text:
Rubik's Cube/Introduction

This revision Ray Calvin Baker 04:30, 10 November 2011 (UTC) was begun after discovering "Speed Cubing" pages in the Wikiversity: I am DELIGHTED to discover that several other people share an interest in "The Cube"! As a (once upon a time) professional Mathematician and Computer Scientist, I would have been embarrassed NOT to find a solution! But how can I prove that I succeeded? Simple! My method is so clumsy that no one else will ever want to dispute that it's mine! But it is a COMPLETE method, so it may be of interest to beginners in Cubology. There are, of course, many other methods of solving The Cube; so I think it reasonable to invite you to find your own personal method. Be careful -- you cannot just pick parts of several methods and always expect them to work together seamlessly. You need ONE COMPLETE METHOD that works for you!

My notation does not agree (yet) with that of the rest of the speed cubing community. I used L and R for Left and Right, T and B for Top and Bottom (This creates a conflict with "back"), and F and K for Front and bacK.

The existing community seems to use L and R for Left and Right, U and D for Up and Down, and F and B for Front and Back. This looks good to me (no conflicts!), so I will begin to change my text to agree with the nore common usage.

 Progress for conversion from LRTBFK to LRUDFB: | Progress for installation: [ ] Chapter Zero                              | [OK] Installed [ ] Chapter One                               | [OK] Installed [ ] Chapter Two                               | [OK] Installed [ ] Chapter Three                             | [OK] Installed [ ] Chapter Four                              | [OK] Installed [ ] Chapter Five                              | [OK] Installed [ ] Chapter Six                               | [OK] Installed [ ] Chapter Seven                             | [OK] Installed [ ] Chapter Eight                             | [OK] Installed [ ] Chapter Nine                              | [OK] Installed [ ] Chapter Ten                               | [OK] Installed [ ] Chapter Eleven                            | [OK] Installed | Initial installation complete.

Empty boxes, "[ ]", indicate "work to be done"; "[OK]" indicates "this conversion complete". I tried to add links for chapters 0 through 3. Several attempts actually worked!    +-+                   </PRE>  <PRE> </PRE>  <PRE> </PRE>  <PRE> </PRE>  <PRE> </PRE>  <PRE> </PRE>  <PRE> </PRE>  <PRE> </PRE> Ray Calvin Baker 15:09, 29 October 2011 (UTC)
 * I have discovered (by experiment) that HTML tags make my formatted text look better.       | |                                                                                |
 * 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         |
 * Several people have written books describing ways to solve Rubik's Cube. Most of my books  |
 * are in storage now, but I thought it might be interesting to write out a solution of my    |
 * own, especially for "middle school" students. The title of this book is NOT "How to Solve  |
 * Rubik's Cube" -- instead of just showing you how to solve the Cube, I want to help you find |
 * your very own personal ways to do that! (I will show you A way, but I am sure that you will |
 * want to find better ways!)                                                                 |
 * This is a journal of my attempts to rediscover A WAY to solve the Cube. There are better,  |
 * faster ways to solve the Cube than this -- I want you to discover some of them for         |
 * yourself. You may find other methods in books, articles by other writers, and on the       |
 * internet. Many of these showcase someone's most sophisticated methods. This journal of mine |
 * showcases some of my most clumsy methods, because I want you to experience the excitement  |
 * of solving a very large problem for yourself. If you read my account carefully, and learn  |
 * to read the diagrams, and understand your Cube, YOU WILL be able to solve the Cube. You may |
 * also be able to better understand what other writers have written about the Cube.          |
 * Although I may mention a few fancy mathematical techniques, for the most part I will try to |
 * SHOW you pictures and diagrams. The greater the number of ways you can find to look care-  |
 * fully at the Cube and visualize it in your mind, the more likely you are to FIND YOUR VERY |
 * OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE.                                                   |
 * Since this is a journal, it can contain a lot of the "internal chatter" that goes on in my |
 * mind as I work on parts of the puzzle. I hope this does not seem stupid to you; I hope it  |
 * helps you to learn to think in some of the ways professional problem solvers use. Even     |
 * though I have tried to express my thoughts in words, to share with you, I believe that most |
 * of my personal thinking is done in pictures and diagrams. Since diagrams and pictures are  |
 * multi-dimensional, you may guess that I am not a linear thinker. (You would be correct.)   |
 * Please bear with me if you think my writing style is tangled and repetitious! (It is!) You |
 * probably have your own personal, best ways of thinking, too. Please learn to use and enjoy |
 * them!                                                                                |
 * This account is dedicated to the students of Christian Center School, in the hope that they |
 * will be encouraged to explore further some of the "mathematics without arithmetic". Please |
 * let me know if you think I can rewrite parts of this article more simply or more clearly!  |
 * This is for YOU!                                                                           |
 * Special thanks are due to the teachers and students who helped me by carefully checking    |
 * every word and diagram, and every step of the solution, as they unscrambled boxes full of  |
 * Cubes!                                                                                |
 * Teachers:                             Students:                                            |
 * Mrs. Baker (Music teacher)            Romeo                                                |
 * Get your name in here, too! Find a mistake, or tell me about your ideas to write a chapter |
 * more clearly.                                                                              |
 * An apology is due to teachers of the English language. After decades of programming        |
 * computers in applications where it is vitally important to quote literal strings exactly, I |
 * still try to make quotations exact, punctuation marks and all. In other words, "This is an |
 * example of non-standard usage of quotation marks.". What shall I do if I wish to refer to, |
 * or quote, passages which contain quotation marks, """", and apostrophes, "''"? I hope you  |
 * can at least understand the extent of some of my technical problems!                       |
 * 1) Some of you may like arithmetic, so I will include a few ways to count and calculate some   #
 * 2) arithmetical stuff about Rubik's Cube. These sections are optional, and are set off by the  #
 * 3) "#" marks. The main use of arithmetic will be simply to show that we have ways to solve ANY #
 * 4) scrambled Rubik's Cube, and to help us track progress toward a solution of this famous      #
 * 5) (notorious?) puzzle.                                                                        #
 * TABLE OF CONTENTS and                                                                      |
 * List of Diagrams                                                                           |
 * Chapter Zero - - - - - - - - - - The Cube                                                  |
 * can at least understand the extent of some of my technical problems!                       |
 * 1) Some of you may like arithmetic, so I will include a few ways to count and calculate some   #
 * 2) arithmetical stuff about Rubik's Cube. These sections are optional, and are set off by the  #
 * 3) "#" marks. The main use of arithmetic will be simply to show that we have ways to solve ANY #
 * 4) scrambled Rubik's Cube, and to help us track progress toward a solution of this famous      #
 * 5) (notorious?) puzzle.                                                                        #
 * TABLE OF CONTENTS and                                                                      |
 * List of Diagrams                                                                           |
 * Chapter Zero - - - - - - - - - - The Cube                                                  |
 * TABLE OF CONTENTS and                                                                      |
 * List of Diagrams                                                                           |
 * Chapter Zero - - - - - - - - - - The Cube                                                  |
 * Chapter Zero - - - - - - - - - - The Cube                                                  |
 * Chapter Zero - - - - - - - - - - The Cube                                                  |
 * Chapter One - - - - - - - - - - Problem Solving Strategies                                 |
 * Chapter One - - - - - - - - - - Problem Solving Strategies                                 |
 * Chapter Two - - - - - - - - - - Using Pictures, Diagrams, Notation, and Abbreviations      |
 * Chapter Two - - - - - - - - - - Using Pictures, Diagrams, Notation, and Abbreviations      |
 * DIAGRAM 2-1. An Intact Rubik's Cube, Showing How Some Parts Are Named                  |
 * DIAGRAM 2-2. Diagram of a Partially Disassembled Cube                                  |
 * DIAGRAM 2-3. Another Perspective of a Partially Disassembled Cube                      |
 * DIAGRAM 2-4. The Physicist's Method -- How to Disassemble the Cube                     |
 * DIAGRAM 2-5. Another View -- How to Disassemble the Cube                               |
 * DIAGRAM 2-6. FRONT Clockwise Move and FRONT Counterclockwise Move                      |
 * DIAGRAM 2-7. The FRONT Twice Move                                                      |
 * DIAGRAM 2-8. BOTTOM Clockwise and BOTTOM Counterclockwise Moves                        |
 * DIAGRAM 2-9. BOTTOM Counterclockwise Move                                              |
 * DIAGRAM 2-10. BOTTOM Clockwise and BOTTOM Counterclockwise Are Invers Operations       |
 * DIAGRAM 2-11. Peeking at the BOTTOM Side to See Bv and B^ More Clearly                 |
 * DIAGRAM 2-12. BACK Clockwise Move and BACK Countertclockwise Move                      |
 * DIAGRAM 2-13. LEFT Clockwise Move and LEFT Counterclockwise Move                       |
 * DIAGRAM 2-14. RIGHT Clockwise Move and RIGHT Counterclockwise Move                     |
 * DIAGRAM 2-15. TOP Clockwise Move and TOP Counterclockwise Move                         |
 * DIAGRAM 2-16. Three Types of Layers                                                    |
 * DIAGRAM 2-17. Each Face of the Cube Takes a Moment on TOP                              |
 * DIAGRAM 2-18. We Return the Cube to its Original Orientation                           |
 * DIAGRAM 2-19. Six More Ways to Rotate the Entire Cube                                  |
 * DIAGRAM 2-20. Four More Ways to Rotate the Entire Cube                                 |
 * DIAGRAM 2-21. An Explanation of an Isometric Drawing With A Cube                       |
 * Chapter Three - - - - - - - - - Some Simple Moves -- Positioning Four Corner Cubies        |
 * Chapter Three - - - - - - - - - Some Simple Moves -- Positioning Four Corner Cubies        |
 * DIAGRAM 3-1. The Goal of Chapter Three.                                                |
 * DIAGRAM 3-2. Correct Position with Correct and Incorrect Orientations                  |
 * Chapter Four - - - - - - - - - - Ignoring Details -- Moving Corner Cubies                  |
 * Chapter Four - - - - - - - - - - Ignoring Details -- Moving Corner Cubies                  |
 * DIAGRAM 4-1. Blank Diagrams for Planning Moves                                         |
 * DIAGRAM 4-2. I Found a Three-Way Swap                                                  |
 * DIAGRAM 4-3. An Experiment Which Seemed Useless                                        |
 * DIAGRAM 4-4. Three More Moves Provide a Useful Two-Cubie Interchange                   |
 * DIAGRAM 4-5. Summary of FRONT RIGHT TOP and BOTTOM FRONT RIGHT SWAP                    |
 * DIAGRAM 4-6. How Do You Swap Diagonally Opposite Cubies?                               |
 * DIAGRAM 4-7. Plan for SWAPping Diagonally Opposite Cubies the Hard Way                 |
 * DIAGRAM 4-8. Ten Stages in SWAP Diagonally Opposite Cubies the Hard Way                |
 * DIAGRAM 4-9. Preparing to Locate the Final Four Corner Cubies                          |
 * DIAGRAM 4-10. Introducing More Abbreviated Diagrams                                    |
 * DIAGRAM 4-11. The First Set of Six Possibilities                                       |
 * DIAGRAM 4-12. The Second Set of Siz Possibilities                                      |
 * DIAGRAM 4-13. The Third Set of Six Possibilities                                       |
 * DIAGRAM 4-14. The Final Set of Six Possibilities                                       |
 * Chapter Five - - - - - - - - - - Keeping Track of Lots of Details -- Rotating Corner Cubies |
 * Chapter Five - - - - - - - - - - Keeping Track of Lots of Details -- Rotating Corner Cubies |
 * DIAGRAM 5-1. Diagram for Orienting Corner Cubies -- Position Zero                      |
 * DIAGRAM 5-2. What FRONT Clockwise Did                                                  |
 * DIAGRAM 5-3. What RIGHT Counterclockwise Did -- Position Two                           |
 * DIAGRAM 5-4. Comparing Position Zero With Position Two                                 |
 * DIAGRAM 5-5. After Six Turns                                                           |
 * DIAGRAM 5-6. Unwrapping the Cube                                                       |
 * DIAGRAM 5-7. Unwrapped Cube After Six Moves                                            |
 * DIAGRAM 5-8. Developing a New Type of Diagram                                          |
 * DIAGRAM 5-9. The Peculiar Arithmetic of Rotate by 120 Degrees                          |
 * Chapter Six - - - - - - - - - - Customize Your Moves -- Commutation                        |
 * Chapter Six - - - - - - - - - - Customize Your Moves -- Commutation                        |
 * DIAGRAM 6-1. Example Step One                                                          |
 * DIAGRAM 6-2. Example Step Two                                                          |
 * DIAGRAM 6-3. Example Step Three                                                        |
 * DIAGRAM 6-4. Summary: How to Undo One Move                                             |
 * Chapter Seven - - - - - - - - - Finishing the Orientation of Corner Cubies                 |
 * Chapter Seven - - - - - - - - - Finishing the Orientation of Corner Cubies                 |
 * DIAGRAM 7-1. Typical Coded Rotation Problems                                           |
 * DIAGRAM 7-2. A Sequence of Moves Which Changes Orientations od Corner Cubies           |
 * DIAGRAM 7-3. Repeating the Sequence of Moves                                           |
 * DIAGRAM 7-4. A Sucessful Experiment                                                    |
 * DIAGRAM 7-5. Solving a Typical Problem                                                 |
 * DIAGRAM 7-6. Solving Three Corners Rotated in Same Direction                           |
 * DIAGRAM 7-7. Find a Solution For a Rotation Problem                                    |
 * DIAGRAM 7-8. Find Second Corner Cubie to Rotate, Then Apply These Moves                |
 * DIAGRAM 7-9. Is the FRONT LEFT TOP Corner Cubie Properly Oriented?                     |
 * DIAGRAM 7-10. X Marks Our Progress                                                     |
 * Chapter Eight - - - - - - - - - Moving Edge Cubies                                         |
 * Chapter Eight - - - - - - - - - Moving Edge Cubies                                         |
 * DIAGRAM 8-1. The Down-Slice Move                                                       |
 * DIAGRAM 8-2. Turn the TOP Layer Twice                                                  |
 * DIAGRAM 8-3. The Up-Slice Move                                                         |
 * DIAGRAM 8-4. Exchanging Three Edge Cubies                                              |
 * DIAGRAM 8-5. Recommended Way to Position Edge Cubies                                   |
 * DIAGRAM 8-6. A New Type of Diagram Emphasizes the Edge Cubies                          |
 * DIAGRAM 8-7. Positioning Cubie #2 Without Disturbing #1                                |
 * DIAGRAM 8-8. Positioning Cubie #3 Without Disturbing #2 Or #1                          |
 * Chapter Nine - - - - - - - - - - Rubik's Maneuver -- How to Flip Two Edge Cubies           |
 * Chapter Nine - - - - - - - - - - Rubik's Maneuver -- How to Flip Two Edge Cubies           |
 * DIAGRAM 9-1. A Cube in Need of Rubik's Maneuver                                        |
 * DIAGRAM 9-2. Instructions for Rubik's Maneuver                                         |
 * DIAGRAM 9-3. Special Case Not Handled by Instuctions in Chapter Eight                  |
 * DIAGRAM 9-4. The Unscrambled Cube                                                      |
 * Chapter Ten - - - - - - - - - - Reference Summary of Useful Operations                     |
 * Chapter Ten - - - - - - - - - - Reference Summary of Useful Operations                     |
 * INTERCHANGE THREE CORNER CUBIES                                                        |
 * INTERCHANGE TWO CORNER CUBIES                                                          |
 * CHANGE ORIENTATION OF TWO CORNER CUBIES                                                |
 * INTERCHANGE THREE EDGE CUBIES                                                          |
 * CHANGE ORIENTATION OF TWO EDGE CUBIES                                                  |
 * Chapter Eleven - - - - - - - - - I GET A TASTE OF MY OWN PRESCRIPTIONS                     |
 * Chapter Eleven - - - - - - - - - I GET A TASTE OF MY OWN PRESCRIPTIONS                     |
 * There is a "hidden" sub-group -- I need to find a way to solve it, too.                |
 * Chapter Twelve - - - - - - - - - MY (YOUR) VERY OWN PERSONAL WAY TO SOLVE RUBIK'S CUBE     |
 * Chapter Twelve - - - - - - - - - MY (YOUR) VERY OWN PERSONAL WAY TO SOLVE RUBIK'S CUBE     |
 * (You need to write this chapter yourself!)                                             |

title: Rubik's Cube/Introduction


text:
Rubik's Cube/Problem Solving Strategies

 This looks best in "Edit" mode!  Ray Calvin Baker 19:24, 18 August 2011 (UTC)

 * HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE                              |
 * (Preliminary April 20, 2007 version)            |
 * by Mr. Ray Calvin Baker                         |
 * FREE public Domain Educational material         |
 * Chapter One - - - - - - - - - - Problem Solving Strategies                                 |
 * These are some generally useful ideas for solving all kinds of problems. I will try to show |
 * you some of the ways I applied them to finding ways to unscramble Rubik's Cube.            |
 * OBSERVE! Any detail could provide an important clue. (But sometimes the most obvious       |
 * feature of a puzzle can be deceptive. Look again and keep looking.)                        |
 * EVALUATE! Some details may be more important than others. Some details may be irrelevant.  |
 * Keep evaluating what you have learned by observation, and keep trying to concentrate on the |
 * important details. Remenber also that what is important may depend on what you are trying  |
 * to accomplish.                                                                             |
 * TRY THINGS! If you are stuck, do something -- anything. But be prepared to observe and     |
 * evaluate what happens.                                                                     |
 * TAKE NOTES! Write down your observations and thoughts. Make diagrams and pictures. I spent |
 * several hours working with pencil and paper and my computer for every few minutes of       |
 * playing with the Cube. And here's why: after I make several moves with the Cube, I may     |
 * forget what I did, and the result is a scrambled Cube. If I write down the moves, I can    |
 * know exactly what I did. If the results were useful, I have a record which I can read and  |
 * use again. Even if the results were not what I had hoped for, I may be able to undo the    |
 * mistake step by step, then try again.                                                      |
 * DRAW PICTURES and DIAGRAMS! They may help you to visualize what happens as you do things to |
 * your Cube. I intend to use pictures and diagrams throughout this entire writing.           |
 * PLAN and PREDICT! With a good plan firmly in mind, you will not be afraid to risk any      |
 * apparent progress. Even if your predictions do not always work out as you expect, you will |
 * learn from the attempts. Keep trying!                                                      |
 * DIVIDE! Break up a big problem into smaller parts if you can see a way to do that.         |
 * SIMPLIFY THE PROBLEM. Sometimes, you can gain insight or experience by solving a simpler   |
 * problem. This may also help you divide the big problem into several smaller problems.      |
 * DON'T BE AFRAID TO START OVER! Several times, I thought I had found a clever move; but it  |
 * didn't work when I tried it on the Cube. I had to start over on that part of the puzzle.   |
 * TAKE A BREAK! After working and studying hard for a while, you have saturated your mind    |
 * with all that you know about a problem. If you are not making any progress, sometimes it   |
 * helps simply to take a break. Rest, or try doing something else. I sometimes have dreamed  |
 * in my sleep about the answer to a problem. I had to check out whether the dream was        |
 * correct, but often a flash of insight is just what you need to solve a problem.            |
 * CHANGE YOUR POINT OF VIEW. Consider how others might view the problem. Seek and consider   |
 * other view points. Try to think about what things would be like if you actually did solve  |
 * the problem. Change your perspective, in any ways that you can. Be flexible! Too many      |
 * people have already said, "It can't be done!" (and been wrong!) for you to take that       |
 * statement seriously.                                                                       |
 * PERSEVERE! KEEP TRYING! Remember "NED" -- Never give up! Encourage others! Do your best!   |
 * At a private Christian school, you have another option -- "If any of you lacks wisdom,     |
 * let him ask God, who gives graciously, and will not scold you."                            |
 * with all that you know about a problem. If you are not making any progress, sometimes it   |
 * helps simply to take a break. Rest, or try doing something else. I sometimes have dreamed  |
 * in my sleep about the answer to a problem. I had to check out whether the dream was        |
 * correct, but often a flash of insight is just what you need to solve a problem.            |
 * CHANGE YOUR POINT OF VIEW. Consider how others might view the problem. Seek and consider   |
 * other view points. Try to think about what things would be like if you actually did solve  |
 * the problem. Change your perspective, in any ways that you can. Be flexible! Too many      |
 * people have already said, "It can't be done!" (and been wrong!) for you to take that       |
 * statement seriously.                                                                       |
 * PERSEVERE! KEEP TRYING! Remember "NED" -- Never give up! Encourage others! Do your best!   |
 * At a private Christian school, you have another option -- "If any of you lacks wisdom,     |
 * let him ask God, who gives graciously, and will not scold you."                            |
 * At a private Christian school, you have another option -- "If any of you lacks wisdom,     |
 * let him ask God, who gives graciously, and will not scold you."                            |
 * let him ask God, who gives graciously, and will not scold you."                            |

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Rubik's Cube/Problem Solving Strategies


text:
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.)                                                               |


text:
Rubik's Cube/The Cube

This looks best in "Edit" mode!

Ray Calvin Baker 19:31, 18 August 2011 (UTC)

  Ray Calvin Baker 15:01, 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 Zero - - - - - - - - - - The Cube                                                  |
 * DO NOT PEEL ANY OF THE COLORED LABELS OFF YOUR CUBE! (I call this "the chemist's method,   |
 * because it involves the breaking of octillions of weak molecular bonds. This is not meant  |
 * to disparage chemists at all -- one of the first books on the Cube that I read was written |
 * by a chemist, and he described elegant geometrical and mathematical moves such as "Rubik's |
 * Maneuver".)                                                                                |
 * There are other, less damaging ways to disassemble a Cube if you really want to see what's |
 * inside it (or if you want to cheat.) -- see Chapter Two, "Using Pictures, Diagrams,        |
 * Notation, and Abbreviations", for "the Physicist's method of disassembling the Cube. Of    |
 * course, if you can follow directions from a mathematician, you won't need to damage or take |
 * apart your Cube at all!                                                                    |
 * From the outside, your Cube looks like it is made of 27 smaller cubes. (There's a picture  |
 * in Chapter Two, "Using Pictures, Diagrams, Notation, and Abbreviations".) Each of these    |
 * smaller cubes is called a cubie. A cube is a shape having eight corners, six flat square   |
 * faces, and twelve edges. A Cube (upper case) is Rubik's Cube (or a cheap imitation of it). |
 * When it came from the factory, each face of your Cube had nine colored labels, all the same |
 * color -- six faces, six different colors. It may not seem likely to you now, but your Cube |
 * (unless it's physically damaged) can be restored to this condition.                        |
 * Your Cube consists of one central core, which holds six axles. Each axle supports one face |
 * cubie, which is free to rotate unless blocked by other components of your Cube. Each face  |
 * cubie has a colored label -- six faces, six different colors. Each of the twelve edge      |
 * cubies is clamped between a pair of face cubies. Each edge cubie has two different colored |
 * labels. Each of eight sets of three edge cubies clamps a corner cubie to the rest of the   |
 * Cube. Each corner cubie has three different colored labels. (There are diagrammatic        |
 * pictures in Chapter Two, "Using Pictures, Diagrams, Notation, and Abbreviations".)         |
 * Even though each of the eight corner cubies and twelve edge cubies is a separate piece,    |
 * they are all clamped together into a secure and stable assembly -- the Cube. And each face |
 * of the Cube can be rotated. Of course, that rearranges the colored labels.                 |
 * YOUR MISSION, should you choose to accept it, IS TO RESTORE YOUR CUBE TO ITS FACTORY       |
 * CONDITION!                                                                                |
 * How will you be able to tell where each cubie belongs when part of your Cube is scrambled? |
 * Look at the center square on each side. This piece is firmly attached to the core of the   |
 * Cube, and is only free to rotate, The central square of each side of the Cube is your      |
 * "fixed landmark" around which you must arrange all of the other cubies. (Once we start     |
 * arranging edge cubies, we will temporarily use other landmarks, but we will always restore |
 * the positions of the center squares.)                                                      |
 * I suggest that you skim through this paper to see how the parts are organized, and to      |
 * convince yourself that there really are ways to solve the Cube. Then, you can pick up a    |
 * scrambled Cube to try to follow the exact steps you will need to solve it. Don't be        |
 * discouraged if your first few attempts seem only to scramble the Cube more! Start again    |
 * from the beginning if you have to. You will become more familiar with your Cube, and you   |
 * will, with practice, become more confident that you can master each necessary operation.   |
 * 1) Just how many ways are there to arrange the cubies of the Cube? Let's calculate! But be     #
 * 2) aware that most calculators cannot hold numbers this large! (Some calculators may use       #
 * 3) scientific notation to express large numbers, but they almost always truncate the           #
 * 4) significant figures of the numbers, which is an approximation, not an exact number.)        #
 * 5) There are 8 factorial (usually written "8!") ways to arrange the eight corner cubies.       #
 * 6) This is 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 = 8 * 7 * 6 * 5 * 4 * 3 * #
 * 7) 2 * 1 =  40,320. (I have used lots of calculators and computer languages which use "*" as   #
 * 8) the sign for "multiply" or "times".)                                                        #
 * 9) Then, There are ( (three to the eighth power) divided by three ) ways to orient the eight   #
 * 10) corner cubies. This is often written (on a computer) as "(3 ^ 8) / 3", which is             #
 * 11) (3 * 3 * 3 * 3 * 3 * 3 * 3 * 3) / 3 =  6,561 / 3  = 2,187. (We need to divide by three      #
 * 12) here, because it is not possible to rotate just one corner cubie. Whenever one corner cubie #
 * 13) is rotated clockwise, another has to rotate counterclockwise, and vice versa -- unless, of  #
 * 14) course, you physically disassemble your Cube.)                                              #
 * 15) Next, there are 12! / 2 ways to arrange the edge cubies. This is (12 * 11 * 10 * 9 * 8 * 7  #
 * 16) * 6 * 5 * 4 * 3 * 2 * 1) / 2 =  479,001,600 / 2 = 239,500,800. ( We need to divide by two   #
 * 17) here, because it is not possible to interchange exactly two edge cubies, unless you         #
 * 18) physically disassemble the Cube.)                                                           #
 * 19) Lastly, there are ( ( 2 ^ 12 ) / 2 ) = ( 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 )    #
 * 20) / 2 = 4,096 / 2 = 2,048 ways to orient the twelve edge cubies. (It is not possible to       #
 * 21) "flip" just one edge cubie, unless you disassemble... )                                     #
 * 22) Multiply these four numbers together, and you get 40,320 * 2,187 *  239,500,800 * 2,048     #
 * 23) = 43,252,003,274,489,856,000, which is exactly the number of ways the cubies of Rubik's     #
 * 24) Cube can be arranged. This is a twenty digit number -- forty-three quintillion, two        #
 * 25) hundred fifty-two quadrillion, three trillion, two hundred seventy-four billion, four       #
 * 26) hundred eighty-nine million, eight hundred fifty-six thousand. (OK, one of these ways is    #
 * 27) actually the unscrambled, pristine Cube, so there are really only                           #
 * 43,252,003,274,489,855,999 different scrambled Cubes, plus exactly ONE unscrambled Cube.)  #
 * 1) Each of those four factors of 43,252,003,274,489.056,000 corresponds to one of our major    #
 * 2) goals, which are discussed in the following chapters.                                       #
 * 3) By the way, if you do dissassemble your Cube, there are twelve distinctively different ways #
 * 4) to put it back together. Twelve = 3 * 2 * 2, and these factors correspond to those numbers  #
 * 5) by which we divided in the previous calculations. Of these twelve ways, only one will allow #
 * 6) you to solve the Cube. There will always (until you disassemble again) be something wrong   #
 * 7) with the other eleven ways of re-assembly -- an edge cubie will be flipped (disoriented),   #
 * 8) or two edge cubies will be interchanged, or a corner cubie will be improperly oriented --   #
 * 9) you get the idea. (The twelve ways to re-assemble the Cube are sometimes called "orbits".)  #
 * 1) Multiply these four numbers together, and you get 40,320 * 2,187 *  239,500,800 * 2,048     #
 * 2) = 43,252,003,274,489,856,000, which is exactly the number of ways the cubies of Rubik's     #
 * 3) Cube can be arranged. This is a twenty digit number -- forty-three quintillion, two        #
 * 4) hundred fifty-two quadrillion, three trillion, two hundred seventy-four billion, four       #
 * 5) hundred eighty-nine million, eight hundred fifty-six thousand. (OK, one of these ways is    #
 * 6) actually the unscrambled, pristine Cube, so there are really only                           #
 * 43,252,003,274,489,855,999 different scrambled Cubes, plus exactly ONE unscrambled Cube.)  #
 * 1) Each of those four factors of 43,252,003,274,489.056,000 corresponds to one of our major    #
 * 2) goals, which are discussed in the following chapters.                                       #
 * 3) By the way, if you do dissassemble your Cube, there are twelve distinctively different ways #
 * 4) to put it back together. Twelve = 3 * 2 * 2, and these factors correspond to those numbers  #
 * 5) by which we divided in the previous calculations. Of these twelve ways, only one will allow #
 * 6) you to solve the Cube. There will always (until you disassemble again) be something wrong   #
 * 7) with the other eleven ways of re-assembly -- an edge cubie will be flipped (disoriented),   #
 * 8) or two edge cubies will be interchanged, or a corner cubie will be improperly oriented --   #
 * 9) you get the idea. (The twelve ways to re-assemble the Cube are sometimes called "orbits".)  #
 * 1) or two edge cubies will be interchanged, or a corner cubie will be improperly oriented --   #
 * 2) you get the idea. (The twelve ways to re-assemble the Cube are sometimes called "orbits".)  #

title: Rubik's Cube/The Cube