Rubik's Cube/Keeping Track of Details

This is where Chapter 05 has been installed. Some diagrams may have been messed up in translation. I will try to fix them as I convert from Top, Bottom, and bacK to Up, Down, and Back.

Ray Calvin Baker 03:43, 10 November 2011 (UTC)

 +-+ +-+ 
 * 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 resources                      |
 * Chapter Five - - - - - - - - - - Keeping Track of Lots of Details -- Rotating Corner Cubies |
 * Goal Two is to get all of the corner cubies properly oriented in their proper corners of   |
 * the Cube.                                                                                  |
 * We will need to consider more details when we try to orient the corner cubies, without     |
 * messing up where they are placed. For now, we are still ignoring the edge cubies. So, let's |
 * start with a diagram of the corner cubies of Rubik's Cube. (Yes, I have still left out some |
 * details which are not essential for what I am trying to do.) I am using arbitrary letters  |
 * and numbers to identify the labels on the visible faces of each corner cubie. And yes, this |
 * is another chapter where we use pencil and paper more than Cube turning.                   |
 * This diagram is like an isometric drawing, with some lines left out. I tried to emphasize  |
 * the three sides of the visible corner cubies, and I added some notes, "(1)", "(C)", "(5)", |
 * "(G)", "(e)", "(f)", and "(g)" to indicate the hidden labels on several of the corner      |
 * cubies. Later on, I will show you several other types of diagrams that I used; I hope a few |
 * of them will also help you.                                                                |
 * d               a, b, c, and d are on the TOP side.                       |
 * a   TOP    c                                                                    |
 * (1)        b        (C)     A, B, E, and F are on the FRONT side.                     |
 * A              3                                                                  |
 * B  2              2, 3, 6, and 7 are on the RIGHT side.                     |
 * (5) FRONT     RIGHT (G)                                                               |
 * E              7        (Items indicated in parentheses are                       |
 * F  6               actually hidden, around the corner.)                     |
 * (e)        (g)                                                                   |
 * (f)        Position 0                                                      |
 * DIAGRAM 5-1. Diagram for Orienting Corner Cubies -- Position Zero                     |
 * (There is a cubie in the back, which would bear the labels "H", "h", and "8".)             |
 * If we are not yet cubemeisters, we will need to do SOMETHING to get started. I propose that |
 * we explore what happens with two simple moves. As you will see, we may be able to find LOTS |
 * of clever operators with just a little bit of VERY CAREFUL analysis.                       |
 * So, here is our first (arbitary) move, Fv. Rotate the FRONT face clockwise one quarter     |
 * turn.                                                                                      |
 * d                                                                         |
 * 5   TOP    c                                                                    |
 * (e)        1        (C)                                                               |
 * E              3                                                                  |
 * A  a                                                                        |
 * (f) FRONT     RIGHT (G)                                                               |
 * F              7                                                                  |
 * B  b                                                                        |
 * (6)        (g)                                                                   |
 * (2)        Position 1                                                      |
 * DIAGRAM 5-2. What FRONT Clockwise Did                                                 |
 * Our second arbitrary move is R^. Rotate the RIGHT face counterclockwise one quarter turn.  |
 * d                                                                         |
 * 5   TOP    G                                                                    |
 * (e)        C        (g)                                                               |
 * E              7                                                                  |
 * c  3                                                                        |
 * (f) FRONT     RIGHT (2)                                                               |
 * F              b                                                                  |
 * 1  a                                                                        |
 * (6)        (B)                                                                   |
 * (A)        Position 2                                                      |
 * DIAGRAM 5-3. What RIGHT Counterclockwise Did -- Position Two                          |
 * Now let's compare position 0 with position 2.                                              |
 * d                                    d                                    |
 * a   TOP    c                         5    TOP    G                              |
 * (1)        b        (C)              (e)         C        (g)                         |
 * A              3                     E               7                            |
 * B  2                                 c   3                                  |
 * (5) FRONT     RIGHT (G)    Fv R^     (f)  FRONT     RIGHT (2)                         |
 * E              7       >         F               b                            |
 * F  6                                 1   a                                  |
 * (e)        (g)                       (6)         (B)                             |
 * (f)                                  (A)                                   |
 * Position 0                           Position 2                               |
 * DIAGRAM 5-4. Comparing Position Zero With Position Two                                |
 * "A" started on the front side of the TOP LEFT FRONT cubie. "A" ended on the bottom side of |
 * the BOTTOM RIGHT FRONT cubie.                                                              |
 * "f" started on the bottom side of the BOTTOM RIGHT FRONT cubie. "f" ended on the left side |
 * of the BOTTOM LEFT FRONT cubie.                                                            |
 * "5" started on the left side of the BOTTOM LEFT FRONT cubie. "5" ended on the top side of  |
 * the TOP LEFT FRONT cubie.                                                                  |
 * We can summarize these facts like this: A -> f -> 5 -> a -> etc. The entire series is:     |
 * A -> f -> 5 -> a -> 6 -> e -> 1 -> F -> E -> A.                                            |
 * "a" started on the top side of the TOP LEFT FRONT cubie. But this means that if we repeat  |
 * these two moves three times, the TOP LEFT FRONT cube will be rotated counterclockwise by   |
 * 120 degrees. Other things will happen also, so we need to think things through.            |
 * "B" started on the front side of the TOP RIGHT FRONT cubie. "B" ended on the bottom side of |
 * the BOTTOM RIGHT BACK cubie.                                                               |
 * "g" started on the bottom side of the BOTTOM RIGHT BACK cubie. "g" ended on the back side  |
 * of the TOP RIGHT BACK cubie.                                                               |
 * "C" started on the back side of the TOP RIGHT BACK cubie. "C" ended on the top side of the |
 * TOP RIGHT FRONT cubie.                                                                     |
 * "b" started on the top side of the TOP RIGHT FRONT cubie. We can summarize these facts like |
 * this: B -> g -> C -> b -> etc. The entire series is:                                       |
 * B -> g -> C -> b -> 7 -> 3 -> 2 -> G -> c -> B.                                            |
 * Repeating the two moves three times will also rotate the TOP RIGHT FROMT cube clockwise by |
 * 120 degrees.                                                                               |
 * You should be able to verify these sequences:                                              |
 * 3 -> 2 -> G -> c -> B -> g -> C -> b -> 7 -> 3 -> 2, and                                   |
 * E -> A -> f -> 5, F -> E -> A -> f, and 7 -> 3 -> 2 -> G.                                  |
 * Notice that three repetitions of the two moves rotates the TOP RIGHT BACK cubie clockwise  |
 * by 120 degrees.                                                                            |
 * d                                    d                                    |
 * a   TOP    c                         5    TOP    G                              |
 * (1)        b        (C)              (e)         C        (g)                         |
 * A              3                     E               7                            |
 * B  2                                 c   3                                  |
 * (5) FRONT     RIGHT (G)    Fv R^     (f)  FRONT     RIGHT (2)    Fv R^                |
 * E              7       >         F               b       >                |
 * F  6                                 1   a                                  |
 * (e)        (g)                       (6)         (B)                             |
 * (f)                                  (A)                                   |
 * Position 0                           Position 2                               |
 * d                                    d                                    |
 * f   TOP    2                         A    TOP    3                              |
 * (6)        g        (B)              (a) +       B      - (c)                         |
 * F              b                     1       -       C                            |
 * G  7                                 2   b                                  |
 * (A) FRONT     RIGHT (3)    Fv R^     (E)  FRONT     RIGHT (7)                         |
 * 1              C       >         e               g                            |
 * e  5                           +     6   f     -                            |
 * (a)        (c)                       (5)    +    (G)                             |
 * (E)                                  (F)                                   |
 * "After Four Turns"                   "After Six Turns"                            |
 * DIAGRAM 5-5. After Six Turns                                                          |
 * In this last picture, "After Six Turns", I have marked clockwise rotations of corner cubies |
 * with "-" signs, and counterclockwise rotations with "+" signs. (The two cubies in the BACK |
 * LEFT column are not rotated at all. I can use the "0" symbol to indicate this, if I need   |
 * to.)                                                                                       |
 * Hmmm. Three cubies are rotated clockwise, three are rotated counterclockwise, and two are  |
 * not rotated at all. Because we noticed that all cubies are returned to their stating       |
 * locations, we will, hopefully, not have to worry about messing up Goal One. Can we do      |
 * something useful with the information we have learned so far? Yes!                         |
 * Let's try to ignore "irrelevant" details, and concentrate on the patterns of rotation of   |
 * the corner cubies. I make a copy of the picture "After Six Turns" (with slightly different |
 * perspective), unroll it, then remove some details. NOTE: These operations are NOT something |
 * you can actually do on a real Cube -- these are conceptual drawings only, intended to help |
 * us visualize more of the surface of the Cube. We can NOT take a Cube apart in this fashion. |
 * but we CAN make drawings of all six sides.                                                 |
 * TOP               |
 * explodes;           |
 * _ *---* _         |
 * *  |corner |   *        |
 * _ * _                        _ * _                  /| ? |cubies | ? |\\      |
 * _ * _ d _ * _                _ * _ d _ * _             / * _ *stretch* _ * \\     |
 * _ * _ ? _ * _ ? _ * _        _ * _ ? _ * _ ? _ * _        /A |   *---*   | 3\\    |
 * * _ a _ * _TOP_ * _ c _ *    * _ A _ * _TOP_ * _ 3 _ *     * _ | ? |       | ? | _ *    |
 * |  * _ ? _ * _ ? _ *   |     |   * _ ? _ * _ ? _ *   |     |   * _ |   B   | _ *   |    |
 * | A |  * _ b _ *   | 3 |     | 1 |   * _ B _ *   | C |     | 1 |   * _   _ *   | C |    |
 * * _ | ? |  *   | ? | _ *     * _ | ? |   *   | ? | _ *     * _ | ? |   *   | ? | _ *    |
 * |  * _ | B | 2 | _ *   |     |   * _ | 2 | b | _ *   |     |   * _ | 2 | b | _ *   |    |
 * | ? |  * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |    |
 * * _ FRONT  *   RIGHT _ *     * _ FRONT   *   RIGHT _ *     * _ FRONT   *   RIGHT _ *    |
 * |  * _ | ? | ? | _ *   |     |   * _ | ? | ? | _ *   |     |   * _ | ? | ? | _ *   |    |
 * | E |  * _ | _ *   | 7 |     | e |   * _ | _ *   | g |     | e |   * _ | _ *   | g |    |
 * * _ | ? |  *   | ? | _ *     * _ | ? |   *   | ? | _ *     * _ | ? |   *   | ? | _ *    |
 * * _ | F | 6 | _ *            * _ | 6 | f | _ *          \\  * _ | 6 | f | _ *  /    |
 * * _ | _ *                    * _ | _ *               \\ |   * _ | _ *   | /     |
 * *                            *            BOTTOM  \\| ? |   *   | ? |/      |
 * explodes * _ |  F   | _ *        |
 * Position 0                   After Six Moves               Exploding the Cube           |
 * DIAGRAM 5-6A.                DIAGRAM 5-6B.                 DIAGRAM 5-6C.                |
 * RIP!              * | Opening   | *     | *                               * |   |
 * _  _              /| *  the      * |\    *d|\\                            /|d*   |
 * _ * \\ /  * _        * |/|   Cube's  |\| *   |\\| *                          * |/|   |
 * _ *  |   V   |   * _    /|?* |   surface | *?|\  | *?|\\                        /|?* |   |
 * ( _ | ? | _ - _ | ? | _ ) ( _/| *          * |\\_) * |\\|  \\_                  _/ |/| *   |
 * |  * _ *   |   * _ *   | |   *-|           |-*   | |\\| *     * _           _ *    * |/|   |
 * |  |   * _ | _ *   |   | |   |   * _   _ *   |   | | *?|\\ A  |   * _   _ *   | 3 /|?* | | |
 * * _ | ? |  -   | ? | _ * * _ | ? |   -   | ? | _ * *?|\\| * _ | ? |   -   | ? | _* |/|?* | |
 * |  * _ |       | _ *   | |   * _ |       | _ *   | |\\| *a|   * _ |   B   | _ *    |c* |/| |
 * |  |   * _   _ *   |   | |   |   * _   _ *   |   | | *L|\\| 1 |   * _   _ *   | C  |/|K* | |
 * * _ | ? |  *   | ? | _ * * _ | ? |   *   | ? | _ * * |\\|?* _ | ? |   *   | ? | _  * |/| * |
 * |  * _ |   |   | _ *   | |   * _ |   |   | _ *   | |\\| * |   * _ | 2 | b | _ *    |?* |/| |
 * | ? |  * _ | _ *   | ? | |   |   * _ | _ *   |   | | *?|\\| ? |   * _ | _ *   | ?  |/|?* | |
 * * _ FRONT  *   RIGHT _ * * _ FRONT   *   RIGHT _ * * |\\| * _ FRONT   *   RIGHT _  * |/| * |
 * |  * _ | ? | ? | _ *   | |   * _ |   |   | _ *   |  \\| *E|   * _ | ? | ? | _ *    |7* |/  |
 * |  |   * _ | _ *   |   | |   |   * _ | _ *   |   |   *?|\\| e |   * _ | _ *   | g  |/|?*   |
 * * _ | ? |  *   | ? | _ * * _ |   |   *   |   | _ *    \\| * _ | ? |   *   | ? | _  * |/    |
 * |  * _ |   |   | _ *   | |   * _ |   |   | _ *   |      *     * _ | 6 | f | _ *     *      |
 * |  |   * _ | _ *   |   | |   |   * _ | _ *   |   |       \\ 5  |   * _ | _ *   |  G /      |
 * ( _ | ? |  *   | ? | _ ) ( _ |   |   *   |   | _ )         \\_ | ? |   *   | ? | _/        |
 * * _ |      | _ *         * _ |       | _ *                * _ |   F   | _ *            |
 * * _  _ *                 * _   _ *                        * _   _ *                |
 * Starting to Rip Apart     Separating the LEFT        We Are Unwrapping the Cube!           |
 * the Surface of the        Side From the BACK                                               |
 * Cube                      Side                                                             |
 * DIAGRAM 5-6D.             DIAGRAM 5-6E.              DIAGRAM 5-6F.                         |
 * DIAGRAM 5-6. Unwrapping the Cube                                                           |
 * There are several ways to show all six sides of the Cube on one diagram. Each way          |
 * introduces some kind of distortion, but sometimes we can live with that if we can gain more |
 * insight into how the Cube works.                                                           |
 * Examples:                                                                                  |
 * Why not choose a way that emphasizes the details we are interested in? For now, we are     |
 * interested primarily in how the corner cubies can be rotated. So, let's take a diagram of  |
 * an "unwrapped" Cube and ignore the parts we don't need now. Later, we may want to emphasize |
 * some other details, instead.                                                               |
 * +---+---+---+      A completely unwrapped Cube                        |
 * --> |TOP|TOP|TOP|                                                         |
 * +---+---+---+      The distortions in this kind of                    |
 * |TOP|TOP|TOP|      diagram separate parts of some                     |
 * +---+---+---+      cubies, often by a large distance.                 |
 * |TOP|TOP|TOP|                                                         |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          For example,           |
 * --> | L | L | L | F | F | F | R | R | R | K | K | K | <---     the three arrows       |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          point to separated     |
 * | L | L | L | F | F | F | R | R | R | K | K | K |          parts of the           |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          BACK LEFT TOP          |
 * | L | L | L | F | F | F | R | R | R | K | K | K |          corner cubie.          |
 * | B | B | B |                                                         |
 * +---+---+---+      This is a diagram of the Cube                      |
 * | B | B | B |      as we would like to see it                         |
 * +---+---+---+      when we are finished working                       |
 * | B | B | B |      with it.                                           |
 * DIAGRAM 5-7. A Completely Unwrapped Cube                                          |
 * (0)--+---+--(-)     A completely unwrapped Cube                        |
 * --> | 4 | ? | 3 |                                                         |
 * +---+---+---+      The distortions in this kind of                    |
 * | ? |TOP| ? |      diagram separate parts of some                     |
 * +---+---+---+      cubies, often by a large distance.                 |
 * | A | ? | B |                                                         |
 * (0)--+---+--(+)--+---+--(-)--+---+--(-)--+---+--(0)         For example,           |
 * --> | D | ? | a | 1 | ? | 2 | b | ? | C | c | ? | d | <---     the three arrows       |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          point to separated     |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |          parts of the           |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          BACK LEFT TOP          |
 * |  | ? | E | e | ? | 6 | f | ? | g | 7 | ? |   |           corner cubie.          |
 * (0)--+---+--(+)--+---+--(+)--+---+--(-)--+---+--(0)                                |
 * | 5 | ? | F |                                                         |
 * +---+---+---+      This is what diagram 5-6F woulf look like          |
 * | ? | B | ? |      if we finished unwrapping it.                      |
 * |  | ? | G |       I emphasize the rotation of each corner            |
 * (0)--+---+--(-)     cubie with "(+)", "(-)", and "(0)" signs.          |
 * DIAGRAM 5-7. Unwrapped Cube After Six Moves                                       |
 * If we ignore portions of diagram 5-7 which do not pertain to the rotations of corner       |
 * cubies, we can make some simplified diagrams, which may be easier to work with.            |
 * | A | ? | B |                                                         |
 * (0)--+---+--(+)--+---+--(-)--+---+--(-)--+---+--(0)  First, we ignore portions     |
 * | D | ? | a | 1 | ? | 2 | b | ? | C | c | ? | d |   of the TOP and BOTTOM.        |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |   Then, we ignore the lines     |
 * +---+---+---+---+---+---+---+---+---+---+---+---+   and the edge cubies.          |
 * |  | ? | E | e | ? | 6 | f | ? | g | 7 | ? |   |                                  |
 * (0)--+---+--(+)--+---+--(+)--+---+--(-)--+---+--(0)                                |
 * | 5 | ? | F |                                                         |
 * DIAGRAM 5-8A.                                                                      |
 * TOP                       For short, we can draw it like this:             |
 * 0      +      -       -       0                                                       |
 * D    a 1    2 b     C c     d       0 + - - 0                                        |
 * LEFT   FRONT  RIGHT   BACK         0 + + - 0                                        |
 * H    E e    6 f     g 7     h         [F] [K]  <-- These notes serve only to remind  |
 * 0      +      +       -       0                     us which are the FRONT and BACK   |
 * BOTTOM                                     sides of the Cube.                |
 * DIAGRAM 5-8B.                        DIAGRAM 5-8C.                                    |
 * DIAGRAM 5-8. Developing a New Type of Diagram                                         |
 * See if you can figure out the rather peculiar arithmetic of "rotate by 120 degrees".       |
 * "0" & "0" = "0"    "0" & "+" = "+"     "0" & "-" = "-"                                |
 * "+" & "0" = "+"    "+" & "+" = "-"     "+" & "-" = "0"                                |
 * "-" & "0" = "-"    "-" & "+" = "0"     "-" & "-" = "+"                                |
 * DIAGRAM 5-9. The Peculiar Arithmetic of Rotate by 120 Degrees                         |
 * Now I remove even more details from the stuff between "TOP" and "BOTTOM" above, and make   |
 * four copies. (By the way, what we are doing here is "paper and pencil" work, organized in a |
 * systematic fashion. It does not yet require that you do anything with a Cube.)             |
 * 0 + - - 0    0 + - - 0     0 + - - 0     0 + - - 0    (I used "[F]" and "[K]" to      |
 * 0 + + - 0    0 + + - 0     0 + + - 0     0 + + - 0     emphasize how the Cube         |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]     is oriented.)                  |
 * Now combine these (using "&" arithmetic) with rotated copies. (There is some important     |
 * information concerning "how to do this" as it is related to working with an actual Cube in |
 * Chapter Six, "Customize Your Moves -- Commutation". Please don't rush into things just     |
 * yet -- leave your Cube on the table just a little while longer.)                           |
 * 0 + - - 0    - 0 + - -     - - 0 + -     + - - 0 +                                    |
 * 0 + + - 0    - 0 + + -     + - 0 + +     + + - 0 +                                    |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]                                    |
 * ... and the results are:                                                                   |
 * 0 - + + 0    - + 0 + -     - 0 - 0 -     + + + + +                                    |
 * 0 - - + 0    - + - 0 -     + 0 + 0 +     - + - 0 -                                    |
 * [F] [K]      [F] [k]       [F] [K]       [F] [K]                                    |
 * These strange-looking diagrams really are coded solutions for problems involving the       |
 * orientation of the corner cubies of the Cube. Chapter Seven, "Finishing the Orientation of |
 * Corner Cubies", will explain how we can use these patterns, and many others, to get all    |
 * eight corners of the Cube perfectly oriented. But first, let me explain in Chapter Six,    |
 * "Customize Your Moves -- Commutation", how to "customize" any series of moves you can learn |
 * to solve lots and lots and lots of Cube problems.                                          |
 * Diagram 5-5 shows us that we can rotate several corner cubies without changing their       |
 * locations. In Chapter Seven, "Finishing the Orientation of Corner Cubies", we will find    |
 * some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, |
 * we need to understand a very helpful principle that will guide all of our work. This will  |
 * be explained in Chapter Six, "Customize Your Moves -- Commutation".                        |
 * If we ignore portions of diagram 5-7 which do not pertain to the rotations of corner       |
 * cubies, we can make some simplified diagrams, which may be easier to work with.            |
 * | A | ? | B |                                                         |
 * (0)--+---+--(+)--+---+--(-)--+---+--(-)--+---+--(0)  First, we ignore portions     |
 * | D | ? | a | 1 | ? | 2 | b | ? | C | c | ? | d |   of the TOP and BOTTOM.        |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |   Then, we ignore the lines     |
 * +---+---+---+---+---+---+---+---+---+---+---+---+   and the edge cubies.          |
 * |  | ? | E | e | ? | 6 | f | ? | g | 7 | ? |   |                                  |
 * (0)--+---+--(+)--+---+--(+)--+---+--(-)--+---+--(0)                                |
 * | 5 | ? | F |                                                         |
 * DIAGRAM 5-8A.                                                                      |
 * TOP                       For short, we can draw it like this:             |
 * 0      +      -       -       0                                                       |
 * D    a 1    2 b     C c     d       0 + - - 0                                        |
 * LEFT   FRONT  RIGHT   BACK         0 + + - 0                                        |
 * H    E e    6 f     g 7     h         [F] [K]  <-- These notes serve only to remind  |
 * 0      +      +       -       0                     us which are the FRONT and BACK   |
 * BOTTOM                                     sides of the Cube.                |
 * DIAGRAM 5-8B.                        DIAGRAM 5-8C.                                    |
 * DIAGRAM 5-8. Developing a New Type of Diagram                                         |
 * See if you can figure out the rather peculiar arithmetic of "rotate by 120 degrees".       |
 * "0" & "0" = "0"    "0" & "+" = "+"     "0" & "-" = "-"                                |
 * "+" & "0" = "+"    "+" & "+" = "-"     "+" & "-" = "0"                                |
 * "-" & "0" = "-"    "-" & "+" = "0"     "-" & "-" = "+"                                |
 * DIAGRAM 5-9. The Peculiar Arithmetic of Rotate by 120 Degrees                         |
 * Now I remove even more details from the stuff between "TOP" and "BOTTOM" above, and make   |
 * four copies. (By the way, what we are doing here is "paper and pencil" work, organized in a |
 * systematic fashion. It does not yet require that you do anything with a Cube.)             |
 * 0 + - - 0    0 + - - 0     0 + - - 0     0 + - - 0    (I used "[F]" and "[K]" to      |
 * 0 + + - 0    0 + + - 0     0 + + - 0     0 + + - 0     emphasize how the Cube         |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]     is oriented.)                  |
 * Now combine these (using "&" arithmetic) with rotated copies. (There is some important     |
 * information concerning "how to do this" as it is related to working with an actual Cube in |
 * Chapter Six, "Customize Your Moves -- Commutation". Please don't rush into things just     |
 * yet -- leave your Cube on the table just a little while longer.)                           |
 * 0 + - - 0    - 0 + - -     - - 0 + -     + - - 0 +                                    |
 * 0 + + - 0    - 0 + + -     + - 0 + +     + + - 0 +                                    |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]                                    |
 * ... and the results are:                                                                   |
 * 0 - + + 0    - + 0 + -     - 0 - 0 -     + + + + +                                    |
 * 0 - - + 0    - + - 0 -     + 0 + 0 +     - + - 0 -                                    |
 * [F] [K]      [F] [k]       [F] [K]       [F] [K]                                    |
 * These strange-looking diagrams really are coded solutions for problems involving the       |
 * orientation of the corner cubies of the Cube. Chapter Seven, "Finishing the Orientation of |
 * Corner Cubies", will explain how we can use these patterns, and many others, to get all    |
 * eight corners of the Cube perfectly oriented. But first, let me explain in Chapter Six,    |
 * "Customize Your Moves -- Commutation", how to "customize" any series of moves you can learn |
 * to solve lots and lots and lots of Cube problems.                                          |
 * Diagram 5-5 shows us that we can rotate several corner cubies without changing their       |
 * locations. In Chapter Seven, "Finishing the Orientation of Corner Cubies", we will find    |
 * some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, |
 * we need to understand a very helpful principle that will guide all of our work. This will  |
 * be explained in Chapter Six, "Customize Your Moves -- Commutation".                        |
 * 0 - + + 0    - + 0 + -     - 0 - 0 -     + + + + +                                    |
 * 0 - - + 0    - + - 0 -     + 0 + 0 +     - + - 0 -                                    |
 * [F] [K]      [F] [k]       [F] [K]       [F] [K]                                    |
 * These strange-looking diagrams really are coded solutions for problems involving the       |
 * orientation of the corner cubies of the Cube. Chapter Seven, "Finishing the Orientation of |
 * Corner Cubies", will explain how we can use these patterns, and many others, to get all    |
 * eight corners of the Cube perfectly oriented. But first, let me explain in Chapter Six,    |
 * "Customize Your Moves -- Commutation", how to "customize" any series of moves you can learn |
 * to solve lots and lots and lots of Cube problems.                                          |
 * Diagram 5-5 shows us that we can rotate several corner cubies without changing their       |
 * locations. In Chapter Seven, "Finishing the Orientation of Corner Cubies", we will find    |
 * some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, |
 * we need to understand a very helpful principle that will guide all of our work. This will  |
 * be explained in Chapter Six, "Customize Your Moves -- Commutation".                        |
 * some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, |
 * we need to understand a very helpful principle that will guide all of our work. This will  |
 * be explained in Chapter Six, "Customize Your Moves -- Commutation".                        |