# Rubik's Cube/Ignoring Details

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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. # | | *---------------------------------------------------------------------------------------------*