Rubik's Cube/Customize Your 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 Educational Material |
| |
| Chapter Six - - - - - - - - - - Customize Your Moves -- Commutation |
| |
| There is a clever trick which can often be used in manipulating Rubik's Cube (and many |
| other puzzles, too). It's called "commutation". When you know how to do something when the |
| cubies are at the "correct" locations, but some cubies are not at the correct locations, |
| try this three step process. First, move the cubies to their "correct" location. (You will |
| need to remember exactly what moves you made to do this for step three, later. Take notes |
| if you must!) Second, do the thing you know how to do. Third, undo whatever you did in the |
| first step. |
| |
| An example: we know how to convert |
| 0 - + + 0 0 0 0 0 0 |
| 0 - - + 0 to 0 0 0 0 0 |
| ^ ^ (all cubies in correct orientation). |
| (Note: I put in the "^" symbol just to emphasize the orientation.) |
| |
| We just use our 0 + - - 0 |
| 0 + + - 0 operator, Fv R^ Fv R^ Fv R^. |
| |
| So, how do we solve |
| + 0 - + + |
| + 0 - - + ? |
| ^ |
| First step: rotate the entire cube to this position: |
| 0 - + + 0 |
| 0 - - + 0 |
| ^ . |
| Second step: Do Fv R^ Fv R^ Fv R^. This converts |
| 0 - + + 0 0 0 0 0 0 |
| 0 - - + 0 to 0 0 0 0 0 |
| ^ ^ . |
| Third step: undo whatever we did in the first step. In this case, we rotate the entire Cube |
| back to its original orientation, from 0 0 0 0 0 0 0 0 0 0 |
| ^ to ^ . |
| |
| Before you say, "This is strange stuff -- it has no relevance for me!", let me point out |
| some uses you have already made using this principle, from previous chapters. |
| |
| |
| You have some understanding of how to rotate the entire Cube to any of its 24 possible |
| orientations. You have used some of these moves to place several corner cubies in their |
| correct locations. In effect, you have learned one series of moves, but you now know 24 |
| ways to use that series of moves. |
| |
| Here is a portion of one of the diagrams from Chapter Four, "Ignoring Details -- Moving |
| Corner Cubies". See if you can find some of the ways this principle has been used (even |
| though I didn't tell you that you were using it). |
| |
| (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. |
| |
| Look at diagram 4-11B. Step one of the useful principle is, "3R^". Step two is, "DO THE |
| SWAP". Step three is, "3Rv", which "undoes" step one. |
| |
| Look at diagram 4-11F. Step one is, "K^ 3R^ 3T^". Step two is, "Fv R^ F^ Rv T2". Step three |
| is, "3Tv 3Rv Kv", which "undoes" step one. |
| |
| Do you remember the clumsy way to exchange two diagonal corner cubies? Here is a much |
| easier way! Starting with diagram 4-11E, but using commutation, step one is shown in |
| diagram 6-1. |
| |
| (L) klt (K) (L)_ ? _(K) (L)_ ? _(K) (L)_ ? _(K) Step one: |
| krt T flt klt T _ ? ? _ T _flt klt T _ ? we have put the |
| F frt R 3R^ | flt | 3T^ | klt | Tv | flt | corner cubies to be |
| diagonal --> | F | R | --> | F | R | --> | F | R | SWAPped into "correct" |
| corners) krt | _ ? ? _ | _frt ? _ | _frt FRT and BFR positions. |
| frt krt krt |
| (Expanding the diagram |
| a little bit) |
| |
| DIAGRAM 6-1. Example Step One |
| |
| (L)_ ? _(K) Step two: (L)_ ? _(K) |
| klt T _ ? DO THE SWAP klt T _ ? |
| | flt | | krt | |
| | F | R | | F | R | |
| ? _ | _frt ? _ | _frt |
| krt flt |
| |
| DIAGRAM 6-2. Example Step Two |
| |
| Step three: (L)_ ? _(K) (L)_ ? _(K) (L)_ ? _(K) (L) klt (K) |
| "Undo" what klt T _ ? ? _ T _krt klt T _ ? flt T krt |
| you did in | krt | T^ | klt | 3Tv | krt | 3Rv | frt | |
| step one. | F | R | --> | F | R | | F | R | --> | F | R | |
| ? _ | _frt ? _ | _frt flt | _ ? ? _ | _ ? |
| flt flt frt ? |
| |
| DIAGRAM 6-3. Example Step Three |
| |
| Wasn't that much faster and easier than the clumsy way shown in Chapter Four, ""? |
| |
| Here is a quick review and summary of how to "undo" one move. (You learned most of this |
| when the moves were introduced.). |
| |
| B^ undoes Bv. B2 undoes B2. Bv undoes B^. |
| F^ undoes Fv. F2 undoes F2. Fv undoes F^. |
| K^ undoes Kv. K2 undoes K2. Kv undoes K^. |
| L^ undoes Lv. L2 undoes L2. Lv undoes L^. |
| B^ undoes Bv. B2 undoes B2. Bv undoes B^. |
| B^ undoes Bv. B2 undoes B2. Bv undoes B^. |
| |
| 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^. |
| 2F^ undoes 2Fv. 2F2 undoes 2F2. 2Fv undoes 2F^. |
| 2K^ undoes 2Kv. 2K2 undoes 2K2. 2Kv undoes 2K^. |
| 2L^ undoes 2Lv. 2L2 undoes 2L2. 2Lv undoes 2L^. |
| 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^. |
| 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^. |
| |
| 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^. |
| 3F^ undoes 3Fv. 3F2 undoes 3F2. 3Fv undoes 3F^. |
| 3K^ undoes 3Kv. 3K2 undoes 3K2. 3Kv undoes 3K^. |
| 3L^ undoes 3Lv. 3L2 undoes 3L2. 3Lv undoes 3L^. |
| 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^. |
| 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^. |
| |
| DIAGRAM 6-4. Summary: How to Undo One Move |
| |
| How does one undo a sequence of moves, such as "F^ Rv T^"? Start by considering the last |
| move made in "step 1" of the "customizing" process ("T^"), then undo it ("Tv"). Then, |
| consider the next- to-last move ("Rv"); undo it ("R^"). Continue this process. Eventually, |
| you will consider the first move made in "step 1" ("F^"); undo that step ("Fv"). You're |
| done undoing the sequence of moves! To summarize the example, undo the sequence "F^ Rv T^" |
| by doing "Tv R^ Fv". |
| |
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