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". | | | +---------------------------------------------------------------------------------------------+