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Rubik's Cube/Keeping Track of Details

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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".                         |
|                                                                                             |
+---------------------------------------------------------------------------------------------+