Rubik's Cube/Ignoring Details

This is where Chapter 04 has been installed. Sorry, folks! Several of the diagrams got messed up in translation!

Ray Calvin Baker 03:38, 10 November 2011 (UTC)

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 * HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE                              |
 * (Preliminary April 20, 2007 version)            |
 * by Mr. Ray Calvin Baker                         |
 * This is a FREE educational resource             |
 * Chapter Four - - - - - - - - - - Ignoring Details -- Moving Corner Cubies                  |
 * Goal One is to get all eight corner cubies properly located at the eight corners of the    |
 * Cube. Our "warming up" exercises positioned four of the corner cubies. Now don't be        |
 * alarmed, but we may have to temporarily give up some of this apparent progress in order to |
 * make real and lasting progress toward the goal of getting all six sides of the Cube        |
 * correct. I propose to set the Cube aside for a little while, do some "paper and pencil"    |
 * work to plan some moves, and then apply some carefully planned moves to the Cube.          |
 * In this chapter, I will first describe how I found an essential move (swapping two corner  |
 * cubies). Then, I will show how to use this swap, if necessary, to locate all eight corner  |
 * cubies. Next, after we have thought things through, I will show you another way to organize |
 * a plan. Finally, after everything is laid out for you, I will invite you to pick up your   |
 * Cube once more, and put all eight corner cubies where they belong.                         |
 * YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL! For now, we are |
 * THINKING about strategic ways to deal with the Cube, not moving cubies!                    |
 * 1) If you were able to position the four corner cubies of the TOP layer correctly, either as   #
 * 2) described in chapter three, or using your own methods, there are now only 4 * 3 * 2 * 1     #
 * 3) = 24 ways to arrange the remaining four corner cubies. I will show you all 24 of these,     #
 * 4) later in this chapter.                                                                      #
 * The best way I have discovered so far on how to find ways to shuffle the corner cubies is  |
 * to make a lot of copies of a simplified diagram of the Cube on scratch paper, then try     |
 * things until something useful shows up. (The brackets "[ ]" above each diagram are used to |
 * indicate which corner cubie is hidden at the BOTTOM BACK LEFT "BKL" location of the Cube.  |
 * The parentheses "" indicate spaces where you can fill in an indication of which cubie   |
 * occupies that corner of the Cube. Use the space between the diagrams to record the         |
 * operation you tried. You may also number the moves, if you want to.)                       |
 * [ ]                 [ ]                  [ ]                   [ ]           |
 * _ _              _  _              _  _               _  _               |
 * _    _        _     _        _     _         _     _            |
 * |       |   __   |        |   __   |        |   __    |        |   __       |
 * _ |  _   #1   _  |  _   #2   _  |  _   #3    _  |  _   #4       |
 * ... and so on.... |
 * This is a sample of part of a work sheet, such as I use to plan moves for corner cubies. |
 * DIAGRAM 4-1. Blank Diagrams for Planning Moves                                           |
 * The first diagrams I made looked like this:                                                |
 * [h]                 [h]                  [h]                                |
 * _ (a) _             _ (a) _              _ (a) _                                    |
 * (b) _    _ (c)      (e) _     _ (c)      (e) _     _ (f)                                |
 * |   (d)    |   Fv   |    (b)    |   R^   |    (c)    |   F^                            |
 * (e) _ |  _ (f)  #1  (g) _  |  _ (f)  #2  (g) _  |  _ (d)  #3                            |
 * (g)       -->       (d)        -->       (b)        -->                           |
 * This is just an                 I began                                                 |
 * arbitrary way                   exploring                                               |
 * to label the                    with a few                                              |
 * corner cubies.                  arbitrary moves.                                        |
 * [h]                 [h]                  [h]                                |
 * _ (a) _             _ (a) _              _ (g) _                                    |
 * (c) _    _ (f)      (c) _     _ (b)      (b) _     _ (c)                                |
 * |   (b)    |   Rv   |    (g)    |   T2   |    (a)    |                                 |
 * (e) _ |  _ (d)  #4  (e) _  |  _ (f)  #5  (e) _  |  _ (f)                                |
 * (g)       -->       (d)        -->       (d)                                      |
 * Comparing this to         T2 looked like an                        |
 * the starting position,    interesting thing                        |
 * b and c have traded       to try, and it was!                      |
 * places, while e and f                                              |
 * are where they started.                                            |
 * DIAGRAM 4-2. I Found a Three-Way Swap                                                   |
 * To evaluate whether or not a series of moves is useful, we need to compare the original    |
 * position of the corner cubies with their final position. Sometimes, we can find "cycles"   |
 * that return many cubies to their original positions, while moving a few cubies in a        |
 * predictable pattern.                                                                       |
 * Comparing this last diagram with the first, we find that five cubies are back at their     |
 * starting position, while three cubies, a, d, and g, have moved. We can diagram this:       |
 * a --> d --> g --> a. (Do you see how and why I wrote this down?) This is called a "cycle"  |
 * (of length three), and it helps us predict what would happen if we did "Fv R^ F^ Rv T2"    |
 * twice, or three times, or more; and we don't have to work out all of the messy details to  |
 * see this! P.S. This is a useful recipe for moving exactly three corner cubies any way we   |
 * want to, if the three corner cubies are in exactly the proper positions. The idea discussed |
 * in Chapter Six, "Customize Your Moves -- Commutation", will allow you to shuffle any three |
 * cubies.                                                                                    |
 * Flush with success at finding a useful result so soon, I guessed that I could find a way to |
 * interchange exactly two corner cubies just as easily. I tried to move one of the three     |
 * cubies "out of the way" with a move of the LEFT side, then I tried using the cycle I had   |
 * discovered.                                                                                |
 * YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL!                |
 * After "Fv R^ F^ Rv T2", the diagrams of the imaginary Cube we are THINKING about looks like |
 * this.                                                                                      |
 * [h]                 [g]                  [g]                   [g]          |
 * _ (g) _             _ (b) _              _ (b) _               _ (b) _              |
 * (b) _    _ (c)      (e) _     _ (c)      (h) _     _ (c)       (h) _     _ (f)          |
 * |   (a)    |   L^   |    (a)    |   Fv   |    (e)    |   R^    |    (c)    |   F^      |
 * (e) _ |  _ (f)  #6  (h) _  |  _ (f)  #7  (d) _  |  _ (f)  #8   (d) _  |  _ (a)  #9      |
 * (d)       -->       (d)        -->       (a)        -->        (e)        -->     |
 * Continuing an experiment....                                           |
 * [g]                 [g]                  [g]                   [g]          |
 * _ (b) _             _ (b) _              _ (d) _               _ (d) _              |
 * (c) _    _ (f)      (c) _     _ (e)      (e) _     _ (c)       (h) _  |  _ (c)          |
 * |   (e)    |   Rv   |    (d)    |   T2   |    (b)    |   Fv    |    (e)    |   R^      |
 * (h) _ |  _ (a)  10  (h) _  |  _ (f)  11  (h) _  |  _ (f)  12   (a) _  |  _ (f)  13      |
 * (d)       -->       (a)        -->       (a)        -->        (b)        -->     |
 * [g]                 [g]                  [g]                   [g]          |
 * _ (d) _             _ (d) _              _ (d) _               _ (a) _              |
 * (h) _    _ (f)      (c) _     _ (f)      (c) _     _ (e)       (e) _  |  _ (c)          |
 * |   (c)    |   F^   |    (e)    |   Rv   |    (a)    |   T2    |    (d)    |   Lv      |
 * (a) _ |  _ (b)  14  (h) _  |  _ (b)  15  (h) _  |  _ (f)  16   (h) _  |  _ (f)  17      |
 * (e)       -->       (a)        -->       (b)        -->        (b)        -->     |
 * [h]                                      [h]                                |
 * _ (g) _                                  _ (a) _                                    |
 * (a) _    _ (c)                           (b) _     _ (c)                                |
 * |   (d)    |                             |    (d)    |                                 |
 * (e) _ |  _ (f)                           (e) _  |  _ (f)                                |
 * (b)                                      (g)                                      |
 * Final position                           Original position                              |
 * .. that seemed to result in nothing useful.                                             |
 * DIAGRAM 4-3. An Experiment Which Seemed Useless                                         |
 * This was disappointing! Not every idea of mine worked! Sometimes I had to scribble for an  |
 * hour or two on scratch paper to find a way to do something important and necessary. But    |
 * after searching for a way to interchange exactly two corner cubies, I eventually found     |
 * three more moves that really did accomplish what I needed.                                 |
 * [h]                 [h]                  [h]                   [h]          |
 * _ (g) _             _ (g) _              _ (a) _               _ (a) _              |
 * (a) _    _ (c)      (a) _     _ (d)      (b) _     _ (g)       (b) _     _ (c)          |
 * |   (d)    |   Rv   |    (b)    |   Tv   |    (d)    |   R^    |    (g)    |           |
 * (e) _ |  _ (f)  18  (e) _  |  _ (c)  19  (e) _  |  _ (c)  20   (e) _  |  _ (f)          |
 * (b)       -->       (f)        -->       (f)        -->        (d)                |
 * DIAGRAM 4-4. Three More Moves Provide a Useful Two-Cubie Interchange                    |
 * YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL!                |
 * Here is a summary of this entire process we have been thinking about.                      |
 * [h]    First,   Fv R^ F^ Rv T2                 [h]    Notice that corner    |
 * _ (a) _        Second,  L^                     _ (a) _        cubies d and g have   |
 * (b) _ T  _ (c)     Third,   Fv R^ F^ Rv T2     (b) _  T  _ (c)    traded places,        |
 * | F (d) R  |      Fourth,  Fv R^ F^ Rv T2      |  F (g) R  |                           |
 * (e) _ |  _ (f)     Fifth,   Lv                 (e) _  |  _ (f)    These are in the      |
 * (g)          Sixth,   Rv Tv R^                 (d)          FRT and BFR           |
 * Original position                              Final position     locations.            |
 * DIAGRAM 4-5. Summary of FRONT RIGHT TOP and BOTTOM FRONT RIGHT SWAP                     |
 * Call this the "FRONT RIGHT TOP and BOTTOM FRONT RIGHT SWAP"!                               |
 * Now that I know a way to interchange two corner cubies, I have completed the difficult part |
 * of the strategic planning which will allow me to accomplish Goal One. Surely you can find  |
 * easier, better, and faster ways to position the corner cubies! (Hint: Why do a three-cycle |
 * twice? Isn't it easier and quicker to do it backwards once instead?)                       |
 * CAUTION! When you try to use a procedure such as this, or any others to be discovered in   |
 * the following chapters, it is EXTREMELY IMPORTANT that EVERY DETAIL and EVERY MOVE be      |
 * EXACTLY CORRECT! Otherwise, you may mess up parts of the Cube you thought you had finished. |
 * You may have to start over! (Oh, well! It's happened to me, LOTS of times!)                |
 * Hey! Not so fast! What if the corner cubies I want to interchange are NOT in the FRONT     |
 * RIGHT TOP and BOTTOM FRONT RIGHT locations? This sounds like it could be a serious problem, |
 * since we correctly positioned all corner cubies of the TOP layer according to directions in |
 * the previous chapter. It is quite possible that two (or more!) corner cubies in the BOTTOM |
 * layer are not correctly positioned for this interchange to work properly.                  |
 * Fortunately, there is an easy solution for this problem, if the corner cubies which need to |
 * be interchanged are adjacent to eache other. (Some examples are: BFR and BFL, BFL and BKL, |
 * and BKL and BKR.) Just use some combination of three-layer moves (like, for example, 3T^,  |
 * 3Tv, 3R^, 3Rv, 3F^, and 3Fv) to rotate the entire Cube until the corner cubies you wish to |
 * switch ARE in the FRONT RIGHT TOP and BOTTOM FRONT RIGHT positions. Then apply the series  |
 * of moves which interchanges those to corner cubies! (This principle will be dicussed more  |
 * thoroughly in Chapter Six, "Customize Your Moves -- Commutation".)                         |
 * What can you do if the corner cubies to be interchanged are at opposite corners of the     |
 * BOTTOM layer?                                                                              |
 * YOUR CUBE SHOULD STILL BE ON THE TABLE!                                                    |
 * You had matched up some        After "3R2" flips the                                     |
 * of the corner cubies on        entire Cube, you MIGHT                                    |
 * the BACK, FRONT, LEFT,         find 2 corner cubies in       If that is what you find,   |
 * RIGHT, and TOP sides.          the correct FRT and KLT       you should be able to       |
 * locations.                   do this!                    |
 * _ * _                          _ * _                         _ * _             |
 * _ * _TOP_ * _                  _ * _klt_ * _                 _ * _klt_ * _         |
 * _ * _ ? _ * _ ? _ * _          _ * _   _ * _   _ * _         _ * _   _ * _   _ * _     |
 * * _TOP_ * _TOP_ * _TOP_ *      * _krt * _   _  * _flt_ *     * _flt_ * _   _ * _krt_ *   |
 * |  * _ ? _ * _ | _ *   |       |   * _   _ * _   _ *   |     |   * _   _ * _   _ *   |   |
 * | F |  * _TOP_ *   | R |       |krt|   * _frt_ *   |flt|     |flt|   * _frt_ *   |krt|   |
 * * _ | ? |  *   | ? | _ *  3R2  * _ |   |   *   |   | _ *     * _ |   |   *   |   | _ *   |
 * |  * _ | F | R | _ *   |  -->  |   * _ |frt|frt| _ *   |     |   * _ |frt|frt| _ *   |   |
 * | ? |  * _ | _ *   | ? |       |   |   * _ | _ *   |   |     |   |   * _ | _ *   |   |   |
 * * _ | F |  *   | R | _ *       * _ |"k"|   *   |"r"| _ *     * _ | F |   *   | R | _ *   |
 * |  * _ | ? | ? | _ *   |       |   * _ |   |   | _ *   |     |   * _ |   |   | _ *   |   |
 * | ? |  * _ | _ *   | ? |       |"k"|   * _ | _ *   |"r"|     | F |   * _ | _ *   | R |   |
 * * _ | ? |  *   | ? | _ *       * _ |   |   *   |   | _ *     * _ |   |   *   |   | _ *   |
 * * _ | ? | ? | _ *              * _ |"k"|"r"| _ *             * _ | F | R | _ *       |
 * * _ | _ *                      * _ | _ *                     * _ | _ *           |
 * What you had at the end        ONE POSSIBILITY after you     What you want.              |
 * of chapter three. (BACK        turn the Cube upside down.    Notice that you had to      |
 * and LEFT sides show a          (I have used "k" and "r"      exchange cubies marked      |
 * similar pattern.)              to indicate that these        "krt" and "klt".            |
 * (This is a repeat of           colors match correctly.)                                  |
 * diagram 3-1B.)                                                                          |
 * DIAGRAM 4-6A.                  DIAGRAM 4-6B.                 DIAGRAM 4-6C.               |
 * DIAGRAM 4-6. How Do You Swap Diagonally Opposite Cubies?                                 |
 * I told you, "You should be able to do this!". Here is "the hard way". (I am going to shrink |
 * the diagrams a bit, leaving out "irrelevant details", so that I can fit more diagrams on   |
 * the page.) Remember, at this stage, we are interested in getting corner cubies to the      |
 * correct LOCATIONS. Their orientation is irrelevant.                                        |
 * YOUR CUBE SHOULD STILL BE ON THE TABLE!                                                    |
 * Here is the plan, using only rotations of the entire Cube, and our newly-discovered SWAP   |
 * sequence. Three SWAPS of adjacent corner cubies results in the SWAp of two diagonally      |
 * opposite corner cubies. (Parentheses, "(" and ")", are used to highlight the pair of corner |
 * cubies about to be SWAPped.)                                                               |
 * 1klt                 1klt                    1klt               1klt              |
 * (2krt    3flt        4frt     3flt)        (4frt     2krt        3flt    2krt          |
 * 4frt)               (2krt                  3flt)                4frt              |
 * DIAGRAM 4-7A.        DIAGRAM 4-7B.         DIAGRAM 4-7C.         DIAGRAM 4-7D.         |
 * DIAGRAM 4-7. Plan for SWAPping Diagonally Opposite Cubies the Hard Way                 |
 * (We are still ignoring the edge cubies at this stage of the solution.)                     |
 * _ * _                _ * _                  _ * _                 _ * _            |
 * _ * _klt_ * _        _ * _ l _ * _          _ * _ l _ * _         _ * _klt_ * _        |
 * * _krt_ T _flt_ *    * _ l _ l _klt_ * DO   * _ l _ l _klt_ *     * _frt_ T _flt_ *      |
 * * _frt_ *  |     |   * _krt_ *   | THE  |   * _frt_ *   |     |   * _krt_ *   |          |
 * |  * _ |
 * |krt|  *   |flt| 3Fv | F |   *   |klt| FRT  | F |   *   |klt| 3F^ |frt|   *   |flt| 3R^  |
 * * _ |frt|frt| _ * --> * _ |krt|krt| _ * and * _ |frt|frt| _ * --> * _ |krt|krt| _ * -->  |
 * |  F _ | _ R   |     |   F _ | _ t   | BFR  |   F _ | _ t   |     |   F _ | _ R   |      |
 * | F |  *   | R |     | F |   *   |flt| SWAP | F |   *   |flt|     | F |   *   | R |      |
 * * _ | F | R | _ *    * _ |frt|frt| _ * ---> * _ |krt|krt| _ *     * _ | F | R | _ *      |
 * * _ | _ *            * _ | _ *              * _ | _ *             * _ | _ *          |
 * You worked hard in   I'm using "l" to       "krt" indicates       After rotating the     |
 * chapter three to     indicate the face      the cubie that        entire Cube, then      |
 * get four corner      which USED to be       will end up in        using the SWAP moves,  |
 * cubies correct, so   on the LEFT (but       the KRT position      we can return the      |
 * I'll show those as   isn't any more         when we are done      Cube to this           |
 * "F" and "R"! (You    because I turned       (we hope!).           position.              |
 * can't see "B", "K"   the Cube).        See                        "frt" and "krt" have   |
 * or "L" -- they're                    diagram                      been SWAPped.          |
 * OK, too.)                              4-5                                               |
 * DIAGRAM 4-8A.        DIAGRAM 4-8B.          DIAGRAM 4-8C.         DIAGRAM 4-8D.          |
 * _ * _                 _ * _                 _ * _                 _ * _            |
 * _ * _klt_ * _         _ * _klt_ * _         _ * _klt_ * _         _ * _klt_ * _        |
 * * _frt_ T _ k _ * DO  * _frt_ T _ k _ *     * _frt_ T _krt_ *     * _ l _ T _krt_ *      |   |                                                                                       |
 * * _flt_ *  | THE  |   * _krt_ *   |     |   * _flt_ *   |     |   * _frt_ *   | THE  |
 * |frt|  *   | R | FRT  |frt|   *   | R | 3Rv |frt|   *   |krt| 3Fv | F |   *   |krt| FRT  |
 * * _ |flt|flt| _ * and * _ |krt|krt| _ * --> * _ |flt|flt| _ * --> * _ |frt|frt| _ * and  |
 * |  F _ | _ R   | BFR  |   F _ | _ R   |     |   F _ | _ R   |     |   F _ | _ R   | BFR  |
 * | F |  *   | R | SWAP | F |   *   | R |     | F |   *   | R |     | F |   *   | R | SWAP |
 * * _ |krt|krt| _ * ---> * _ |flt|flt| _ *    * _ | F | R | _ *     * _ |flt|flt| _ * -->  |
 * * _ | _ *             * _ | _ *             * _ | _ *             * _ | _ *          |
 * This may not     See                        ... now the "krt"                       See  |
 * look like much  diagram                     cubie is in place,                  siagram  |
 * progress yet,    4-5                        and we know how to                      4-5  |
 * but ...                                     SWAP "frt" and "flt".                        |
 * DIAGRAM 4-8E.         DIAGRAM 4-8F.         DIAGRAM 4-8G.         DIAGRAM 4-8H.          |
 * _ * _                 _ * _                                                        |
 * _ * _klt_ * _         _ * _klt_ * _                                                    |
 * * _ l _ T _krt_ *     * _flt_ T _krt_ *                                                  |                                                                                       |   |   * _flt_ *   |      |   * _frt_ *   |                                                  |
 * | F |  *   |krt| 3F^  |flt|   *   |krt|                                                  |
 * * _ |flt|flt| _ * --? * _ |frt|frt| _ *                                                  |
 * |  F _ | _ R   |      |   F _ | _ R   |                                                  |
 * | F |  *   | R |      | F |   *   | R |                                                  |
 * * _ |frt|frt| _ *     * _ | F | R | _ *                                                  |
 * * _ | _ *             * _ | _ *                                                      |
 * Rotate the Cube       ... and now we have all eight                                      |
 * back to position...   corner cubiea at their proper locations.                           |
 * DIAGRAM 4-8I.         DIAGRAM 4-8J.                                                      |
 * DIAGRAM 4-8. Ten Stages in SWAP Diagonally Opposite Cubies the Hard Way                  |
 * That was a real mental workout! I have shown several examples of ways we can use the SWAP, |
 * and, if you are clever, you should be able to get all eight corner cubies into their proper |
 * locations.                                                                                 |
 * We still need to orient all eight corner cubies. Then we will be able to finish off all    |
 * twelve edge cubies. Believe me, you are about to learn how to do these things. You will    |
 * also learn a general principle which will make your planning and Cube turning MUCH simpler |
 * and easier (It's in Chapter Six)!                                                          |
 * I promised that I would show you all 24 possible ways the corner cubies on the VOTTOM layer |
 * of your Cube could be arranged, after you have finished the work in Chapter Three, "Some   |
 * Simple Moves -- Positioning Four Corner Cubies". This will complete all of the strategic   |
 * planning we will need to fininsh Chapter Four.                                             |
 * _ * _                          _ * _                         _ * _             |
 * _ * _TOP_ * _                  _ * _ ? _ * _           (LEFT)  * _???_ *  (BACK)   |
 * _ * _ ? _ * _ ? _ * _          _ * _ ? _ * _   _ * _         _ * _     *     _ * _     |
 * * _TOP_ * _TOP_ * _TOP_ *      * _ ? _ * _"b"_ * _ ? _ *     * _???_ *  TOP  * _???_ *   |
 * |  * _ ? _ * _ | _ *   |       |   * _ ? _ * _ ? _ *   |         *     _ * _     *       |
 * | F |  * _TOP_ *   | R |       | ? |   * _ ? _ *   | ? |      FRONT  * _???_ *  RIGHT    |
 * * _ | ? |  *   | ? | _ *  3R2  * _ | ? |   *   | ? | _ *                 *               |
 * |  * _ | F | R | _ *   |  -->  |   * _ | ? | ? | _ *   |     This diagram indicates the  |
 * | ? |  * _ | _ *   | ? |       | ? |   * _ | _ *   | ? |     four corner cubies which    |
 * * _ | F |  *   | R | _ *       * _ |"k"|   *   |"r"| _ *     are now of interest to us.  |
 * |  * _ | ? | ? | _ *   |       |   * _ | ? | ? | _ *   |                                 |
 * | ? |  * _ | _ *   | ? |       |"k"|   * _ | _ *   |"r"|            (L) ??? (K)          |
 * * _ | ? |  *   | ? | _ *       * _ | ? |   *   | ? | _ *             ??? T ???           |
 * * _ | ? | ? | _ *              * _ |"k"|"r"| _ *                 F  ???  R           |
 * * _ | _ *                      * _ | _ *                     Condensed           |
 * *                              *                          version            |
 * DIAGRAM 4-9A.                  DIAGRAM 4-9B.                 DIAGRAM 4-9C.               |
 * DIAGRAM 4-9. Preparing to Locate the Final Four Corner Cubies                            |
 * PICK UP YOUR CUBE! It should have the TOP side showing your chosen color on at least five  |
 * colored labels, as shown in diagram 4-9A. Turn your Cube upside down as shown in diagrams  |
 * 4-9A and 4-9B.                                                                             |
 * I am going to refocus on what is now the FRONT, BACK, LEFT, RIGHT, and TOP, so I'll show   |
 * you how things are now named, and I'll show you what we are looking for. Then I'll tell you |
 * what we need to do for each of the 24 possibilities.                                       |
 * First, the names (and abbreviations) of the LOCATIONS for the last four corner cubies are  |
 * shown in diagram 4-10.                                                                     |
 * _ * _                          The last four corner cubies        |
 * (LEFT)     _ - BACK  - _      (BACK)           are indicated like this:           |
 * * _LEFT TOP  _ *                                                        |
 * -      - _KLT_ -       -                 klt -- the cubie which belongs     |
 * _ * _            *             _ * _                  at location KLT             |
 * _ - FRONT - _    -       -     _ - BACK  - _                                          |
 * * _LEFT  TOP _ *      TOP      * _RIGHT  TOP _ *     flt -- the cubie which belongs     |
 * |  - _FLT_ -       -       -       - _KRT_ -   |            at location FLT             |
 * *            _ * _             *                                                |
 * |          -     _ - FRONT - _     -           |     krt -- the cubie which belongs     |
 * FRONT    * _RIGHT  TOP _ *     RIGHT                  at location KRT             |
 * |                  - _FRT_ -                   |                                        |
 * * (This is an enlargement    frt -- the cubie which belongs     |
 * | of part of diagram 4-9C.)         at location KLT             |
 * DIAGRAM 4-10. Introducing More Abbreviated Diagrams                                    |
 * Now, as promised, here are the 24 possbilities of what you will see on what is now the TOP |
 * of your Cube (diagrams 4-11A through 4-14F)                                                |
 * (L) klt (K)  : (L) klt (K) : (L) klt (K) : (L) klt (K)    : (L) klt (K)   : (L) klt (K)    |
 * flt T krt   :  flt T frt  :  frt T krt  :  frt T flt     :  krt T flt    :  krt T frt     |
 * F frt  R    :  F  krt  R  :  F  flt  R  :  F  krt  R     :  F  frt  R    :  F  flt  R     |
 * :            :             :  (3-cycle)     :  (diagonal    :  (3-cycle)     |
 * :            :             :                :   corners)    :                |
 * WOW! All 8   : 3R^         : 3Fv         : K^ 3R^ 3T^     : This is done  : K^ 3R^ 3T^     |
 * corner cubies : DO THE SWAP : DO THE SWAP : Diagram 4-2:  : exactly like  : Diagram 4-2:   |
 * are properly : diagram 4-5 : diagram 4-5 : Fv R^ F^ Rv T2 : diagrams 4-8A : Fv R^ F^ Rv T2 |
 * located!     : 3Rv         : 3Fv         : 3Tv 3Rv Kv     : through 4-8J. : 3Tv 3Rv Kv     |
 * :            :             : Then look at   :               :                |
 * :            :             : diagram 4-11F. :               :                |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-11A.       : 4-11B.      : 4-11C.      : 4-11D.         : 4-11E.        : 4-11F.         |
 * DIAGRAM 4-11. The First Set of Six Possibilities                                           |
 * (L) krt (K)  : (L) frt (K) : (L) krt (K) : (L) flt (K)    : (L) flt (K)   : (L) frt (K)    |
 * klt T frt   :  klt T krt  :  klt T flt  :  klt T krt     :  klt T frt    :  klt T flt     |
 * F flt  R    :  F  flt  R  :  F  frt  R  :  F  frt  R     :  F  krt  R    :  F  krt  R     |
 * :            :             :  (3=cycle)     :  (diagonal    :  (3-cycle)     |
 * :            :             :                :   corners)    :                |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-12A.       : 4-12B.      : 4-12C.      : 4-12D.         : 4-12E.        : 4-12F.         |
 * DIAGRAM 4-12. The Second Set of Siz Possibilities                                          |
 * For diagrams 4-12A through 4-12F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-11A through 4-11F, then follow the directions for the correct diagram.     |
 * (L) frt (K)  : (L) krt (K) : (L) flt (K) : (L) krt (K)    : (L) frt (K)   : (L) flt (K)    |
 * krt T flt   :  frt T flt  :  krt T frt  :  flt T frt     :  flt T krt    :  frt T krt     |
 * F klt  R    :  F  klt  R  :  F  klt  R  :  F  klt  R     :  F  klt  R    :  F  klt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-13A.       : 4-13B.      : 4-13C.      : 4-13D.         : 4-13E.        : 4-13F.         |
 * DIAGRAM 4-13. The Third Set of Six Possibilities                                           |
 * For diagrams 4-13A through 4-13F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-12A through 4-12F. Follow the directions for those diagrams.               |
 * (L) flt (K)  : (L) flt (K) : (L) frt (K) : (L) frt (K)    : (L) krt (K)   : (L) krt (K)    |
 * frt T klt   :  krt T klt  :  flt T klt  :  krt T klt     :  frt T klt    :  flt T klt     |
 * F krt  R    :  F  frt  R  :  F  krt  R  :  F  flt  R     :  F  flt  R    :  F  frt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-14A.       : 4-14B.      : 4-14C.      : 4-14D.         : 4-14E.        : 4-14F.         |
 * DIAGRAM 4-14. The Final Set of Six Possibilities                                           |
 * For diagrams 4-14A through 4-14F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-13A through 4-13F. Tricky, huh? I think you see a pattern!                 |
 * Some of these moves may have altered the orientation of some corner cubies, but that is not |
 * a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
 * BE CORRECT before you proceed any further. If the location of any corner cubie is          |
 * incorrect, you will have to go back and try again.                                         |
 * 1) For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) *     #
 * 2) (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800    #
 * 3) (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three        #
 * 4) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 5) arrangements of the cubies left.                                                            #
 * First, the names (and abbreviations) of the LOCATIONS for the last four corner cubies are  |
 * shown in diagram 4-10.                                                                     |
 * _ * _                          The last four corner cubies        |
 * (LEFT)     _ - BACK  - _      (BACK)           are indicated like this:           |
 * * _LEFT TOP  _ *                                                        |
 * -      - _KLT_ -       -                 klt -- the cubie which belongs     |
 * _ * _            *             _ * _                  at location KLT             |
 * _ - FRONT - _    -       -     _ - BACK  - _                                          |
 * * _LEFT  TOP _ *      TOP      * _RIGHT  TOP _ *     flt -- the cubie which belongs     |
 * |  - _FLT_ -       -       -       - _KRT_ -   |            at location FLT             |
 * *            _ * _             *                                                |
 * |          -     _ - FRONT - _     -           |     krt -- the cubie which belongs     |
 * FRONT    * _RIGHT  TOP _ *     RIGHT                  at location KRT             |
 * |                  - _FRT_ -                   |                                        |
 * * (This is an enlargement    frt -- the cubie which belongs     |
 * | of part of diagram 4-9C.)         at location KLT             |
 * DIAGRAM 4-10. Introducing More Abbreviated Diagrams                                    |
 * Now, as promised, here are the 24 possbilities of what you will see on what is now the TOP |
 * of your Cube (diagrams 4-11A through 4-14F)                                                |
 * (L) klt (K)  : (L) klt (K) : (L) klt (K) : (L) klt (K)    : (L) klt (K)   : (L) klt (K)    |
 * flt T krt   :  flt T frt  :  frt T krt  :  frt T flt     :  krt T flt    :  krt T frt     |
 * F frt  R    :  F  krt  R  :  F  flt  R  :  F  krt  R     :  F  frt  R    :  F  flt  R     |
 * :            :             :  (3-cycle)     :  (diagonal    :  (3-cycle)     |
 * :            :             :                :   corners)    :                |
 * WOW! All 8   : 3R^         : 3Fv         : K^ 3R^ 3T^     : This is done  : K^ 3R^ 3T^     |
 * corner cubies : DO THE SWAP : DO THE SWAP : Diagram 4-2:  : exactly like  : Diagram 4-2:   |
 * are properly : diagram 4-5 : diagram 4-5 : Fv R^ F^ Rv T2 : diagrams 4-8A : Fv R^ F^ Rv T2 |
 * located!     : 3Rv         : 3Fv         : 3Tv 3Rv Kv     : through 4-8J. : 3Tv 3Rv Kv     |
 * :            :             : Then look at   :               :                |
 * :            :             : diagram 4-11F. :               :                |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-11A.       : 4-11B.      : 4-11C.      : 4-11D.         : 4-11E.        : 4-11F.         |
 * DIAGRAM 4-11. The First Set of Six Possibilities                                           |
 * (L) krt (K)  : (L) frt (K) : (L) krt (K) : (L) flt (K)    : (L) flt (K)   : (L) frt (K)    |
 * klt T frt   :  klt T krt  :  klt T flt  :  klt T krt     :  klt T frt    :  klt T flt     |
 * F flt  R    :  F  flt  R  :  F  frt  R  :  F  frt  R     :  F  krt  R    :  F  krt  R     |
 * :            :             :  (3=cycle)     :  (diagonal    :  (3-cycle)     |
 * :            :             :                :   corners)    :                |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-12A.       : 4-12B.      : 4-12C.      : 4-12D.         : 4-12E.        : 4-12F.         |
 * DIAGRAM 4-12. The Second Set of Siz Possibilities                                          |
 * For diagrams 4-12A through 4-12F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-11A through 4-11F, then follow the directions for the correct diagram.     |
 * (L) frt (K)  : (L) krt (K) : (L) flt (K) : (L) krt (K)    : (L) frt (K)   : (L) flt (K)    |
 * krt T flt   :  frt T flt  :  krt T frt  :  flt T frt     :  flt T krt    :  frt T krt     |
 * F klt  R    :  F  klt  R  :  F  klt  R  :  F  klt  R     :  F  klt  R    :  F  klt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-13A.       : 4-13B.      : 4-13C.      : 4-13D.         : 4-13E.        : 4-13F.         |
 * DIAGRAM 4-13. The Third Set of Six Possibilities                                           |
 * For diagrams 4-13A through 4-13F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-12A through 4-12F. Follow the directions for those diagrams.               |
 * (L) flt (K)  : (L) flt (K) : (L) frt (K) : (L) frt (K)    : (L) krt (K)   : (L) krt (K)    |
 * frt T klt   :  krt T klt  :  flt T klt  :  krt T klt     :  frt T klt    :  flt T klt     |
 * F krt  R    :  F  frt  R  :  F  krt  R  :  F  flt  R     :  F  flt  R    :  F  frt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-14A.       : 4-14B.      : 4-14C.      : 4-14D.         : 4-14E.        : 4-14F.         |
 * DIAGRAM 4-14. The Final Set of Six Possibilities                                           |
 * For diagrams 4-14A through 4-14F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-13A through 4-13F. Tricky, huh? I think you see a pattern!                 |
 * Some of these moves may have altered the orientation of some corner cubies, but that is not |
 * a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
 * BE CORRECT before you proceed any further. If the location of any corner cubie is          |
 * incorrect, you will have to go back and try again.                                         |
 * 1) For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) *     #
 * 2) (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800    #
 * 3) (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three        #
 * 4) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 5) arrangements of the cubies left.                                                            #
 * DIAGRAM 4-13. The Third Set of Six Possibilities                                           |
 * For diagrams 4-13A through 4-13F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-12A through 4-12F. Follow the directions for those diagrams.               |
 * (L) flt (K)  : (L) flt (K) : (L) frt (K) : (L) frt (K)    : (L) krt (K)   : (L) krt (K)    |
 * frt T klt   :  krt T klt  :  flt T klt  :  krt T klt     :  frt T klt    :  flt T klt     |
 * F krt  R    :  F  frt  R  :  F  krt  R  :  F  flt  R     :  F  flt  R    :  F  frt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-14A.       : 4-14B.      : 4-14C.      : 4-14D.         : 4-14E.        : 4-14F.         |
 * DIAGRAM 4-14. The Final Set of Six Possibilities                                           |
 * For diagrams 4-14A through 4-14F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-13A through 4-13F. Tricky, huh? I think you see a pattern!                 |
 * Some of these moves may have altered the orientation of some corner cubies, but that is not |
 * a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
 * BE CORRECT before you proceed any further. If the location of any corner cubie is          |
 * incorrect, you will have to go back and try again.                                         |
 * 1) For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) *     #
 * 2) (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800    #
 * 3) (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three        #
 * 4) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 5) arrangements of the cubies left.                                                            #
 * Some of these moves may have altered the orientation of some corner cubies, but that is not |
 * a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
 * BE CORRECT before you proceed any further. If the location of any corner cubie is          |
 * incorrect, you will have to go back and try again.                                         |
 * 1) For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) *     #
 * 2) (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800    #
 * 3) (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three        #
 * 4) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 5) arrangements of the cubies left.                                                            #
 * 1) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 2) arrangements of the cubies left.                                                            #