# Rubik's Cube/Some Simple Moves

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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. | | | # 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. # # # # The number of possibilities for each side that I described above are: 6, 1, 4, 1, 1, and 1. # # The product of these numbers is 24, and this is the number of ways a cube can be oriented. # # Check it out! # # # # How is this fact useful? Turning the entire Cube does nothing to unscramble the Cube, but # # it never scrambles the Cube any worse, either. This means that, by positioning the entire # # Cube properly, you can multiply your Cube solving methods by a factor of 24. Why learn 24 # # different methods, when you can freely rotate the Cube to a desired position, apply just # # ONE method, then return the Cube to its original orientation? This principle will be # # 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.) | | | # We have now positioned four corner cubies correctly. For those who like arithmetic, this # # means we now have only (4 factorial) * ((3 to the eighth power) / 3) * ((12 factorial) / 2) # # * (2 to the twelfth power) / 2) ways to arrange the Cube. This is (24) * (6,561 / 3) * # # (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! # # # # Although we also oriented the four corner cubies in the TOP layer, I am not going to count # # this as progress, because moves in the next chapters may (temporarily) mess up the # # orientation of some cubies. I will count up more progress later, at the proper time, when # # 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. | | |

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