * _ | _ * / * _ | _ * | | (4) * (4) * | | | | DIAGRAM 8-2. Turn the TOP Layer Twice | | | | _ * _ (2) _ * _ (8) _ * _ (8) | | _ * _ T _ * _ / _ * _ T _ * _ / _ * _ T _ * _ / | | _ * _ ? _ * _ 1 _ * _ _ * _ ? _ * _ 7 _ * _ _ * _ ? _ * _ 7 _ * _ | | * _ T _ * _ k _ * _ T _ * * _ T _ * _ T _ * _ k _ * * _ T _ * _ T _ * _ T _ * | | | * _ 8 _ * _ ? _ * | | * _ 3 _ * _ ? _ * | | * _ 3 _ * _ ? _ * | | | | k | * _ T _ * | l | | k | * _ f _ * | l | | k | * _ T _ * | l | | | * _ | 7 | * | ? | _ * * _ | 4 | * | ? | _ * * _ | 4 | * | ? | _ * | | | * _ | k | l | _ * | | * _ | b | R | _ * | | * _ | k | l | _ * | | | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | | * _ | t | * | R | _ * 2Rv * _ | F | * | R | _ * R^ * _ | F | * | R | _ * | | | * _ | ? | ? | _ * | --> | * _ | ? | ? | _ * | --> | * _ | ? | ? | _ * | | | | F | * _ | _ * | R | | F | * _ | _ * | l | | F | * _ | _ * | R | | | * _ | 3 | * | ? | _ * * _ | 5 | * | ? | _ * * _ | 5 | * | ? | _ * | | * _ | F | R | _ * * _ | b | R | _ * * _ | F | R | _ * | | / * _ | _ * / * _ | _ * / * _ | _ * | | (4) * (6) * (6) * | | | | DIAGRAM 8-3. The Up-Slice Move | | | | _ * _ (8) _ * _ (4) _ * _ (1) | | _ * _ T _ * _ / _ * _ T _ * _ / _ * _ T _ * _ / | | _ * _ ? _ * _ 7 _ * _ _ * _ ? _ * _ 3 _ * _ _ * _ ? _ * _ 2 _ * _ | | * _ T _ * _ T _ * _ T _ * * _ T _ * _ T _ * _ T _ * * _ T _ * _ T _ * _ T _ * | | | * _ 3 _ * _ ? _ * | | * _ 7 _ * _ ? _ * | | * _ 3 _ * _ ? _ * | | | | k | * _ T _ * | l | | F | * _ T _ * | R | | F | * _ T _ * | R | | | * _ | 4 | * | ? | _ * * _ | 8 | * | ? | _ * * _ | 4 | * | ? | _ * | | | * _ | k | l | _ * | | * _ | F | R | _ * | | * _ | F | R | _ * | | | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | | * _ | F | * | R | _ * T2 * _ | F | * | R | _ * * _ | F | * | R | _ * | | | * _ | ? | ? | _ * | --> | * _ | ? | ? | _ * | | * _ | ? | ? | _ * | | | | F | * _ | _ * | R | | F | * _ | _ * | R | | F | * _ | _ * | R | | | * _ | 5 | * | ? | _ * * _ | 5 | * | ? | _ * * _ | 5 | * | ? | _ * | | * _ | F | R | _ * * _ | F | R | _ * * _ | F | R | _ * | | / * _ | _ * / * _ | _ * / * _ | _ * | | (6) * (6) * (6) * | | Compare this... ... with the starting position. | | | | DIAGRAM 8-4. Exchanging Three Edge Cubies | | | | After doing T2 a second time, all corner cubies have been restored to their places, with | | the correct orientation. The edge cubie marked "5" and "6" is also back where it started. | | But the edge cubie marked "1" and "2" has gone to the BOTTOM BACK location. The edge cubie | | marked "7" and "8" has gone to the FRONT TOP location, and the cubie marked "3" and "4" has | | gone to the BACK TOP location. We have a very useful 3-cycle! | | | | In fact, this three-way shuffle is so simple, easy, and effective that you should memorize | | it (forwards and backwards, if you would like!). | | | | You may use customization to shuffle any three edge cubies, but be careful -- many a Cube | | has gotten re-scrambled because someone forgot (or messed up) step 3 of the customization | | process. You can proceed one edge cubie at a time, until only three edge cubies are out of | | place. (One final three-way shuffle will put these three cubies into their proper places.) | | I suggest that you line up (1) one edge cubie (#1) which is out of place, (2) the edge | | cubie which belongs in the space occupied by edge cubie #1, and (3) any other edge cubie | | which is out of place (#3). Then the three way shuffle (if done in the proper direction) | | will restore one edge cubie (#2) to its proper location. Do not forget step 3 of the | | customizing process! | | | | _ * _ (1) | | _ * _ T _ * _ /...........Locate the edge cubie which belongs | | _ * _ ? _ * _ 2 _ * _ in the BOTTOM BACK location "#2" | | * _ T _ * _ T _ * _ T _ * here at the BACK TOP edge. | | Locate any other ........._ 3 _ * _ ? _ * | | | out-of-place edge | F | * _ T _ * | R | | | cubie "#3" here * _ | 4 | * | ? | _ * | | at the FRONT TOP | * _ | F | R | _ * | | | edge. | ? | * _ | _ * | ? | | | * _ | F | * | R | _ * ..... Locate an out-of-place edge cubie "#1" | | | * _ | ? | ? | _ * | at the BOTTOM BACK edge. | | | F | * _ | _ * | R | (it's hidden in back.) | | The edge cubie * _ | 5 | * | ? | _ * | | at BOTTOM FRONT..........._ | F | R | _ * This is another view of the starting | | gets returned to / * _ | _ * position for shuffling three edge | | this position. (6) * cubies. | | | | DIAGRAM 8-5. Recommended Way to Position Edge Cubies | | | | Let's try to develop another type of diagram to showw all twelve edge cubies. This may make | | "customization" easier to visualize, as we put all twelve edge cubies into their proper | | places. I start with diagram 5-6, but, instead of emphasizing the corner cubies, I | | concentrate on the twelve edge cubies. The result is diagram 8-6. Leaving out a few more | | lines results in diagram 8-7. | | | | * This is a repeat * | | |\ of diagram 5-6F /| | | | * from Chapter Five. * | | | *d|\ /|d* | | | | * * |/| | | | *?| /|?* | | | * |\| \_ _/ |/| * | | |\| * * _ _ * * |/| Exploded TOP Side | | | *?|\ A | * _ _ * | 3 /|?* | | | *?|\| * _ | ? | - | ? | _ * |/|?* vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv | | |\| *a| * _ | B | _ * |c* |/| | d | ? | A | ? | B | | | 3 | ? | d | | | | *L|\| 1 | * _ _ * | C |/|K* | +---+---+---+---+---+---+---+---+---+---+---+---+ | | *\|?* _ | ? | * | ? | _ * |/| * | | ? | a | 1 | ? | 2 | b | ? | C | c | ? | | | | | | \| * | * _ | 2 | b | _ * |?* |/| +---+---+---+---+---+---+---+---+---+---+---+---+ | | | *?|\| ? | * _ | _ * | ? |/|?* | | ? | L | ? | ? FRONT ? | ? RIGHT ? | ? | K | ? | | | * |\| * _ FRONT * RIGHT _ * |/| * +---+---+---+---+---+---+---+---+---+---+---+---+ | | \| *E| * _ | ? | ? | _ * |7* |/ | | ? | E | e | ? | 6 | f | ? | g | 7 | ? | | | | *?|\| e | * _ | _ * | g |/|?* +---+---+---+---+---+---+---+---+---+---+---+---+ | | \| * _ | ? | * | ? | _ * |/ | | ? | 5 | ? | F | ? | G | ? | | | | * * _ | 6 | f | _ * * ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ | | \ 5 | * _ | _ * | G / | | \_ | ? | * | ? | _/ Exploded BOTTOM Side | | * _ | F | _ * | | * _ _ * This time, we will ignore the corner cubies... | | - | | | | We Are Unwrapping the Cube (Again!) | | | | DIAGRAM 8-6A. DIAGRAM 8-6B. | | | | (exploded TOP) | | +----+ +----+ +----+ +----+ | | | lt | | ft | | rt | | kt | | | + --+----+-- + --+----+-- + --+----+-- + --+----+-- + | | | | lt | | | ft | | | rt | | | kt | | | | +----+----+----+----+----+----+----+----+----+----+----+----+----+----+ | | | kl | kl |LEFT| fl | fl |FRONT fr | fr |RIGHT kr | kr |BACK| kl | kl | | | +----+----+----+----+----+----+----+----+----+----+--=-+----+----+----+ | | / | | bl | | | bf | | | br | | | bk | | \ | | Overlap + --+----+-- + --+----+-- + --+----+-- + --+----+-- + Note | | | bl | | bf | | br | | bk | the | | +----+ +----+ +----+ +----+ overlap. | | (exploded BOTTOM) | | | | DIAGRAM 8-6C. ... and emphasize the edge cubies. | | | | DIAGRAM 8-6. A New Type of Diagram Emphasizes the Edge Cubies | | | | Eventually, you will line up the last three corner cubies, shuffle them into place, and | | undo the moves you made to line up those three edge cubies for the shuffle. That should | | complete the accomplishment of goal three for you! (Yes, it is possible that all twelve | | edge cubies were already in their correct locations. You don't need to use Chapter Eight at | | all when that happens!) | | | | For the sake of completeness, here is a discussion of a way to position the three cubies | | (#1,#2, and #3) as indicated in diagram 8-5. I am going to assume that you can identify one | | edge cubie (#1) which is not in the correct location, then rotate the entire Cube until #1 | | is positioned at the BOTTOM BACK edge of the Cube. You may want to keep careful notes as | | you proceed. You will need to remember how to undo both parts of this two-part | | customization process. | | | | After we have rotated the Cube to get the #1 edge cubie into the BOTTOM BACK | | position, there are eleven possible places to find edge cubie #2. | | | | Part 1 Case 1. +----+ +----+ +----+ +----+ | | | #2 | | | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Move: | | +----+----+----+----+----+----+----+----+----+ "Tv". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 2. +----+ +----+ +----+ +----+ | | | | | #2 | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Move: | | +----+----+----+----+----+----+----+----+----+ "T2". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 3. +----+ +----+ +----+ +----+ | | | | | | | #2 | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Move: | | +----+----+----+----+----+----+----+----+----+ "T^". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 4. +----+ +----+ +----+ +----+ | | | | | | | | | #2 |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ none needed for part one. | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 5. +----+ +----+ +----+ +----+ | | | | | | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | #2 | L | | F | | R | | K | #2 | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "Lv Tv". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 6. +----+ +----+ +----+ +----+ | | | | | | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | #2 | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "L^ Tv". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 7. +----+ +----+ +----+ +----+ | | | | | | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | #2 | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "F^ T2". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 8. +----+ +----+ +----+ +----+ | | | | | | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | #2 | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "R^ T^". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 9. +----+ +----+ +----+ +----+ | | | | | | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "L2 Tv". | | | #2 | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 10. +----+ +----+ +----+ +----+ | | | | | | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "F2 T2". | | | | | #2 | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | Part 1 Case 11. +----+ +----+ +----+ +----+ | | | | | | | | | |<--This is where we want #2. | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "R2 T^". | | | | | | | #2 | | #1 | | | +----+ +----+ +----+ +----+ | | | | DIAGRAM 8-7. Positioning Cubie #2 Without Disturbing #1 | | | | Now we need to position any other out-of-place edge cubie (#3), without disturbing either | | edge cubie #1 or edge cubie #2. This is the second part of the two-part customization | | process. You should be able to find an out-of-place edge cubie (#3) in one of the ten | | locations shown in diagram 8-8. | | | | After we have rotated the Cube to get the #1 edge cubie into the BOTTOM BACK | | position, and we have done Part 1 customizing moves to get the #2 edge cubie | | into the BACK TOP position, there are ten possible places to find edge cubie #3. | | | | __ This is where we want #3. | | Part 2 Case 1. / | | +----+ +----+ +----+ +----+ | | | #3 | | | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "Lv Fv". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 2. / | | +----+ +----+ +----+ +----+ | | | | | #3 | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ none needed for part two. | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 3. / | | +----+ +----+ +----+ +----+ | | | | | | | #3 | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "R^ F^". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 4. / | | +----+ +----+ +----+ +----+ | | | | | | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | #3 | L | | F | | R | | K | #3 | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "L2 Fv". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 5. / | | +----+ +----+ +----+ +----+ | | | | | | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | #3 | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "Fv". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 6. / | | +----+ +----+ +----+ +----+ | | | | | | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | #3 | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "F^". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 7. / | | +----+ +----+ +----+ +----+ | | | | | | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | #3 | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "R2 F^". | | | | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 8. / | | +----+ +----+ +----+ +----+ | | | | | | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "L^ Fv". | | | #3 | | | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 9. / | | +----+ +----+ +----+ +----+ | | | | | | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "F2". | | | | | #3 | | | | #1 | | | +----+ +----+ +----+ +----+ | | | | __ This is where we want #3. | | Part 2 Case 10. / | | +----+ +----+ +----+ +----+ | | | | | | | | | #2 | | | +----+----+----+----+----+----+----+----+----+ | | | | L | | F | | R | | K | | Customizing Moves: | | +----+----+----+----+----+----+----+----+----+ "Rv F^". | | | | | | | #3 | | #1 | | | +----+ +----+ +----+ +----+ | | | | DIAGRAM 8-8. Positioning Cubie #3 Without Disturbing #2 Or #1 | | | | When edge cubies #1, #2, and #3 are in position, do the three-way shuffle of edge cubies, | | just as we developed it earlier in this chapter. Then undo the customization process you | | used to get the three edge cubies into position. Be sure to undo part two (the part which | | positioned edge cubie #3) first, then undo part one (the part which positioned edge | | cubie #2) second. | | | | Let's just check to see if you really understand how to "undo" the "customization" process | | whenn there are two parts to it, such as we have in this chapter. Here is an example. | | Suppose, after manipulating the entire Cube to get edge cubie #1 into position, we find | | that "Part 1 Case 11" is our situation, according to diagram 8-7. We apply the customizing | | moves, "R2 T^". Then suppose we find that "Part 2 Case 10" is our situation for positioning | | cubie #3. The customizing moves are "Rv F^", so we make these moves. Having gotten all | | three edge cubies where we want them, we do the three-way shuffle, "Down-slice, TOP twice, | | Up-slice, TOP twice", or "2R^ Rv T2 2R^ Rv T2". How do we "undo" the customization? | | | | Answer for the example: the entire customization, both part one and part two, is this: | | "R2 T^ Rv F^". So, to undo this. the moves are: "Fv R^ Tv R2". | | | | Common sense should allow you to correctly position all twelve edge cubies, using all of | | the directions which I have given you in this chapter. However, you will need to be | | careful, especially in selecting ways to "customize", and in "undoing" your customizations. | | | | Careful attention to the details of the 3-way shuffle may help you orient all edge cubies | | as you go along, but Chapter Nine, "Rubik's Maneuver -- How to Flip Two Edge Cubies", is | | provided just in case some edge cubies do not have the correct orientation. | | | # If you suceeded in getting all twelve edge cubies properly located, there are only # # ((2 to the 12th power) / 2) = 4,096 / 2 = 2,046 ways these edge cubies can be oriented. We # # are closing in on the final, complete solution -- all siz sides -- of the Cube! # | | +---------------------------------------------------------------------------------------------+