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

 This looks best in "Edit" mode.

Ray Calvin Baker 19:28, 18 August 2011 (UTC)

  Ray Calvin Baker 15:03, 29 October 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.)                                                               |
 * 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.)                                                               |
 * 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.)                                                               |
 * 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.)                                                               |