Rubik's Cube/Some Simple Moves
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| HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE |
| (Preliminary April 20, 2007 version) |
| by Mr. Ray Calvin Baker |
| 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.
|
| |
- ---------------------------------------------------------------------------------------------*