Rubik's Cube/Ignoring Details
This is where Chapter 04 has been installed. Sorry, folks! Several of the diagrams got messed up in translation!
Ray Calvin Baker 03:38, 10 November 2011 (UTC)
+---------------------------------------------------------------------------------------------+
| HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE |
| (Preliminary April 20, 2007 version) |
| by Mr. Ray Calvin Baker |
| This is a FREE educational resource |
| |
| Chapter Four - - - - - - - - - - Ignoring Details -- Moving Corner Cubies |
| |
| Goal One is to get all eight corner cubies properly located at the eight corners of the |
| Cube. Our "warming up" exercises positioned four of the corner cubies. Now don't be |
| alarmed, but we may have to temporarily give up some of this apparent progress in order to |
| make real and lasting progress toward the goal of getting all six sides of the Cube |
| correct. I propose to set the Cube aside for a little while, do some "paper and pencil" |
| work to plan some moves, and then apply some carefully planned moves to the Cube. |
| |
| In this chapter, I will first describe how I found an essential move (swapping two corner |
| cubies). Then, I will show how to use this swap, if necessary, to locate all eight corner |
| cubies. Next, after we have thought things through, I will show you another way to organize |
| a plan. Finally, after everything is laid out for you, I will invite you to pick up your |
| Cube once more, and put all eight corner cubies where they belong. |
| |
| YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL! For now, we are |
| THINKING about strategic ways to deal with the Cube, not moving cubies! |
| |
# If you were able to position the four corner cubies of the TOP layer correctly, either as #
# described in chapter three, or using your own methods, there are now only 4 * 3 * 2 * 1 #
# = 24 ways to arrange the remaining four corner cubies. I will show you all 24 of these, #
# later in this chapter. #
| |
| The best way I have discovered so far on how to find ways to shuffle the corner cubies is |
| to make a lot of copies of a simplified diagram of the Cube on scratch paper, then try |
| things until something useful shows up. (The brackets "[ ]" above each diagram are used to |
| indicate which corner cubie is hidden at the BOTTOM BACK LEFT "BKL" location of the Cube. |
| The parentheses "( )" indicate spaces where you can fill in an indication of which cubie |
| occupies that corner of the Cube. Use the space between the diagrams to record the |
| operation you tried. You may also number the moves, if you want to.) |
| |
| [ ] [ ] [ ] [ ] |
| _ ( ) _ _ ( ) _ _ ( ) _ _ ( ) _ |
| ( ) _ _ ( ) ( ) _ _ ( ) ( ) _ _ ( ) ( ) _ _ ( ) |
| | ( ) | __ | ( ) | __ | ( ) | __ | ( ) | __ |
| ( ) _ | _ ( ) #1 ( ) _ | _ ( ) #2 ( ) _ | _ ( ) #3 ( ) _ | _ ( ) #4 |
| ( ) --> ( ) --> ( ) --> ( ) --> |
| ... and so on.... |
| This is a sample of part of a work sheet, such as I use to plan moves for corner cubies. |
| |
| DIAGRAM 4-1. Blank Diagrams for Planning Moves |
| |
| The first diagrams I made looked like this: |
| |
| [h] [h] [h] |
| _ (a) _ _ (a) _ _ (a) _ |
| (b) _ _ (c) (e) _ _ (c) (e) _ _ (f) |
| | (d) | Fv | (b) | R^ | (c) | F^ |
| (e) _ | _ (f) #1 (g) _ | _ (f) #2 (g) _ | _ (d) #3 |
| (g) --> (d) --> (b) --> |
| This is just an I began |
| arbitrary way exploring |
| to label the with a few |
| corner cubies. arbitrary moves. |
| |
| [h] [h] [h] |
| _ (a) _ _ (a) _ _ (g) _ |
| (c) _ _ (f) (c) _ _ (b) (b) _ _ (c) |
| | (b) | Rv | (g) | T2 | (a) | |
| (e) _ | _ (d) #4 (e) _ | _ (f) #5 (e) _ | _ (f) |
| (g) --> (d) --> (d) |
| Comparing this to T2 looked like an |
| the starting position, interesting thing |
| b and c have traded to try, and it was! |
| places, while e and f |
| are where they started. |
| |
| DIAGRAM 4-2. I Found a Three-Way Swap |
| |
| To evaluate whether or not a series of moves is useful, we need to compare the original |
| position of the corner cubies with their final position. Sometimes, we can find "cycles" |
| that return many cubies to their original positions, while moving a few cubies in a |
| predictable pattern. |
| |
| Comparing this last diagram with the first, we find that five cubies are back at their |
| starting position, while three cubies, a, d, and g, have moved. We can diagram this: |
| a --> d --> g --> a. (Do you see how and why I wrote this down?) This is called a "cycle" |
| (of length three), and it helps us predict what would happen if we did "Fv R^ F^ Rv T2" |
| twice, or three times, or more; and we don't have to work out all of the messy details to |
| see this! P.S. This is a useful recipe for moving exactly three corner cubies any way we |
| want to, if the three corner cubies are in exactly the proper positions. The idea discussed |
| in Chapter Six, "Customize Your Moves -- Commutation", will allow you to shuffle any three |
| cubies. |
| |
| Flush with success at finding a useful result so soon, I guessed that I could find a way to |
| interchange exactly two corner cubies just as easily. I tried to move one of the three |
| cubies "out of the way" with a move of the LEFT side, then I tried using the cycle I had |
| discovered. |
| |
| YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL! |
| |
| After "Fv R^ F^ Rv T2", the diagrams of the imaginary Cube we are THINKING about looks like |
| this. |
| |
| [h] [g] [g] [g] |
| _ (g) _ _ (b) _ _ (b) _ _ (b) _ |
| (b) _ _ (c) (e) _ _ (c) (h) _ _ (c) (h) _ _ (f) |
| | (a) | L^ | (a) | Fv | (e) | R^ | (c) | F^ |
| (e) _ | _ (f) #6 (h) _ | _ (f) #7 (d) _ | _ (f) #8 (d) _ | _ (a) #9 |
| (d) --> (d) --> (a) --> (e) --> |
| Continuing an experiment.... |
| |
| [g] [g] [g] [g] |
| _ (b) _ _ (b) _ _ (d) _ _ (d) _ |
| (c) _ _ (f) (c) _ _ (e) (e) _ _ (c) (h) _ | _ (c) |
| | (e) | Rv | (d) | T2 | (b) | Fv | (e) | R^ |
| (h) _ | _ (a) 10 (h) _ | _ (f) 11 (h) _ | _ (f) 12 (a) _ | _ (f) 13 |
| (d) --> (a) --> (a) --> (b) --> |
| |
| [g] [g] [g] [g] |
| _ (d) _ _ (d) _ _ (d) _ _ (a) _ |
| (h) _ _ (f) (c) _ _ (f) (c) _ _ (e) (e) _ | _ (c) |
| | (c) | F^ | (e) | Rv | (a) | T2 | (d) | Lv |
| (a) _ | _ (b) 14 (h) _ | _ (b) 15 (h) _ | _ (f) 16 (h) _ | _ (f) 17 |
| (e) --> (a) --> (b) --> (b) --> |
| |
| [h] [h] |
| _ (g) _ _ (a) _ |
| (a) _ _ (c) (b) _ _ (c) |
| | (d) | | (d) | |
| (e) _ | _ (f) (e) _ | _ (f) |
| (b) (g) |
| Final position Original position |
| |
| .. that seemed to result in nothing useful. |
| |
| DIAGRAM 4-3. An Experiment Which Seemed Useless |
| |
| This was disappointing! Not every idea of mine worked! Sometimes I had to scribble for an |
| hour or two on scratch paper to find a way to do something important and necessary. But |
| after searching for a way to interchange exactly two corner cubies, I eventually found |
| three more moves that really did accomplish what I needed. |
| |
| [h] [h] [h] [h] |
| _ (g) _ _ (g) _ _ (a) _ _ (a) _ |
| (a) _ _ (c) (a) _ _ (d) (b) _ _ (g) (b) _ _ (c) |
| | (d) | Rv | (b) | Tv | (d) | R^ | (g) | |
| (e) _ | _ (f) 18 (e) _ | _ (c) 19 (e) _ | _ (c) 20 (e) _ | _ (f) |
| (b) --> (f) --> (f) --> (d) |
| |
| DIAGRAM 4-4. Three More Moves Provide a Useful Two-Cubie Interchange |
| |
| YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL! |
| |
| Here is a summary of this entire process we have been thinking about. |
| |
| [h] First, Fv R^ F^ Rv T2 [h] Notice that corner |
| _ (a) _ Second, L^ _ (a) _ cubies d and g have |
| (b) _ T _ (c) Third, Fv R^ F^ Rv T2 (b) _ T _ (c) traded places, |
| | F (d) R | Fourth, Fv R^ F^ Rv T2 | F (g) R | |
| (e) _ | _ (f) Fifth, Lv (e) _ | _ (f) These are in the |
| (g) Sixth, Rv Tv R^ (d) FRT and BFR |
| Original position Final position locations. |
| |
| DIAGRAM 4-5. Summary of FRONT RIGHT TOP and BOTTOM FRONT RIGHT SWAP |
| |
| Call this the "FRONT RIGHT TOP and BOTTOM FRONT RIGHT SWAP"! |
| |
| Now that I know a way to interchange two corner cubies, I have completed the difficult part |
| of the strategic planning which will allow me to accomplish Goal One. Surely you can find |
| easier, better, and faster ways to position the corner cubies! (Hint: Why do a three-cycle |
| twice? Isn't it easier and quicker to do it backwards once instead?) |
| |
| CAUTION! When you try to use a procedure such as this, or any others to be discovered in |
| the following chapters, it is EXTREMELY IMPORTANT that EVERY DETAIL and EVERY MOVE be |
| EXACTLY CORRECT! Otherwise, you may mess up parts of the Cube you thought you had finished. |
| You may have to start over! (Oh, well! It's happened to me, LOTS of times!) |
| |
| Hey! Not so fast! What if the corner cubies I want to interchange are NOT in the FRONT |
| RIGHT TOP and BOTTOM FRONT RIGHT locations? This sounds like it could be a serious problem, |
| since we correctly positioned all corner cubies of the TOP layer according to directions in |
| the previous chapter. It is quite possible that two (or more!) corner cubies in the BOTTOM |
| layer are not correctly positioned for this interchange to work properly. |
| |
| Fortunately, there is an easy solution for this problem, if the corner cubies which need to |
| be interchanged are adjacent to eache other. (Some examples are: BFR and BFL, BFL and BKL, |
| and BKL and BKR.) Just use some combination of three-layer moves (like, for example, 3T^, |
| 3Tv, 3R^, 3Rv, 3F^, and 3Fv) to rotate the entire Cube until the corner cubies you wish to |
| switch ARE in the FRONT RIGHT TOP and BOTTOM FRONT RIGHT positions. Then apply the series |
| of moves which interchanges those to corner cubies! (This principle will be dicussed more |
| thoroughly in Chapter Six, "Customize Your Moves -- Commutation".) |
| |
| What can you do if the corner cubies to be interchanged are at opposite corners of the |
| BOTTOM layer? |
| |
| YOUR CUBE SHOULD STILL BE ON THE TABLE! |
| |
| You had matched up some After "3R2" flips the |
| of the corner cubies on entire Cube, you MIGHT |
| the BACK, FRONT, LEFT, find 2 corner cubies in If that is what you find, |
| RIGHT, and TOP sides. the correct FRT and KLT you should be able to |
| locations. do this! |
| _ * _ _ * _ _ * _ |
| _ * _TOP_ * _ _ * _klt_ * _ _ * _klt_ * _ |
| _ * _ ? _ * _ ? _ * _ _ * _ _ * _ _ * _ _ * _ _ * _ _ * _ |
| * _TOP_ * _TOP_ * _TOP_ * * _krt * _ _ * _flt_ * * _flt_ * _ _ * _krt_ * |
| | * _ ? _ * _ | _ * | | * _ _ * _ _ * | | * _ _ * _ _ * | |
| | F | * _TOP_ * | R | |krt| * _frt_ * |flt| |flt| * _frt_ * |krt| |
| * _ | ? | * | ? | _ * 3R2 * _ | | * | | _ * * _ | | * | | _ * |
| | * _ | F | R | _ * | --> | * _ |frt|frt| _ * | | * _ |frt|frt| _ * | |
| | ? | * _ | _ * | ? | | | * _ | _ * | | | | * _ | _ * | | |
| * _ | F | * | R | _ * * _ |"k"| * |"r"| _ * * _ | F | * | R | _ * |
| | * _ | ? | ? | _ * | | * _ | | | _ * | | * _ | | | _ * | |
| | ? | * _ | _ * | ? | |"k"| * _ | _ * |"r"| | F | * _ | _ * | R | |
| * _ | ? | * | ? | _ * * _ | | * | | _ * * _ | | * | | _ * |
| * _ | ? | ? | _ * * _ |"k"|"r"| _ * * _ | F | R | _ * |
| * _ | _ * * _ | _ * * _ | _ * |
| * * * |
| What you had at the end ONE POSSIBILITY after you What you want. |
| of chapter three. (BACK turn the Cube upside down. Notice that you had to |
| and LEFT sides show a (I have used "k" and "r" exchange cubies marked |
| similar pattern.) to indicate that these "krt" and "klt". |
| (This is a repeat of colors match correctly.) |
| diagram 3-1B.) |
| |
| DIAGRAM 4-6A. DIAGRAM 4-6B. DIAGRAM 4-6C. |
| |
| DIAGRAM 4-6. How Do You Swap Diagonally Opposite Cubies? |
| |
| I told you, "You should be able to do this!". Here is "the hard way". (I am going to shrink |
| the diagrams a bit, leaving out "irrelevant details", so that I can fit more diagrams on |
| the page.) Remember, at this stage, we are interested in getting corner cubies to the |
| correct LOCATIONS. Their orientation is irrelevant. |
| |
| YOUR CUBE SHOULD STILL BE ON THE TABLE! |
| |
| Here is the plan, using only rotations of the entire Cube, and our newly-discovered SWAP |
| sequence. Three SWAPS of adjacent corner cubies results in the SWAp of two diagonally |
| opposite corner cubies. (Parentheses, "(" and ")", are used to highlight the pair of corner |
| cubies about to be SWAPped.) |
| |
| 1klt 1klt 1klt 1klt |
| (2krt 3flt 4frt 3flt) (4frt 2krt 3flt 2krt |
| 4frt) (2krt 3flt) 4frt |
| |
| DIAGRAM 4-7A. DIAGRAM 4-7B. DIAGRAM 4-7C. DIAGRAM 4-7D. |
| |
| DIAGRAM 4-7. Plan for SWAPping Diagonally Opposite Cubies the Hard Way |
| |
| (We are still ignoring the edge cubies at this stage of the solution.) |
| |
| _ * _ _ * _ _ * _ _ * _ |
| _ * _klt_ * _ _ * _ l _ * _ _ * _ l _ * _ _ * _klt_ * _ |
| * _krt_ T _flt_ * * _ l _ l _klt_ * DO * _ l _ l _klt_ * * _frt_ T _flt_ * |
| | * * *
| * _frt_ * | | * _krt_ * | THE | * _frt_ * | | * _krt_ * | |
| | * _ |
| |krt| * |flt| 3Fv | F | * |klt| FRT | F | * |klt| 3F^ |frt| * |flt| 3R^ |
| * _ |frt|frt| _ * --> * _ |krt|krt| _ * and * _ |frt|frt| _ * --> * _ |krt|krt| _ * --> |
| | F _ | _ R | | F _ | _ t | BFR | F _ | _ t | | F _ | _ R | |
| | F | * | R | | F | * |flt| SWAP | F | * |flt| | F | * | R | |
| * _ | F | R | _ * * _ |frt|frt| _ * ---> * _ |krt|krt| _ * * _ | F | R | _ * |
| * _ | _ * * _ | _ * * _ | _ * * _ | _ * |
| * * * * |
| You worked hard in I'm using "l" to "krt" indicates After rotating the |
| chapter three to indicate the face the cubie that entire Cube, then |
| get four corner which USED to be will end up in using the SWAP moves, |
| cubies correct, so on the LEFT (but the KRT position we can return the |
| I'll show those as isn't any more when we are done Cube to this |
| "F" and "R"! (You because I turned (we hope!). position. |
| can't see "B", "K" the Cube). See "frt" and "krt" have |
| or "L" -- they're diagram been SWAPped. |
| OK, too.) 4-5 |
| |
| DIAGRAM 4-8A. DIAGRAM 4-8B. DIAGRAM 4-8C. DIAGRAM 4-8D. |
| |
| _ * _ _ * _ _ * _ _ * _ |
| _ * _klt_ * _ _ * _klt_ * _ _ * _klt_ * _ _ * _klt_ * _ |
| * _frt_ T _ k _ * DO * _frt_ T _ k _ * * _frt_ T _krt_ * * _ l _ T _krt_ * | | |
| * _flt_ * | THE | * _krt_ * | | * _flt_ * | | * _frt_ * | THE |
| |frt| * | R | FRT |frt| * | R | 3Rv |frt| * |krt| 3Fv | F | * |krt| FRT |
| * _ |flt|flt| _ * and * _ |krt|krt| _ * --> * _ |flt|flt| _ * --> * _ |frt|frt| _ * and |
| | F _ | _ R | BFR | F _ | _ R | | F _ | _ R | | F _ | _ R | BFR |
| | F | * | R | SWAP | F | * | R | | F | * | R | | F | * | R | SWAP |
| * _ |krt|krt| _ * ---> * _ |flt|flt| _ * * _ | F | R | _ * * _ |flt|flt| _ * --> |
| * _ | _ * * _ | _ * * _ | _ * * _ | _ * |
| * * * * |
| This may not See ... now the "krt" See |
| look like much diagram cubie is in place, siagram |
| progress yet, 4-5 and we know how to 4-5 |
| but ... SWAP "frt" and "flt". |
| |
| DIAGRAM 4-8E. DIAGRAM 4-8F. DIAGRAM 4-8G. DIAGRAM 4-8H. |
| |
| _ * _ _ * _ |
| _ * _klt_ * _ _ * _klt_ * _ |
| * _ l _ T _krt_ * * _flt_ T _krt_ * | | | * _flt_ * | | * _frt_ * | |
| | F | * |krt| 3F^ |flt| * |krt| |
| * _ |flt|flt| _ * --? * _ |frt|frt| _ * |
| | F _ | _ R | | F _ | _ R | |
| | F | * | R | | F | * | R | |
| * _ |frt|frt| _ * * _ | F | R | _ * |
| * _ | _ * * _ | _ * |
| * * |
| Rotate the Cube ... and now we have all eight |
| back to position... corner cubiea at their proper locations. |
| |
| DIAGRAM 4-8I. DIAGRAM 4-8J. |
| |
| DIAGRAM 4-8. Ten Stages in SWAP Diagonally Opposite Cubies the Hard Way |
| |
| That was a real mental workout! I have shown several examples of ways we can use the SWAP, |
| and, if you are clever, you should be able to get all eight corner cubies into their proper |
| locations. |
| |
| We still need to orient all eight corner cubies. Then we will be able to finish off all |
| twelve edge cubies. Believe me, you are about to learn how to do these things. You will |
| also learn a general principle which will make your planning and Cube turning MUCH simpler |
| and easier (It's in Chapter Six)! |
| |
| I promised that I would show you all 24 possible ways the corner cubies on the VOTTOM layer |
| of your Cube could be arranged, after you have finished the work in Chapter Three, "Some |
| Simple Moves -- Positioning Four Corner Cubies". This will complete all of the strategic |
| planning we will need to fininsh Chapter Four. |
| |
| _ * _ _ * _ _ * _ |
| _ * _TOP_ * _ _ * _ ? _ * _ (LEFT) * _???_ * (BACK) |
| _ * _ ? _ * _ ? _ * _ _ * _ ? _ * _ _ * _ _ * _ * _ * _ |
| * _TOP_ * _TOP_ * _TOP_ * * _ ? _ * _"b"_ * _ ? _ * * _???_ * TOP * _???_ * |
| | * _ ? _ * _ | _ * | | * _ ? _ * _ ? _ * | * _ * _ * |
| | F | * _TOP_ * | R | | ? | * _ ? _ * | ? | FRONT * _???_ * RIGHT |
| * _ | ? | * | ? | _ * 3R2 * _ | ? | * | ? | _ * * |
| | * _ | F | R | _ * | --> | * _ | ? | ? | _ * | This diagram indicates the |
| | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | four corner cubies which |
| * _ | F | * | R | _ * * _ |"k"| * |"r"| _ * are now of interest to us. |
| | * _ | ? | ? | _ * | | * _ | ? | ? | _ * | |
| | ? | * _ | _ * | ? | |"k"| * _ | _ * |"r"| (L) ??? (K) |
| * _ | ? | * | ? | _ * * _ | ? | * | ? | _ * ??? T ??? |
| * _ | ? | ? | _ * * _ |"k"|"r"| _ * F ??? R |
| * _ | _ * * _ | _ * Condensed |
| * * version |
| |
| DIAGRAM 4-9A. DIAGRAM 4-9B. DIAGRAM 4-9C. |
| |
| DIAGRAM 4-9. Preparing to Locate the Final Four Corner Cubies |
| |
| PICK UP YOUR CUBE! It should have the TOP side showing your chosen color on at least five |
| colored labels, as shown in diagram 4-9A. Turn your Cube upside down as shown in diagrams |
| 4-9A and 4-9B. |
| |
| I am going to refocus on what is now the FRONT, BACK, LEFT, RIGHT, and TOP, so I'll show |
| you how things are now named, and I'll show you what we are looking for. Then I'll tell you |
| what we need to do for each of the 24 possibilities. |
| |
| First, the names (and abbreviations) of the LOCATIONS for the last four corner cubies are |
| shown in diagram 4-10. |
| |
| _ * _ The last four corner cubies |
| (LEFT) _ - BACK - _ (BACK) are indicated like this: |
| * _LEFT TOP _ * |
| - - _KLT_ - - klt -- the cubie which belongs |
| _ * _ * _ * _ at location KLT |
| _ - FRONT - _ - - _ - BACK - _ |
| * _LEFT TOP _ * TOP * _RIGHT TOP _ * flt -- the cubie which belongs |
| | - _FLT_ - - - - _KRT_ - | at location FLT |
| * _ * _ * |
| | - _ - FRONT - _ - | krt -- the cubie which belongs |
| FRONT * _RIGHT TOP _ * RIGHT at location KRT |
| | - _FRT_ - | |
| * (This is an enlargement frt -- the cubie which belongs |
| | of part of diagram 4-9C.) at location KLT |
| |
| DIAGRAM 4-10. Introducing More Abbreviated Diagrams |
| |
| Now, as promised, here are the 24 possbilities of what you will see on what is now the TOP |
| of your Cube (diagrams 4-11A through 4-14F) |
| |
|---------------------------------------------------------------------------------------------|
| |
| (L) klt (K) : (L) klt (K) : (L) klt (K) : (L) klt (K) : (L) klt (K) : (L) klt (K) |
| flt T krt : flt T frt : frt T krt : frt T flt : krt T flt : krt T frt |
| F frt R : F krt R : F flt R : F krt R : F frt R : F flt R |
| : : : (3-cycle) : (diagonal : (3-cycle) |
| : : : : corners) : |
| WOW! All 8 : 3R^ : 3Fv : K^ 3R^ 3T^ : This is done : K^ 3R^ 3T^ |
| corner cubies : DO THE SWAP : DO THE SWAP : Diagram 4-2: : exactly like : Diagram 4-2: |
| are properly : diagram 4-5 : diagram 4-5 : Fv R^ F^ Rv T2 : diagrams 4-8A : Fv R^ F^ Rv T2 |
| located! : 3Rv : 3Fv : 3Tv 3Rv Kv : through 4-8J. : 3Tv 3Rv Kv |
| : : : Then look at : : |
| : : : diagram 4-11F. : : |
| : : : : : |
| DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM |
| 4-11A. : 4-11B. : 4-11C. : 4-11D. : 4-11E. : 4-11F. |
| |
| DIAGRAM 4-11. The First Set of Six Possibilities |
| |
|---------------------------------------------------------------------------------------------|
| |
| (L) krt (K) : (L) frt (K) : (L) krt (K) : (L) flt (K) : (L) flt (K) : (L) frt (K) |
| klt T frt : klt T krt : klt T flt : klt T krt : klt T frt : klt T flt |
| F flt R : F flt R : F frt R : F frt R : F krt R : F krt R |
| : : : (3=cycle) : (diagonal : (3-cycle) |
| : : : : corners) : |
| : : : : : |
| DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM |
| 4-12A. : 4-12B. : 4-12C. : 4-12D. : 4-12E. : 4-12F. |
| |
| DIAGRAM 4-12. The Second Set of Siz Possibilities |
| |
| For diagrams 4-12A through 4-12F, simply turn the TOP layer (Tv) and compare the result |
| with diagrams 4-11A through 4-11F, then follow the directions for the correct diagram. |
| |
|---------------------------------------------------------------------------------------------|
| |
| (L) frt (K) : (L) krt (K) : (L) flt (K) : (L) krt (K) : (L) frt (K) : (L) flt (K) |
| krt T flt : frt T flt : krt T frt : flt T frt : flt T krt : frt T krt |
| F klt R : F klt R : F klt R : F klt R : F klt R : F klt R |
| : : : : : |
| DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM |
| 4-13A. : 4-13B. : 4-13C. : 4-13D. : 4-13E. : 4-13F. |
| |
| DIAGRAM 4-13. The Third Set of Six Possibilities |
| |
| For diagrams 4-13A through 4-13F, simply turn the TOP layer (Tv) and compare the result |
| with diagrams 4-12A through 4-12F. Follow the directions for those diagrams. |
| |
|---------------------------------------------------------------------------------------------|
| |
| (L) flt (K) : (L) flt (K) : (L) frt (K) : (L) frt (K) : (L) krt (K) : (L) krt (K) |
| frt T klt : krt T klt : flt T klt : krt T klt : frt T klt : flt T klt |
| F krt R : F frt R : F krt R : F flt R : F flt R : F frt R |
| : : : : : |
| DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM : DIAGRAM |
| 4-14A. : 4-14B. : 4-14C. : 4-14D. : 4-14E. : 4-14F. |
| |
| DIAGRAM 4-14. The Final Set of Six Possibilities |
| |
| For diagrams 4-14A through 4-14F, simply turn the TOP layer (Tv) and compare the result |
| with diagrams 4-13A through 4-13F. Tricky, huh? I think you see a pattern! |
| |
|---------------------------------------------------------------------------------------------|
| |
| Some of these moves may have altered the orientation of some corner cubies, but that is not |
| a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
| BE CORRECT before you proceed any further. If the location of any corner cubie is |
| incorrect, you will have to go back and try again. |
| |
# For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) * #
# (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800 #
# (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three #
# hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible #
# arrangements of the cubies left. #
| |
*---------------------------------------------------------------------------------------------*