Rubik's Cube/Some Simple Moves

This looks best in "Edit" mode! I have discovered (by experiment) that the proper HTML tags solve the problem. Ray Calvin Baker 14:57, 29 October 2011 (UTC) Ray Calvin Baker 19:33, 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 Three - - - - - - - - - Some Simple Moves -- Positioning Four Corner Cubies        |
 * You could call this "Goal Zero -- some warming-up moves". Although the individual moves are |
 * all (ridiculously) simple, the ways to organize those simple moves is more complicated.    |
 * It's the organization of these simple things that makes Rubik's Cube a MATHEMATICAL puzzle. |
 * Many people have been able to get one of the six sides of the Cube unscrambled; but        |
 * considerably fewer have managed to get all six sides completely unscrambled. If you are one |
 * of those who can unscramble one side, you may not need to read the rest of this chapter at |
 * all -- EXCEPT you may still need to learn how to organize what you know into a complete    |
 * solution. One way to do this is shown in the example of "programmed learning" demonstrated |
 * in this chapter. The other major thing in this chapter is guided practice in using the     |
 * notation developed in Chapter Two, "Using Pictures, Diagrams, Notation, and Abbreviations". |
 * My notes can not help you unless you know how to use them!                                 |
 * Now, although you are welcome to use my notes, I do NOT encourage you to memorize my       |
 * solution! I encourage you to find your own personal, better methods. Then memorize those!  |
 * You may already have found your own personal way to accomplish the goal of this chapter.   |
 * But please check diagram 3-1B carefully, to make sure you have all four corner cubes in    |
 * their proper places. Please be aware that having gotten one side correctly arranged ONCE is |
 * not quite the same as being able to get one side correctly arranged EVERY TIME!            |
 * It would be very nice if you had an unscrambled ("pristine") Cube so you could see how the |
 * cubies move around as you perform the operations I intend to show you, but you probably    |
 * have only a scrambled Cube. So, I will try to describe some very simple moves which will   |
 * unscramble part of one side of your Cube, to get you started, and to build up your         |
 * confidence.                                                                                |
 * 1) There are 24 ways to orient a Cube (or even a plain cube). One of the six faces is on TOP.  #
 * Another one of the faces is on the BOTTOM. One of the four remaining sides can be in FRONT. #
 * The positions and orientations of all other sides are already determined, once the TOP and #
 * FRONT are chosen.                                                                          #
 * 1) The number of possibilities for each side that I described above are: 6, 1, 4, 1, 1, and 1. #
 * 2) The product of these numbers is 24, and this is the number of ways a cube can  be oriented. #
 * 3) Check it out!                                                                               #
 * 4) How is this fact useful? Turning the entire Cube does nothing to unscramble the Cube,  but  #
 * 5) it never scrambles the Cube any worse, either. This means that, by positioning the entire   #
 * 6) Cube properly, you can multiply your Cube solving methods by a factor of 24. Why learn 24   #
 * 7) different methods, when you can freely rotate the Cube to a desired position, apply just    #
 * 8) ONE method, then return the Cube to its original orientation? This principle will be        #
 * 9) discussed more fully in Chapter Six, "Customize Your Moves -- Commutation".                 #
 * Here is the way I would begin. Pick one color on your Cube. Turn your Cube so that the     |
 * center square of this color is on TOP. We will now try to arrange the four corner cubies   |
 * with this color, so that five cubies all show this color on the TOP of your Cube.          |
 * _ * _                                        _ * _                          |
 * _ * _ ? _ * _                                _ + _TOP_ * _                      |
 * _ * _ ? _ * _ ? _ * _                        _ * _ ? _ * _ ? _ * _                  |
 * * _ ? _ * _TOP_ * _ ? _ *                    * _TOP_ * _TOP_ * _TOP_ *                |
 * |  * _ ? _ * _ ? _ *   |                     |   * _ ? _ * _ ? _ *   |                |
 * | ? |  * _ ? _ *   | ? |                     | F |   * _TOP_ *   | R |                |
 * * _ | ? |  *   | ? | _ *                     * _ | ? |   *   | ? | _ *                |
 * |  * _ | ? | ? | _ *   |                     |   * _ | F | R | _ *   |                |
 * | ? |  * _ | _ *   | ? |                     | ? |   * _ | _ *   | ? |                |
 * * _ | F |  *   | R | _ *                     * _ | F |   *   | R | _ *                |
 * |  * _ | ? | ? | _ *   |                     |   * _ | ? | ? | _ *   |                |
 * | ? |  * _ | _ *   | ? |                     | ? |   * _ | _ *   | ? |                |
 * * _ | ? |  *   | ? | _ *                     * _ | ? |   *   | ? | _ *                |
 * * _ | ? | ? | _ *                            * _ | ? | ? | _ *                    |
 * * _ | _ *                                    * _ | _ *                        |
 * What we start with.                         What we want to end with at this stage    |
 * "?" means "I don't know,                    of the solution. (BACK and LEFT sides     |
 * but it doesn't matter".                     should show a similar pattern.)           |
 * We are ignoring the edge cubies at this stage of the solution.          |
 * DIAGRAM 3-1A.                               DIAGRAM 3-1B.                             |
 * DIAGRAM 3-1. The Goal of Chapter Three.                                               |
 * One of the first things we need to learn is how to determine if a cubie is in the correct  |
 * position, even if it's not properly oriented. Another thing to learn is how to determine if |
 * a cubie is in the correct position, and also properly oriented.                            |
 * Let's learn these two things by focusing for a little while on just the FRONT RIGHT TOP    |
 * position of the "What we want to end with..." diagram above (Diagram 3-1B). Can you imagine |
 * a diagonal connection from the TOP side of the FRONT RIGHT TOP position to the TOP central |
 * cubie? I hope so!                                                                          |
 * If you can also imagine a diagonal connection from the FRONT side of the FRONT RIGHT TOP   |
 * cubie to the FRONT central cubie, you are two thirds of the way toward learning how to     |
 * recognize "cubie in correct position, and also properly oriented". If you can also imagine |
 * a diagonal connection from the RIGHT side of the FRONT RIGHT TOP cubie to the RIGHT central |
 * cubie, you have understood this lesson.                                                    |
 * The next three diagrams (Diagrams 3-2 A through C)) show you the six places you need to    |
 * check on your Cube, to correctly master this stage of a solution. These three diagrams also |
 * show you the possible patterns you might find.                                             |
 * _ * _                        _ * _                         _ * _              |
 * _ * _ ? _ * _                _ * _ ? _ * _                 _ * _ ? _ * _          |
 * _ * _ ? _ * _ ? _ * _        _ * _ ? _ * _ ? _ * _         _ * _ ? _ * _ ? _ * _      |
 * * _ ? _ * _TOP_ * _ ? _ *    * _ ? _ * _TOP_ * _ ? _ *     * _ ? _ * _TOP_ * _ ? _ *    |
 * |  * _ ? _ * _ ? _ *   |     |   * _ ? _ * _ ? _ *   |     |   * _ ? _ * _ ? _ *   |    |
 * | ? |  * _TOP_ *   | ? |     | ? |   * _ F _ *   | ? |     | ? |   * _ R _ *   | ? |    |
 * * _ | ? |  *   | ? | _ *     * _ | ? |   *   | ? | _ *     * _ | ? |   *   | ? | _ *    |
 * |  * _ | F | R | _ *   |     |   * _ | R |TOP| _ *   |     |   * _ |TOP| F | _ *   |    |
 * | ? |  * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |    |
 * * _ | F |  *   | R | _ *     * _ | F |   *   | R | _ *     * _ | F |   *   | R | _ *    |
 * |  * _ | ? | ? | _ *   |     |   * _ | ? | ? | _ *   |     |   * _ | ? | ? | _ *   |    |
 * | ? |  * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |    |
 * * _ | ? |  *   | ? | _ *     * _ | ? |   *   | ? | _ *     * _ | ? |   *   | ? | _ *    |
 * * _ | ? | ? | _ *            * _ | ? | ? | _ *             * _ | ? | ? | _ *        |
 * * _ | _ *                    * _ | _ *                     * _ | _ *            |
 * FRONT RIGHT TOP cubie        FRONT RIGHT TOP cubie         FRONT RIGHT TOP cubie        |
 * in correct position,         in correct position, but      in correct position, but     |
 * with correct orientation.    with incorrect orientation.   with incorrect orientation.  |
 * (It needs to be rotated 120  (It needs to be rotated 120  |
 * degrees counterclockwise.)   degrees clockwise.)          |
 * DIAGRAM 3-2A.                DIAGRAM 3-2B.                 DIAGRAM 3-2C.                |
 * DIAGRAM 3-2. Correct Position with Correct and Incorrect Orientations                   |
 * The only other possibility is that the cubie at the FRONT RIGHT TOP position is INCORRECTLY |
 * positioned. In this case, the orientation doesn't matter -- the cubie is in the wrong      |
 * place! Can you recognize this siuation when it occurs? Hint: there will be at least one    |
 * color on the corner cubie at the FRT position which matches NONE of the TOP center square, |
 * the FRONT center square, or the RIGHT center square.                                       |
 * You may need to turn the entire Cube so that each of the corner cubies can be examined in  |
 * turn. At this time, you will want to make all four corners of the TOP side correctly       |
 * positioned and correctly oriented. In later chapters, you will want to make all eight      |
 * corners of the Cube properly positioned and properly oriented.                             |
 * Hopefully, you now know what I would be looking for with respect to the four corners of    |
 * your chosen side of the Cube. Let's do some looking and make some moves to get the first   |
 * four corner cubies of our Cube into proper order. BE ALERT! You should be able to          |
 * accomplish these "warm up" moves on your own, with much less hassle and worry. But as your |
 * teacher, I need to spell out a method which is guaranteed to work. You, as the student,    |
 * have the opportunity to find the best way that works for you!                              |
 * Please be patient with me if the following discussion seems long and tedious! I want you to |
 * develop your own "common sense" appreciation for your Cube! But not everyone has the       |
 * cleverness to solve the Cube without help, so I need to give a full explanation of one way |
 * to solve the Cube for the benefit of those people.                                         |
 * For your information, an "ASSERTION" is a statement which is supposed to be true. Check the |
 * statement carefully, because a mistake has been made somewhere if the "ASSERTION" is NOT   |
 * true! Sometimes, the programmer has made a mistake; that's why programs need to be checked |
 * very carefully. Sometimes, someone misinterpreted the instructions. That's why it's hard   |
 * work to write instructions which are easy to follow. Sometimes, someone failed to perform  |
 * the instructions properly. That's why YOU may need to start over again.                    |
 * You are about to experience an experiment in what is called "programmed learning". The     |
 * steps are all supposed to be very simple, but they do not always go in "1, 2, 3" order.    |
 * Sometimes, you will be asked a question. The answer will usually be quite simple, like     |
 * "YES!" or "NO!" The next step you must take will depend on how you answer the question. Be |
 * sure to follow instructions like "Go to step 3-6." and "Continue with step 3-3." very      |
 * carefully!                                                                                 |
 * BEGIN: STEP 3-1. Determine and record the starting position for your Cube. Here's how:     |
 * Write down on scrap paper, "STARTING POSITION:                                             |
 * The color of the BOTTOM center square is _____.                |
 * The color of the FRONT center square is _____.                 |
 * The color of the BACK  center square is _____.                 |
 * The color of the LEFT  center square is _____.                 |
 * The color of the RIGHT center square is _____.                 |
 * The color of the TOP   center square is _____."                |
 * Fill in the blanks.                                                                        |
 * Now you have a record of the starting position.                                            |
 * You may need to return your Cube to this position several times during the "warm up"       |
 * process. You may also often need to determine whether or not your Cube is in this starting |
 * position.                                                                                  |
 * Now that you have a record of the starting position, the first "warm up" task is to put    |
 * four corner cubies at their correct location.                                              |
 * STEP 3-2. Is the correct cubie in the FRONT RIGHT TOP location? Check for this by          |
 * comparing the three colors of the three sides of the cubie with the central square of the  |
 * FRONT, RIGHT, and TOP sides of the Cube. (The orientation of this cubie may be wrong at    |
 * this time -- I'm only interested in whether or not the correct cubie is in this location.) |
 * YES! Go to step 3-6.            NO! Continue with step 3-3.                                |
 * STEP 3-3. Is the cubie which belongs at FRONT RIGHT TOP actually at the FRONT LEFT TOP     |
 * location?                                                                                  |
 * YES! Go to step 3-7.            NO! Continue with step 3-4.                                |
 * STEP 3-4. Is the cubie which belongs at FRONT RIGHT TOP actually at the BACK LEFT TOP      |
 * location?                                                                                  |
 * YES! Go to step 3-8.            NO! Continue with step 3-5.                                |
 * STEP 3-5. Is the cubie which belongs at FRONT RIGHT TOP actually at the BACK RIGHT TOP     |
 * location?                                                                                  |
 * YES! Go to step 3-9.            NO! Continue with step 3-11.                               |
 * STEP 3-6. ASSERTION: The correct cubie is at the FRONT RIGHT TOP position.                 |
 * Rotate the entire Cube, using the "3T^" move.                                              |
 * Then continue with step 3-10.                                                              |
 * STEP 3-7. ASSERTION: The cubie which belongs at FRONT RIGHT TOP is actually at the FRONT   |
 * LEFT TOP location.                                                                         |
 * Perform this series of moves,  "Lv B^ L^ B2 Fv B^ F^ Lv B^ L^ 3T^",                        |
 * then go to step 3-10.                                                                      |
 * STEP 3-8. ASSERTION: The cubie which belongs at FRONT RIGHT TOP is actually at the BACK    |
 * LEFT TOP location.                                                                         |
 * Perform this series of moves,  "L^ Bv Lv R^ B2 Rv B2 L^ B2 Lv 3T^",                        |
 * then go to step 3-10.                                                                      |
 * STEP 3-9. ASSERTION: The cubie which belongs at FRONT RIGHT TOP is actually at the BACK    |
 * RIGHT TOP location.                                                                        |
 * Perform this series of moves,  "K^ Fv B^ F^ Bv Kv 3T^,                                     |
 * then go to step 3-10.                                                                      |
 * STEP 3-10. Is the Cube in its starting position?                                           |
 * YES! Go to step 3-15.           NO!  Go back to step 3-2.                                  |
 * STEP 3-11. ASSERTION: The cubie which belongs at FRONT RIGHT TOP is actually on the        |
 * BOTTOM layer.                                                                              |
 * (You will need to find it there, then move it into the correct position.)                  |
 * Rotate the BOTTOM of the Cube until the cubie which belongs in the FRONT RIGHT TOP         |
 * position is at the BOTTOM BACK RIGHT position. (I trust you to know how to do this.)       |
 * Then perform this sequence of moves, "Fv B2 F^ 3T^".                                       |
 * Finally, go to step 3-10.                                                                  |
 * STEP 3-12. ASSERTION: The cubie in the FRONT RIGHT TOP position is correctly oriented.     |
 * Rotate the entire Cube, using the "3T^" move.                                              |
 * Go to step 3-14.                                                                           |
 * STEP 3-13. ASSERTION: The cubie in the FRONT RIGHT TOP position is NOT correctly           |
 * oriented.                                                                                  |
 * Perform this sequence of moves, "Fv B2 F^ R^ B2 Rv".                                       |
 * Then continue on to step 3-15.                                                             |
 * STEP 3-14. Is the Cube in its starting position?                                           |
 * YES! Go to step 3-16.           NO! Continue on to step 3-15.                              |
 * STEP 3-15. ASSERTION: All four corner cubes in the TOP layer are correctly positioned.     |
 * Is the cubie in the FRONT RIGHT TOP position correctly oriented?                           |
 * YES! Go back to step 3-12.      NO! Go back to step 3-13.                                  |
 * STEP 3-16. ASSERTION: The Cube is in its starting position.                                |
 * ASSERTION: All four corner cubies in the TOP layer are correctly positioned and            |
 * properly oriented.                                                                         |
 * You have completed the warm-up exercise. Your Cube should look like DIAGRAM 3-1B.          |
 * If it does, CONGRATULATIONS!                                                               |
 * If it doesn't, you need to try again, more carefully this time.                            |
 * I am not especially fond of "programmed learning", because it is so slow and tedious, and  |
 * because it is often very difficult to get a good, global understanding of what is really   |
 * going on. (Too many trees, not enough view of the forest!) Sometimes, for fairly simple    |
 * things, it does work reasonably well, and it does tell you how to accomplish some things.  |
 * Please don't be disappointed that I only arranged the corner cubies -- Chapter Eight,      |
 * "Moving Edge Cubies", and Chapter Nine, "Rubik's Maneuver -- How to Flip Two Edge Cubies", |
 * show you ways to arrange the edge cubies as well, and you can try to do that now, if you   |
 * really want to. But I must warn you, some of the moves in the chapters before Chapter Eight |
 * may mess up those cubies again, so it could be a waste of time to try to arrange them now. |
 * But feel free to do what you want -- I'm trying to help you find your own ways to solve the |
 * Cube! You may even find ways better than mine, which do NOT mess up the other cubies!      |
 * CAUTION! Although the methods used in this chapter appear to have interchanged two corner  |
 * cubies, they may have messed up several other corner cubies. We will need to explore more  |
 * carefully to find methods which will work without undesirable side effects. (Hint: You may |
 * start with the methods used in this chapter, find out what the side effects really are,    |
 * then explore variations of these methods. Sometimes, you may be able to use some of the    |
 * side effects. Other times, you may need to find ways to avoid unwanted side effects.)      |
 * 1) We have now positioned four corner cubies correctly. For those who like arithmetic, this    #
 * 2) means we now have only (4 factorial) * ((3 to the eighth power) / 3) * ((12 factorial) / 2) #
 * 3) * (2 to the twelfth power) / 2) ways to arrange the Cube. This is (24) * (6,561  / 3) *     #
 * 4) (479,001,600 / 2) * (4,096 / 2) =  24 * 2,187 *  239,500,800 * 2,048 =                      #
 * 25,745,240,044,339,200. I told you it is possible to use arithmetic to show progress!      #
 * 1) Although we also oriented the four corner cubies in the TOP layer, I am not going to count  #
 * 2) this as progress, because moves in the next chapters may (temporarily) mess up the          #
 * 3) orientation of some cubies. I will count up more progress later, at the proper time, when   #
 * 4) the corner cubies have been securely oriented.                                              #
 * Ironically, this chapter about some of the simplest moves has been the hardest for me to   |
 * write! It has been about as frustrating as trying to define "common sense". Many of you    |
 * readers are already quite comfortable with your own knowledge of how to get the cubies on  |
 * one face of the Cube correct. My advice to you is -- stick with your own methods if you are |
 * confident they will always work.                                                           |
 * For those less confident, I have tried to give sufficiently clear and complete directions  |
 * so that you, too, will be able to find a complete solution. My chief difficulty is that I  |
 * must be satisfied that I am giving you directions which will always work. But I also need  |
 * to encourage you to find your own, better ways to solve the Cube.
 * STEP 3-14. Is the Cube in its starting position?                                           |
 * YES! Go to step 3-16.           NO! Continue on to step 3-15.                              |
 * STEP 3-15. ASSERTION: All four corner cubes in the TOP layer are correctly positioned.     |
 * Is the cubie in the FRONT RIGHT TOP position correctly oriented?                           |
 * YES! Go back to step 3-12.      NO! Go back to step 3-13.                                  |
 * STEP 3-16. ASSERTION: The Cube is in its starting position.                                |
 * ASSERTION: All four corner cubies in the TOP layer are correctly positioned and            |
 * properly oriented.                                                                         |
 * You have completed the warm-up exercise. Your Cube should look like DIAGRAM 3-1B.          |
 * If it does, CONGRATULATIONS!                                                               |
 * If it doesn't, you need to try again, more carefully this time.                            |
 * I am not especially fond of "programmed learning", because it is so slow and tedious, and  |
 * because it is often very difficult to get a good, global understanding of what is really   |
 * going on. (Too many trees, not enough view of the forest!) Sometimes, for fairly simple    |
 * things, it does work reasonably well, and it does tell you how to accomplish some things.  |
 * Please don't be disappointed that I only arranged the corner cubies -- Chapter Eight,      |
 * "Moving Edge Cubies", and Chapter Nine, "Rubik's Maneuver -- How to Flip Two Edge Cubies", |
 * show you ways to arrange the edge cubies as well, and you can try to do that now, if you   |
 * really want to. But I must warn you, some of the moves in the chapters before Chapter Eight |
 * may mess up those cubies again, so it could be a waste of time to try to arrange them now. |
 * But feel free to do what you want -- I'm trying to help you find your own ways to solve the |
 * Cube! You may even find ways better than mine, which do NOT mess up the other cubies!      |
 * CAUTION! Although the methods used in this chapter appear to have interchanged two corner  |
 * cubies, they may have messed up several other corner cubies. We will need to explore more  |
 * carefully to find methods which will work without undesirable side effects. (Hint: You may |
 * start with the methods used in this chapter, find out what the side effects really are,    |
 * then explore variations of these methods. Sometimes, you may be able to use some of the    |
 * side effects. Other times, you may need to find ways to avoid unwanted side effects.)      |
 * 1) We have now positioned four corner cubies correctly. For those who like arithmetic, this    #
 * 2) means we now have only (4 factorial) * ((3 to the eighth power) / 3) * ((12 factorial) / 2) #
 * 3) * (2 to the twelfth power) / 2) ways to arrange the Cube. This is (24) * (6,561  / 3) *     #
 * 4) (479,001,600 / 2) * (4,096 / 2) =  24 * 2,187 *  239,500,800 * 2,048 =                      #
 * 25,745,240,044,339,200. I told you it is possible to use arithmetic to show progress!      #
 * 1) Although we also oriented the four corner cubies in the TOP layer, I am not going to count  #
 * 2) this as progress, because moves in the next chapters may (temporarily) mess up the          #
 * 3) orientation of some cubies. I will count up more progress later, at the proper time, when   #
 * 4) the corner cubies have been securely oriented.                                              #
 * Ironically, this chapter about some of the simplest moves has been the hardest for me to   |
 * write! It has been about as frustrating as trying to define "common sense". Many of you    |
 * readers are already quite comfortable with your own knowledge of how to get the cubies on  |
 * one face of the Cube correct. My advice to you is -- stick with your own methods if you are |
 * confident they will always work.                                                           |
 * For those less confident, I have tried to give sufficiently clear and complete directions  |
 * so that you, too, will be able to find a complete solution. My chief difficulty is that I  |
 * must be satisfied that I am giving you directions which will always work. But I also need  |
 * to encourage you to find your own, better ways to solve the Cube.
 * 1) orientation of some cubies. I will count up more progress later, at the proper time, when   #
 * 2) the corner cubies have been securely oriented.                                              #
 * Ironically, this chapter about some of the simplest moves has been the hardest for me to   |
 * write! It has been about as frustrating as trying to define "common sense". Many of you    |
 * readers are already quite comfortable with your own knowledge of how to get the cubies on  |
 * one face of the Cube correct. My advice to you is -- stick with your own methods if you are |
 * confident they will always work.                                                           |
 * For those less confident, I have tried to give sufficiently clear and complete directions  |
 * so that you, too, will be able to find a complete solution. My chief difficulty is that I  |
 * must be satisfied that I am giving you directions which will always work. But I also need  |
 * to encourage you to find your own, better ways to solve the Cube.
 * must be satisfied that I am giving you directions which will always work. But I also need  |
 * to encourage you to find your own, better ways to solve the Cube.