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Rubik's Cube/Using Pictures, Diagrams, Notation, and Abbreviations

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Ray Calvin Baker 19:28, 18 August 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 Public Domain Educational Material          |
|                                                                                             |
| Chapter Two  - - - - - - - - - - Using Pictures, Diagrams, Notation, and Abbreviations      |
|                                                                                             |
| I am using a monospaced font on my word processor in order to keep more control of the      |
| spacing and the formatting of the characters. When I was young, I used to enjoy trying to   |
| make pictures on a typewriter. So, let's see if I can make a picture of Rubik's Cube. (The  |
| limitations of a typewriter sometimes mean that I can't quite draw a line segment exactly   |
| the way I would like to. You may trace my drawings with a pencil, making those line         |
| segments exactly correct, if you want to.)                                                  |
|                                                                                             |
|                                    __ * __                The BACK side                     |
|     The LEFT side            __ --         -- __              is hidden                     |
|     is hidden           __ * __               __ * __              here                     |
|        here ...   __ --         -- __   __ --         -- __     ...                         |
|              __ * __               __ * __               __ * __               For your     |
|        __ --         -- __   __ --   TOP   -- __   __ --         -- __         information, |
|      * __               __ * __      side     __ * __               __ *       this is an   |
|      |    -- __   __ --         -- __   __ --         -- __   __ --    |       "isometric"  |
|      |          * __               __ * __               __ *          |       diagram.     |
|      |          |    -- __   __ -- This is -- __   __ --    |          |                    |
|      |          |          * __the FRT position_ * This is  |          |       I will say   |
|      |          |          |    -- __   __ --    | a place  |          |       more about   |
|      * __       |          |          *          | for an   |       __ *       what that    |
|      |    -- __ |          |          |          | edge     | __ --    |       means later, |
|      |          * __       |          |          | cubie __ *    This  |       on diagram   |
|      |          |    -- __ |          |          | __ --    | is the   |       2-21.        |
|      |          |          * __       |       __ *          | KR       |                    |
|      |          |          |    -- __ | __ --    |          | position |                    |
|      * __       |  FRONT   |          *          |  RIGHT   |       __ *       For now,     |
|      |    -- __ |  side    |          |          |  side    | __ --    |       it is just   |
|      |          * __       |          |          |       __ *          |       a way to     |
|      |          |    -- __ |          |          | __ --    |          |       "picture"    |
|      |          |          * __       |       __ *          |          |       a 3-D object |
|      |          |          |    -- __ | __ --    |          |          |       on a 2-D     |
|      * __       |          | This is  *          |          |       __ *       surface.     |
|           -- __ |          | a place  |          |          | __ --                         |
|    Diagram      * __       | for a    |          |       __ *                               |
|    of an intact      -- __ | corner   |          | __ --            The                     |
|    Rubik's Cube,           * __ cubie |       __ *           BOTTOM side                    |
|    showing how some parts       -- __ | __ --    ... is hidden under here                   |
|    are named.                         *                      somewhere                      |
|                                                                                             |
|    DIAGRAM 2-1. An Intact Rubik's Cube, Showing How Some Parts Are Named                    |
|                                                                                             |
| Notice especially the parts of the Cube labelled, "TOP side", "FRONT side", and "RIGHT      |
| side". These will be your "fixed landmarks", at least until you turn the entire Cube, to    |
| expose some "new" sides.                                                                    |
|                                                                                             |
| Yes, it is an awkward thing that at least three faces of the Cube are always hidden, when-  |
| ever I try to make an accurate picture. Even if I try to make a drawing with parts of the   |
| Cube removed so that we can see "the innards", I still can't show everything all at once.   |
| However, I can still try to make conceptual DIAGRAMS that distort some details so that I    |
| can emphasize others. You will see some of these diagrams in later chapters.                |
|                                                                                             |
|                                    __ * __               __ * __                            |
|                              __ -- The TOP -- __   __ -- Corner  -- __                      |
|                            * __ side rotates  __ * _cubies have studs_ *                    |
|                            |    -- __ on__ --    |    -- __that_ --    |                    |
|                         __ * __   the * axle. __ * __       *fit into  |                    |
|                   __ --      __ -- __ | __ --____    \      |grooves   |                    |
|              __ * __   __ --    | |   *   | |    \    \     |in the    |                    |
|        __ --  Edge   *    __* __| |  Axle | |     \    |    |edge      |                    |
|      * __cubies fit  |___/       -|       | |      |   |    |cubies.__ *                    |
|      |    -- __   __ * __|   __ --    |     * _    |__ * __ | __ --    |                    |
|      | between  * __      -- __     /The\    _- --       __ *          |                    |
|      | the core |    -- __   __ *  Central  * __   __ --    |          |                    |
|      | and the  |          *    | Core looks|    *          |          |                    |
|      | corner   |          |    |__ -   - __|    |          |          |                    |
|      * __cubies.|          |    | | like a| |    |          |       __ *                    |
|      |    -- __ |          |    | |_jack._| |    |          | __ --    |                    |
|      |          * __       |    | -- --- -- |    |       __ *          |                    |
|      |          |    -- __ | __ * __ ___ __ * __ | __ --    |          |                    |
|      |          |          *    |           |    *          |          |                    |
|      |          |          |    |           |    |          |          |                    |
|      * __       |          |     \ __ * __ /     |          |       __ *                    |
|           -- __ |          |          |          |          | __ --                         |
|    Diagram of   * __       |       __ * __       |       __ *                               |
|    a partially       -- __ | __ --         -- __ | __ --                                    |
|    disassembled Cube       *        .            *                                          |
|                                                                                             |
|    Another view of some parts of a Cube which are usually hidden...                         |
|                                                                                             |
|    DIAGRAM 2-2. Diagram of a Partially Disassembled Cube                                    |
|                                                                                             |
|                  +-----------+                                    Another                   |
|                  |    TOP    |                                    perspective               |
|                  +-----------+                                    of a                      |
|                      |   |                                        partially                 |
|                      |   | ... another axle                       disassembled              |
|      +---+           |   |                                        Cube                      |
|      | L |       +-----------+-----------+                        and some                  |
|      | E |       |           |           |                        of its                    |
|      | F |-------|   FRONT   | FR edge   |                        component                 |
|      | T | Axle  |   side    | cubie     |                        Cubies                    |
|      |   |-------|           | in place  |                                                  |
|      +---+       |           |           |                        Frontal                   |
|      |   |       +-----------+-----------+                        View                      |
|      +---+           |   |   |           |                                                  |
|        :             |   |   | BFR       |                                                  |
|        :             |   |   | corner    |                                                  |
|      Rotated     +-----------| cubie     |                                                  |
|      slightly,   |  BOTTOM   | in place  |                                                  |
|      to show     +-----------+-----------+                                                  |
|      how the                                                                                |
|      center                                                                                 |
|      square         +-----------+                                    +-----+-----+          |
|      rotates on     |           |                Looking at the      |     |     |          |
|      its axle.      |           |                BACK RIGHT edge +-----------+   |          |
|                     |           |                cubie from the  |           |   |          |
|                 +-- _           |                FRONT           +-----------+   |          |
|                 |    \          |                                    |     |     |          |
|                 | Stud \--------+                                    +-----+-----+          |
|                 |       |                                                                   |
|                 +-------+                                +-----------+-----------+          |
|                                           Looking at the |||||||||||||           |          |
|             Looking at the                RIGHT TOP edge +-----------|           |          |
|             BACK RIGHT TOP                cubie from the    This +- _|           |          |
|             corner cubie                  FRONT.            This |    -_         |          |
|             from the FRONT.                                 edge |      \        |          |
|                                           Shaded parts      rubs |        \------+          |
|                                           show how the   against.|         | |///|          |
|                                           axles and      an axle.+---------+ |///|          |
|                                           face plates             This edge  |///|          |
|                                           clamp an                also rubs  |///|          |
|                                           edge cubie        against an axle, |///|          |
|                                           in place                           +---+          |
|                                                                                             |
|      DIAGRAM 2-3. Another Perspective of a Partially Disassembled Cube                      |
|                                                                                             |
| If you are desperate to see for yourself what the "innards" of the Cube really look like,   |
| or if you want to make your Cube look like it is fresh out of the store, here is a way to   |
| physically disassemble your Cube. (I call this "the Physicist's method".) It is much less   |
| damaging to your Cube than the Chemist's method, but it may make your Cube weak, wobbly,    |
| and stretched out of shape if you try this very often.                                      |
|                                                                                             |
| Turn the top layer of the Cube 45 degrees. Slip a butter knife (NOT a sharp knife!) between |
| the bottom edge of an edge cubie on the top layer and the edge cubie it rests upon. Pry up  |
| gently, and the cubie above the butter knife should pop out. Two corner cubies can then be  |
| removed from the top layer, and further disassembly should be easy, simple, and obvious.    |
|                                                                                             |
| To reassemble your Cube, follow these steps in reverse order (although you will not need to |
| use a butter knife blade to re-insert the final cubie). Use common sense to restore your    |
| Cube to its (almost) pristine condition, or be bold and explore one of the eleven other     |
| ways to re-assemble the Cube. (However, if you try one of these eleven alternative orbits,  |
| you will never be able to solve your Cube using only mathematical moves, until you          |
| disassemble the Cube and put it back together "properly".)                                  |
|                                                                                             |
|        1. The    _______________________________________________                            |
|        TOP      |               |               |               |                           |
|    /   layer    |               |               |               |                           |
|   /    has been |_______________|_______________|_______________|                           |
|   |    turned   |               |     TOP       | Pry this      |                           |
|   |    45       |               |     side      | cubie out.    |                           |
|   \    degrees  |_______________|_______________|_______________|                           |
|    \ _          |               |               |               |       ___________________ |
|        -->      |               |               |               |  <---/                    |
|                _|_______________|_______________|_______________|_    / 2. Slide the blade  |
|          __ --  |               |               |               |  --|  of a butter knife   |
|        * __     |               | FRONT side of |               |    |  through here,       |
|        |    -- _|               | FRONT TOP     |               |_ -- \ then pry up.        |
|        |        |               | edge cubie    |               |      \___________________ |
|        |        |               |               |               |        |                  |
|        |        |_______________|_______________|_______________|        |                  |
|        |          |   FRONT  |    -- __   __ --    | RIGHT    |          |                  |
|        * __       |   side   |          *          | side     |       __ *                  |
|        |    -- __ |   center |          |          | center   | __ --    |                  |
|        |          * __cubie  |          |          | cubie __ *          |                  |
|        |          |    -- __ |          |          | __ --    |          |                  |
|        |          |          * __       |       __ *          |          |                  |
|        |          |          |    -- __ | __ --    |          |          |                  |
|        * __       |          |          *          |          |       __ *                  |
|             -- __ |          |          |          |          | __ --                       |
|                   * __       |          |          |       __ *                             |
|                        -- __ |          |          | __ --                                  |
|                              * __       |       __ *                                        |
|                                   -- __ | __ --                                             |
|                                         *                                                   |
|                                                                                             |
|   DIAGRAM 2-4. The Physicist's Method -- How to Disassemble the Cube                        |
|                                                                                             |
|                                           _                                                 |                                   
|                                    __ ---  |    Prying out the                              |  
|         +-----------+-----------+-----------+   TOP RIGHT                                   |
|         |           | FRONT     |           |   edge cubie ...                              |
|         |           | side of   |           |                                               |
|         |           | FRONT TOP |           ||                                              |
|         |           | edge      |           |_|                                             |
|         |           | cubie     |           |-- __          ... using the flat, dull        |
|   +-------+-------+-------+-------+-------+-------+-- __        blade of a                  |
|   |       |       |       |       |       |       |      -- __  butter knife.               |
|   |       | FRONT |       |       | RIGHT |       |            -- __   __                   |
|   |       | side  |       |       | side  |       |                  -/   -- __             |
|   |       | center|       |       | center|       |                   \ _       -- __       |
|   |       | cubie |       |       | cubie |       |                       -- __       --    |
|   +-------+-------+-------+-------+-------+-------+                             -- __       |
|   |       |       |       |       |       |       |                                   --    |
|   |       |       |       |       |       |       |                                         |
|   |       |       |       |       |       |       |                                         |
|   |       |       |       |       |       |       |   Another view -- how to                |
|   |       |       |       |       |       |       |   disassemble the Cube                  |
|   +-------+-------+-------+-------+-------+-------+                                         |
|                                                                                             |
|   DIAGRAM 2-5. Another View -- How to Disassemble the Cube                                  |
|                                                                                             |
| It may be useful to have some standard abbreviations. I propose these:                      |
|                                                                                             |
|    For indicating locations and cubies:            For indicating colors:                   |
|       use B for BOTTOM                                use b for BLUE                        |
|       use F for FRONT                                 use g for GREEN                       |
|       use K for BACK                                  use k for BLACK                       |
|       use L for LEFT                                  use o for ORANGE                      |
|       use R for RIGHT                                 use p for PURPLE                      |
|       use T for TOP                                   use r for RED                         |
|                                                       use w for WHITE                       |
|                                                       use y for YELLOW                      |
|                                                                                             |
| To avoid confusion with "B" for "BOTTOM", I have used "K" for "BACK". To avoid confusion    |
| with "b" for "blue", I have used "k" for "black". "BACK" is the only key word which         |
| contains the letter "K", and "black" is the only key word which contains the letter "k".    |
|                                                                                             |
| I have arranged these key words alphabetically. I will try to be fairly consistent in how I |
| use these words and their abbreviations. (Yes, there are only six colors on any one         |
| individual good Cube. But different Cubes may have different colors, so we may need some    |
| extra abbreviations for the extra colors.) Since I cannot see your Cube as I write this     |
| book, I will not be able to use the lower case letters as abbreviations for the colors on   |
| your Cube.                                                                                  |
|                                                                                             |
| It is often important to distinguish between locations on a Cube, independently of the      |
| cubie which occupies that location. It makes sense to say, "The orange, purple, and yellow  |
| (opy) cubie is at the FRONT RIGHT TOP (FRT) location. I want to move that cubie to the      |
| FRONT RIGHT BOTTOM (FRB) location." Of course, you must move a lot of other cubies as well  |
| when you move the "opy" cubie.                                                              |
|                                                                                             |
| I will try to use CAPITAL LETTERS to abbreviate locations, and lower case letters to        |
| abbreviate colors. Actually, since I don't know what colors your cube is, I won't be        |
| referring to colors much at all. But feel free to use colors in your personal notes. It's   |
| much easier and more reliable than trying always to refer to the position and orientation   |
| of each cubie, as I must do as I write this book.                                           |
|                                                                                             |
| Corner locations can be uniquely identified by using three of the key words, for example,   |
| BACK FRONT TOP. Edge locations can be uniquely identified by using two of the key words,    |
| for eample BOTTOM LEFT, A face can be uniquely determined by using only one of the key      |
| words, for example, BOTTOM. By the way, you should remember that the face cubie -- the      |
| central square of any side -- can only rotate in its place, so it can often be used as a    |
| "landmark" to identify a face of the Cube.                                                  |
|                                                                                             |
| I sometimes use letters and numbers as arbitrary symbols to identify the various faces of   |
| cubies when I try to describe things to do, such as interchanging two corner cubies. (These |
| are often called OPERATORS or OPERATIONS.)                                                  |
|                                                                                             |
| You probably know that the simplest operations which can be done on a Rubik's Cube are to   |
| turn one layer of the cube 90 degrees. (Turning the entire Cube is sometimes done for       |
| special reasons, but that particular operation does not change the essential position or    |
| orientation of the cubies with respect to each other.)                                      |
|                                                                                             |
| Suppose we have an arbitrarily labeled Cube. (You may use small Post-It notes if you want   |
| to label your Cube this way. And you can peel off Post-It notes easily, and reuse them.)    |
|                                                                                             |
| I am going to depart from alphabetic sequence to show you "FRONT" before I show you         |
| "BOTTOM". If you hold your Cube so that it looks somewhat like the diagram, you can see the |
| "FRONT", but the "BOTTOM" is hidden. It helps to be able to see clearly what you are going  |
| to do, especially when you are learning to do something new (and possibly puzzling).        |
|                                                                                             |
| We can rotate the FRONT face clockwise, an operation abbreviated "Fv", and pronounced,      |
| "FRONT clockwise". We can rotate the FRONT face counterclockwise, abbreviated "F^", and     |
| pronounced, "FRONT counterclockwise". You may want to draw a curved arrow around the symbol |
| for the face of the Cube to emphasize that these notations mean "rotate clockwise" and      |
| "rotate counterclockwise".                                                                  |
|                                                                                             |
|      _ a _                _ a _                _ a _     These are supposed to be           |
|    b _   _ c      )     e _   _ c            b _   _ c   tiny little isometric              |
|    | F d   |     Fv     | F b   |     F^     | F d   |   drawings of a Cube, with           |
|    e _ | _ f            g _ | _ f      )     e _ | _ f   the corner cubies labeled          |
|        g                    d                    g       with lower case letters.           |
|   "before Fv"          "after Fv"                                                           |
|                        "before F^"          "after F^"   I can't draw the curved            |
|                                                          arrows very well with a            |
|                                                          typewriter, so I will              |
|                                                          usually just type "Fv" or "F^".    |
|                                                                                             |
|   DIAGRAM 2-6. FRONT Clockwise Move and FRONT Counterclockwise Move                         |
|                                                                                             |
| Since we can usually see the FRONT side, there should be no confusion here.                 |
|                                                                                             |
| I want you to think of "v" and "^" as little arrows that show you how to rotate a face of   |
| the Cube -- either clockwise or counterclockwise. This should work nicely when you can see  |
| the side of the Cube you want to turn.                                                      |
|                                                                                             |
| I want to show you another simple move -- rotating a face of the Cube by 180 degrees. You   |
| can do this to any face of the Cube. It doesn't matter whether you turn 180 degrees         |
| clockwise or counterclockwise -- the result is the same!                                    |
|                                                                                             |
|      _ a _                 _ a _                     _ a _                                  |
|    b _   _ c             e _   _ c                 g _   _ c                                |
|    | F d   |     Fv      | F b   |       Fv        | F e   |                                |
|    e _ | _ f             g _ | _ f     (again)     d _ | _ f                                |
|        g         -->         d           -->           b                                    |
|                                                                 Either way,                 |
|      _ a _                                           _ a _      clockwise or                |
|    b _   _ c                                       g _   _ c    counterclockwise,           |
|    | F d   |                 F2                    | F e   |    the results                 |
|    e _ | _ f        (Think, "FRONT twice".)        d _ | _ f    are exactly                 |
|        g        ------------------------------>        b        the same!                   |
|                                                                                             |
|      _ a _                 _ a _                     _ a _                                  |
|    b _   _ c             d _   _ c                 g _   _ c                                |
|    | F d   |     F^      | F g   |       F^        | F e   |                                |
|    e _ | _ f             b _ | _ f     (again)     d _ | _ f                                |
|        g         -->         e           -->           b                                    |
|                                                                                             |
|    DIAGRAM 2-7. The FRONT Twice Move                                                        |
|                                                                                             |
| I abbreviate this operation on the FRONT of the Cube as "F2" (think of this move as "FRONT  |
| twice".). If you turn the BOTTOM, BACK, LEFT, RIGHT, or TOP, the abbreviations are "B2",    |
| "K2", "L2", "R2", and "T2". (You can guess that I would call these, "BOTTOM twice", "BACK   |
| twice", "LEFT twice", "RIGHT twice", and "TOP twice".)                                      |
|                                                                                             |
|---------------------------------------------------------------------------------------------|
|                                                                                             |
| If the face of the Cube is hidden (like the "BOOTOM", "BACK", and "LEFT" sides), things can |
| seem somewhat confusing. So, I intend to introduce a different notation to show how to move |
| the parts of the Cube you CAN see. If you feel you must use this alternative notation, be   |
| sure to draw it differently from the symbols "Bv" and "B^", which indicate "rotate          |
| clockwise (when looking at the face)" and "rotate counterclockwise (when looking at the     |
| face)". I use straight line segments with arrow heads instead of curved arrows, so I can    |
| tell the difference.                                                                        |
|                                                                                             |
| We can rotate the BOTTOM face clockwise, an operation abbreviated "Bv", and pronounced as   |
| "BOTTOM clockwise".. The alternate notation for this is:                                    |
|                                                                                             |
|                             )                                                               |
|        _ a _               Bv              _ a _                                            |
|      b _   _ c             or            b _   _ c       (The BOTTOM                        |
|      |   d   |                           |   d   |        face is                           |
|      e _ | _ f     - _     B   _ - >     h _ | _ g        usually                           |
|          g             - >   -               e            hidden.)                          |
|       "Before"                            "After"                                           |
|                                                                                             |
|      DIAGRAM 2-8. BOTTOM Clockwise Move                                                     |
|                                                                                             |
| This is messy to draw on a typewriter, though I hope it suggests to you how the "BOTTOM"    |
| cubies move. I hope you can learn to live with the "Bv" notation.                           |
|                                                                                             |
| We can rotate the BOTTOM face counterclockwise, abbreviated "B^". (I call this "BOTTOM      |
| counterclockwise",) The alternate notation for this is:                                     |
|                                                                                             |
|        _ a _               B^              _ a _                                            |
|      b _   _ c             or            b _   _ c       (The BOTTOM                        |
|      |   d   |                           |   d   |        face is                           |
|      e _ | _ f     < - _   B     _ -     g _ | _ h        usually                           |
|          g               -   < -             f            hidden.)                          |
|       "Before"                            "After"                                           |
|                                                                                             |
|   DIAGRAM 2-9. BOTTOM Counterclockwise Move                                                 |
|                                                                                             |
| You may have noticed that, since the BOTTOM face is usually hidden, we are looking at it    |
| from the "wrong" side. You may turn the Cube in your hand temporarily to "peek" and see     |
| that we are turning the BOTTOM layer the way we say, when we are actually looking at that   |
| layer from the "correct" direction.                                                         |
|                                                                                             |
|        _ a _            _ a _                         _ a _       Every operation has       |
|      b _   _ c        b _   _ c    Note that        b _   _ c     an "inverse"              |
|      |   d   |   Bv   |   d   |       B^            |   d   |     operation which           |
|      e _ | _ f        h _ | _ g    restores         e _ | _ f     "undoes" it; the          |
|          g                e        the Cube to          g         combination               |
|                                    its starting                   Bv -> B^                  |
|                                    position.                      is an example of a        |
|                                                                   "do nothing" identity     |
|                                                                   operation.                |
|                                                                                             |
|   DIAGRAM 2-10. BOTTOM Clockwise and BOTTOM Counterclockwise Are Invers Operations          |
|                                                                                             |
| OK. You got confused. I'll try again to show you the BOTTOM side, turning it first          |
| clockwise, then counterclockwise.                                                           |
|                                                                                             |
|          _ a _       Rotate the         _ d _                 _ d _             _ d _       |
|        b _ T _ c     entire Cube      b F | R c             b F | R c         b F | R c     |
|     ---| F d R |---  to peek       ---| _ g _ |---    Bv    | _ e _ |    B^   | _ g _ |     |
|        e _ | _ f     at the           e _ B _ f             h _ B _ g         e _ B _ f     |
|            g         BOTTOM side          h                     f                 h         |
|                                                                                             |
|     DIAGRAM 2-11. Peeking at the BOTTOM Side to See Bv and B^ More Clearly                  |
|                                                                                             |
| Don't forget to rotate the entire Cube back to its original position, so you can see the    |
| TOP side, the FRONT side, and the RIGHT side, as I will usually draw them.                  |
|                                                                                             |
|---------------------------------------------------------------------------------------------|
|                                                                                             |
| We can rotate the BACK face clockwise, an operation abbreviated "Kv". (Think, "BACK         |
| clockwise"!) We can rotate the BACK face counterclockwise, abbreviated "K^". (Think, "BACK  |
| counterclockwise".) Again, I will also ehow the alternate notation for this:                |
|                                                                                             |
|                       Kv                         K^                                         |
|         _ a _         or           _ c _         or           _ a _                         |
|       b _   _ c     < - _        b _   _ f     - _          b _   _ c   (The BACK           |
|       |   d   |           -      |   d   |         - >      |   d   |    face is            |
|       e _ | _ f          K       e _ | _ h          K       e _ | _ f    usually            |
|           g                ^         g                |         g        hidden.)           |
|                            |                          v                                     |
|                                                                                             |
|       DIAGRAM 2-12. BACK Clockwise Move and BACK Countertclockwise Move                     |
|                                                                                             |
| The BACK side is viewed here from the "wrong" direction, so don't get confused!             |
|                                                                                             |
|---------------------------------------------------------------------------------------------|
|                                                                                             |
| We can rotate the LEFT face clockwise, an operation abbreviated "Lv". We can rotate the     |
| LEFT face counterclockwise, abbreviated "L^". Again, I will also ehow the alternate         |
| notation for this:                                                                          |
|                                                                                             |
|                       Lv                         L^                                         |
|         _ a _         or           _ h _         or           _ a _                         |
|       b _   _ c          _ -     a _   _ c        _ - >     a _   _ c   (The LEFT           |
|       |   d   |      < -         |   d   |      -           |   d   |    face is            |
|       e _ | _ f       L          b _ | _ f        L         e _ | _ f    usually            |
|           g         |                g         ^                g        hidden.)           |
|                     v                          |                                            |
|                                                                                             |
|       DIAGRAM 2-13. LEFT Clockwise Move and LEFT Counterclockwise Move                      |
|                                                                                             |
| The LEFT side is viewed here from the "wrong" direction, so don't get confused!             |
|                                                                                             |
|---------------------------------------------------------------------------------------------|
|                                                                                             |
| We can rotate the RIGHT face clockwise, an operation abbreviated "Rv". We can rotate the    |
| RIGHT face counterclockwise, abbreviated "R^".                                              |
|                                                                                             |
|         _ a _                _ a  _                _ a _                                    |
|       b _   _ c            b _   _ d             b _   _ c                                  |
|       |   d R |     Rv     |   g R |      R^     |   d R |                                  |
|       e _ | _ f            e _ | _ c             e _ | _ f                                  |
|           g                    f                     g                                      |
|      "before Rv"           "after Rv"                                                       |
|                            "before R^"          "after R^"                                  |
|                                                                                             |
|       DIAGRAM 2-14. RIGHT Clockwise Move and RIGHT Counterclockwise Move                    |
|                                                                                             |
| Since we can usually see the RIGHT side, there should be no confusion here.                 |
|                                                                                             |
|---------------------------------------------------------------------------------------------|
|                                                                                             |
| We can rotate the TOP face clockwise, an operation abbreviated "Tv". We can rotate the TOP  |
| face counterclockwise, abbreviated "T^".                                                    |
|                                                                                             |
|         _ a _                _ b _                _ a _                                     |
|       b _ T _ c            d _ T _ a            b _ T _ c                                   |
|       |   d   |     Tv     |   c   |     T^     |   d   |                                   |
|       e _ | _ f            e _ | _ f            e _ | _ f                                   |
|           g                    g                    g                                       |
|      "before Tv"           "after Tv"                                                       |
|                            "before T^"          "after T^"                                  |
|                                                                                             |
|      DIAGRAM 2-15. TOP Clockwise Move and TOP Counterclockwise Move                         |
|                                                                                             |
| Since we can usually see the TOP side, there should be no confusion here.                   |
|                                                                                             |
|---------------------------------------------------------------------------------------------|
|                                                                                             |
| If it's any consolation to you, we will usually be working with the TOP side, the FRONT     |
| side, and the RIGHT side.                                                                   |
|                                                                                             |
| Like Shrek, onions, parfaits, and fancy cakes, the Cube has LAYERS! Here is a diagram to    |                        | help you visualize the Cube in "layers".                                                    |
|                                                                                             |
|               _ * _                        _ * _                        _ * _               |
|           _ * _     - _                _ -     _ * _                _ -       - _           |
|       _ * _     - _     - _        _ -     _ -     _ * _        _ -     _ * _     - _       |
|     * _     - _     - _   _ *    * _   _ -     _ -     _ *    * _     * _TOP_ *     _ *     |
|     |   - _     - _   _ *   |    |   * _   _ -     _ -   |    |   - _     *     _ -   |     |
|     |       - _   _ *   |   |    |   |   * _   _ -       |    |       - _   _ -       |     |
|     |           *   |   |   |    |   |   |   *           |    * _         *         _ *     |
|     |   * _     |   |   |   |    |   |   |   |     _ *   |    |   - _     |     _ -   |     |
|     |   |   *   |   |   |   |    |   |   |   |   *   |   |    |       - _ | _ -       |     |
|     |   | F |   |   |   |   |    |   |   |   |   | R |   |    * _         *         _ *     |
|     |   * _ |   |   |   |   |    |   |   |   |   | _ *   |    |   - _     |     _ -   |     |
|     |       *   |   |   |   |    |   |   |   |   *       |    |       - _ | _ -       |     |
|     * _         |   |   | _ *    * _ |   |   |         _ *    * _         *         _ *     |
|         - _     |   | _ *            * _ |   |     _ *            - _     |     _ -         |
|             - _ | _ *                    * _ | _ *                    - _ | _ -             |
|                 *                            *                            *                 |
|     3 layers between BACK        3 layers between LEFT        3 layers between BOTTOM       |
|     and FRONT                    and RIGHT                    and TOP                       |
|                                                                                             |
|     DIAGRAM 2-16A.               DIAGRAM 2-16B.               DIAGRAM 2-16C.                |
|                                                                                             |
|     DIAGRAM 2-16. Three Types of Layers                                                     |
|                                                                                             |
| All of the moves described so far in this chapter (B^, B2, Bv, F^, F2, Fv, K^, K2, Kv, L^,  |
| L2, Lv, R^, R2, Rv, T^, T2, and Tv) have involved just one single layer -- one layer moves; |
| the other two layers stay fixed in space. During all of these moves, the center squares of  |
| the Cube remain fixed in space. If you wanted to emphasize that you were moving just one    |
| layer, you could write, "1B^", "1B2", "1Bv", "1F^", "1F2", "1Fv", "1K^", "1K2", "1Kv",      |
| "1L^", "1L2", "1Lv", "1R^", "1R2", "1Rv", "1T^", "1T2", and "1Tv". But, just as in algebra  |
| there is no real need to write "1x" when "x" will do, so there is no urgent need to write   |
| "1B^", since "B^" will do just as well.                                                     |
|                                                                                             |
| However, there have been some hints that other moves are possible. (Diagram 2-11 was a very |
| temporary "sneak peek" at the BOTTOM of the Cube. We immediately returned the Cube to its   |
| normal orientation after the "peek".) Indeed there are some other important moves.          |
|                                                                                             |
| From time to time, it may become necessary to rotate the entire Cube. I can indicate these  |
| moves as 3F^, 3F2, 3Fv, 3R^, 3R2, 3Rv, 3T^, 3T2, and 3Tv. Rotate all three layers (instead  |
| of just one layer) and you have rotated the entire Cube. I think of these moves as "three   |
| layer FRONT counterclockwise", "three layer FRONT twice". "three layer FRONT clockwise",    |
| and so on.                                                                                  |
|                                                                                             |
| Do you see why I do not need to mention 3B^, 3B2, 3Bv, 3K^, 3K2, 3Kv, 3L^, 3L2, or 3Lv?     |
| (Partial answer: 3B^ = 3Tv, and 3L2 = 3R2.) It should be obvious to you that each rotation  |
| of the entire Cube causes the center square of four sides to move (and each rotation causes |
| the center square of the remaining two sides to rotate). If you move all three layers, you  |
| do need to write the "3", unless you can work out your own complete system of notation.     |
| Then, you would be writing the book, and I would be reading it!                             |
|                                                                                             |
| Here is a sequence of rotations of the entire Cube which will cause each of the six sides   |
| to enjoy a moment on TOP. (Each side also takes a turn on the BOTTOM.)                      |
|                                                                                             |
|    _ * _           _ * _           _ * _           _ * _           _ * _           _ * _    |
|  * _ t _ *  3Rv  * _ f _ *  3Fv  * _ l _ *  3Fv  * _ k _ *  3Fv  * _ r _ *  3Rv  * _ b _ *  |
|  | f * r |       | b * r |       | b * f |       | b * l |       | b * k |       | l * k |  |
|  * _ | _ |  -->  * _ | _ |  -->  * _ | _ *  -->  * _ | _ *  -->  * _ | _ *  -->  * _ | _ *  |
|      *               *               *               *               *               *      |
|                                                                                             |
|  DIAGRAM 2-17. Each Face of the Cube Takes a Moment on TOP                                  |
|                                                                                             |
| It may be helpful to practice other ways to rotate the Cube. Here we return it to its       |
| original postion.                                                                           |
|                                                                                             |
|       _ * _           _ * _           _ * _           _ * _                                 |
|     * _ b _ *  3F^  * _ k _ *  3F^  * _ t _ *  3Tv  * _ t _ *                               |
|     | l * k |       | l * t |       | l * f |       | f * r |                               |
|     * _ | _ *  -->  * _ | _ *  -->  * _ | _ *  -->  * _ | _ *                               |
|         *               *               *               *                                   |
|                                                                                             |
|     DIAGRAM 2-18. We Return the Cube to its Original Orientation                            |
|                                                                                             |
| If you want to get really fluent in moving your Cube, try these rotations of 180 degrees    |
| about an axis through the midpoint of two opposite edges. (There are six ways to rotate a   |
| Cube in this fashion. You may recall diagram 2-11 , where I showed you a "sneak peek" of    |
| the BOTTOM of the Cube. I do not intend to make any further use of these methods, or to     |
| develop a notation for them. But you can if you want to!)                                   |
|                                                                                             |
|                                                   \                                         | 
|                  __ * __                           \   __ * __                              |
|           __ ---         --- __                 __ -O-         --- __                       |
|         * __       TOP       __ *             * __       TOP       __ *                     |
|         |    --- __   __ ---    |             |    --- __   __ ---    |                     |
|         |           *           |             |           *           |                     |
|      ---O           |           O---          |           |           |                     |
|         |   FRONT   |   RIGHT   |             |   FRONT   |   RIGHT   |                     |
|         |           |           |             |           |           |                     |
|         * __        |        __ *             * __        |        __ *                     |
|              --- __ | __ --                        --- __ | __ -O-                          |
|                     *                                     *      \                          |
|                                                                   \                         |
|         An axis through                       An axis through                               |
|         the centers of the                    the centers of the                            |
|         FRONT LEFT and                        LEFT TOP and                                  |
|         BACK RIGHT                            BOTTOM RIGHT                                  |
|         edge cubies.                          cubies.                                       |
|                                                                                             |
|         DIAGRAM 2-19A.                        DIAGRAM 2-19B.                                |
|                                                                                             |
|                             /                                                               | 
|                  __ * __   /                           __ * __                              |
|           __ ---         -O- __       --- __    __ ---         --- __    (The place         |
|         * __       TOP       __ *            --- __      TOP       __ *   where the         |
|         |    --- __   __ ---    |             |     O- __   __ ---    |   axis comes        |
|         |           *           |             |           *           |   out is hidden     |
|         |           |           |             |           |           |   behind and        |
|         |   FRONT   |   RIGHT   |             |   FRONT   |   RIGHT   |   underneath        |
|         |           |           |             |           |    ( )    |   the Cube.)        |
|         * __        |        __ *             * __        |        __ *  __                 |
|              -O- __ | __ --                        --- __ | __ ---          ---             |
|              /      *                                     *                                 |
|             /                                                                               |
|         An axis through                       An axis through                               |
|         the centers of the                    the centers of the                            |
|         BACK TOP and                          FRONT TOP and                                 |
|         BOTTOM FRONT                          BOTTOM BACK                                   |
|         edge cubies.                          edge cubies.                                  |
|                                                                                             |
|         DIAGRAM 2-19C.                        DIAGRAM 2-19D.                                |
|                                                                                             |
|                                                                |                            |
|                                                                |                            | 
|                         __ * __                             __ * __                         |
|                  __ ---         --- __    __ ---     __ ---         --- __                  |
|   (The place   * __       TOP      __ ---          * __       TOP       __ *  (The place    |
|    where the   |    --- __   __ -O     |           |    --- __( )__ ---    |   where the    |
|    axis comes  |           *           |           |           *           |   axis comes   |
|    out is      |           |           |           |           |           |   out is       |
|    hidden.)    |   FRONT   |   RIGHT   |           |   FRONT   |   RIGHT   |   hidden.)     |
|                |    ( )    |           |           |           O           |                |
|            __  * __        |        __ *           * __        |        __ *                |
|        ---          --- __ | __ --                      --- __ | __ ---                     |
|                            *                                   |                            |
|                                                                |                            |
|                An axis through                     An axis through                          |
|                the centers of the                  the centers of the                       |
|                RIGHT TOP and                       FRONT RIGHT and                          |
|                BOTTOM LEFT                         BACK LEFT                                |
|                edge cubies.                        edge cubies.                             |
|                                                                                             |
|                DIAGRAM 2-19E.                      DIAGRAM 2-19F.                           |
|                                                                                             |
|     DIAGRAM 2-19. Six More Ways to Rotate the Entire Cube                                   |
|                                                                                             |
| Can you find sequences of three-layer moves, using our usual notation, which accomplish the |
| same results?                                                                               |
|                                                                                             |
| There are also four ways to rotate the Cube by 120 degrees, about an axis through opposite  |
| corners of the Cube. You can rotate either clockwise or counterclockwise. (Again, I do not  |
| intend to make much use of these methods -- but you can!)                                   |     
|                                                                                             |
|                              |                                               (You are       |
|    _                         |                              _                 looking       |
|      - _   _ * _           _ * _                _ * _   _ -          _ * _    straight      |
|          * _ T _ *       * _ T _ *            * _ T _ *            * _ T _ *  along the     |
|          | F * R |       | F * R |            | F * R |            | F(*)R |  axis -- it    |
|          * _ | _ * _     * _ | _ *          _ * _ | _ *            * _ | _ *  goes through  |
|              *       - _     *          _ -       *                    *      the center    |
|                              |                                                of the Cube.) |
|                              |                                                              |
|     An axis through      An axis through      An axis through      An axis through          |
|     centers of           centers of           centers of           centers of               |
|     FRONT LEFT TOP and   BACK LEFT TOP and    BACK RIGHT TOP and   FRONT RIGHT TOP and      |
|     BACK RIGHT           BOTTOM FRONT RIGHT   BOTTOM FRONT LEFT    BOTTOM BACK LEFT         |
|     cubies.              cubies.              cubies.              cubies.                  |
|                                                                                             |
|     DIAGRAM 2-20A.       DIAGRAM 2-20B.       DIAGRAM 2-20C.       DIAGRAM 2-20D.           |
|                                                                                             |
|     DIAGRAM 2-20. Four More Ways to Rotate the Entire Cube                                  |
|                                                                                             |
| Again, can you find sequences of moves, using our usual notation, which accomplish the same |
| results?                                                                                    |
|                                                                                             |
| Just as a hint of what's coming up, Chapter Eight, "Moving Edge Cubies" and Chapter Nine,   |
| "Rubik's Maneuver -- How to Flip Two Edge Cubies", will move two layers (and keep the third |
| layer fixed in space). These moves will be called "slice up" and "slice down". They are     |
| useful for moving edge cubies around. Like the "three layer" moves, they also cause the     |
| center square of several sides to move around.                                              |
|                                                                                             |
| Always be careful when you try to use the two-layer moves and the three-layer moves. The    |
| cubies move, but the locations (FRONT, RIGHT, TOP, etc.) do not. When I show you an         |
| isometric diagram of the Cube like this...                                                  |
|                                                                                             |
|               _ * _                                                                         |
|           _ -  ...  - _                                                                     |
|       _ -   this side   - _                                                                 |
|     * _is always called   _ *     This way of representing a cube on a two-dimensional      |
|     |   - the TOP side. -   |     surface is called an "isometric" view, or isometric       |
|     |       - _   _ -       |     projection.  If drawn with drafting instruments           |
|     |  ... this *  ... this |     (instead of a typewriter), a diagram like this has        |
|     |  side is  |  side is  |     equal measurements along each of the three main           |
|     |  alwayss  |  always   |     directions (iso = same, metric = measure).                |
|     |  called   |  called   |                                                               |
|     |  the      |  the      |              ^ height                                         |
|     |  FRONT    |  RIGHT    |              |             The three main directions          |
|     * _side.    |  side.  _ *      < _     |     _ >     of an isometric drawing            |
|         - _     |     _ -              - _ | _ -                                            |
|             - _ | _ -              width   *   depth                                        |
|                 *                                                                           |
|                                                                                             |
|     DIAGRAM 2-21. An Explanation of an Isometric Drawing With a Cube                        |
|                                                                                             |
| As you know, a cube is really a three-dimensional shape. My two-dimensional diagrams cannot |
| really do justice to that essential fact. (Here's another fact about dimensions -- when you |
| move part of your Cube, you are actually explaoring a shape with FOUR dimensions! At        |
| different moments of TIME, different parts of your Cube are in different places. It would   |
| take (at least) four different numbers to describe truly and completely the position of     |
| each particle of your Cube.)                                                                |
|                                                                                             |
*---------------------------------------------------------------------------------------------*

Ray Calvin Baker 15:03, 29 October 2011 (UTC)