Rubik's Cube/Finishing the Orientation of Corner Cubies

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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 Materials                       |
|                                                                                             |
| Chapter Seven  - - - - - - - - - Finishing the Orientation of Corner Cubies                 |
|                                                                                             |
| In Chapter Five, we found some new patterns for rotating corner cubies. Each pattern is a   |
| (coded) solution for \i some\i0  Rubik's Cube problem.                                      | 
|                                                                                             |
| Can you find these patterns?                                                                |
|                                                                                             |
|      + 0 + 0 +     - + - + -     0 + 0 + 0     0 - 0 - 0     0 0 0 0 0     Some sample      |
|      - 0 - 0 -     + + + 0 +     0 + 0 0 0     0 0 0 - 0     0 + 0 - 0     problems.        |
|        [F] [K]       [F] [K]       [F] [K]       [F] [K]       [F] [K]                      |
|                                                                                             |
|      DIAGRAM 7-1. Typical Coded Rotation Problems                                           |
|                                                                                             |
| These last two patterns, or variations of them, will solve ALL problems af rotated corner   |
| cubies (unless your cube has been physically damaged.) Yes, there are simpler ways -- can   |
| you find some?                                                                              |
|                                                                                             |
| We now have tools which will enable us to find a way to rotate just TWO corner cubes (in    |
| opposite directions). Let's work out a detailed plan to do this.                            |
|                                                                                             |
| Starting with a Cube with all eight corner cubies in their proper locations is essential    |
| for completing this chapter. The orientations of these eight corner cubies has not yet      |
| determined -- it can be completely arbitrary, within the constraints imposed by the         |
| essential geometry of the Cube itself. (If seven corner cubies are properly located and     |
| oriented, then the eighth corner cubie is also properly located and oriented -- unless your |
| Cube has been physically damaged or is "out of orbit".)                                     |                                           |                                                                                             |
| Since we are still doing "paper and pencil" work, we can start with an arbitrary config-    |
| uration of properly located corner cubies. Let's start with                                 |
|       0 0 0 0 0                                                                             |
|       0 0 0 0 0                                                                             |
|         [F] [K].                                                                            |
|                                                                                             |
| From our work in chapter five, we know that this sequence of moves changes the orientation  |
| of six corner cubies (but it doesn't change their locations).                               |
|                                                                                             |
|       0 0 0 0 0     Fv R^ Fv R^ Fv R^     0 + - - 0                                         |
|       0 0 0 0 0     -> -> -> -> -> ->     0 + + - 0                                         |
|         [F] [K]                             [F] [K]                                         |
|                                                                                             |
|       DIAGRAM 7-2. A Sequence of Moves Which Changes Orientations od Corner Cubies          |
|                                                                                             |
| If we do that again, we have this.                                                          |
|                                                                                             |
|       0 + + - 0     Fv R^ Fv R^ Fv R^     0 - + + 0                                         |
|       0 + - - 0     -> -> -> -> -> ->     0 - - + 0                                         |
|         [F] [K]                             [F] [K]                                         |
|                                                                                             |
|       DIAGRAM 7-3. Repeating the Sequence of Moves                                          |
|                                                                                             |
| We expect that, if we do it a third time, everything will cancel out, and we will be right  |
| back at the starting configuration -- not very useful! But what if we could fix things so   |
| that not everything cancels out? Let's try an experiment.                                   |
|                                                                                             |
|     0 - + + 0    Fv    0 - - + 0    Fv R^ Fv R^ Fv R^    0 0 + 0 0    F^    0 + - 0 0       |
|     0 - - + 0    ->    0 - + + 0    -> -> -> -> -> ->    0 0 - 0 0    ->    0 0 0 0 0       |
|       [F] [K]            [F] [K]                           [F] [K]            [F] [K]       |
|                                                                                             |
|     DIAGRAM 7-4. A Sucessful Experiment                                                     |
|                                                                                             |
| This can be extremely useful! Just be careful to have your Cube precisely oriented before   |
| you start the sequence of moves. The corner cubie at FRONT LEFT TOP will be rotated         |
| counterclockwise, and the corner cubie at FRONT RIGHT TOP will be rotated clockwise. (Hint: |
| there may be a useful way to turn your Cube 180 degrees if you need to do this.)            |
|                                                                                             |
| How do you solve this problem?                                                              |
|                                                                                             |
|               Step               Step two:                            Step                  |
|               one:               (You know how!)                      three:                |
|   0 0 + 0 0    L^    0 - + 0 0      Fv R^ Fv R^ Fv R^      0 0 0 0 0   Lv    0 0 0 0 0      |
|   0 - 0 0 0   --->   0 0 0 0 0      Fv R^ Fv R^ Fv R^      0 0 0 0 0  --->   0 0 0 0 0      |
|     [F] \{K\}            [F] [K]   Fv Fv R^ Fv R^ Fv R^ F^     [F] [K]           [F] [K]    |
|   A Problem!                                                                   Solved!      |
|                                                                                             |
|   DIAGRAM 7-5. Solving a Typical Problem                                                    |
|                                                                                             |
| NOTE: Step one, a one-layer turn, moves four corner cubies away from their proper           |
| locations. Therefore, it is very important not to forget to do step three, another one-     |
| layer move, in order to return those four corner cubies to their proper places. There are a |
| lot of details not shown in the above simplified diagrams!                                  |
|                                                                                             |
| We can also demonstrate that knowing how to rotate two corner cubies in opposite directions |
| will allow us to solve the problem of three corner cubies all rotated in the same           |
| direction. See if you can follow this sequence of moves.                                    |
|                                                                                             |
|  0 0 0 0 0  ROTATE  0 - + 0 0  ROTATE  0 + - 0 0  3Tv  + - 0 0 0  ROTATE   + + + 0 0        |
|  0 0 0 0 0    TWO   0 0 0 0 0    TWO   0 0 0 0 0  -->  0 0 0 0 0    TWO    0 0 0 0 0        |
|    [F] [K]  CORNERS   [F] [K]  CORNERS   [F] [K]         [F] [K]  CORNERS    [F] [K]        |
|                                                                                             |
|  DIAGRAM 7-6. Solving Three Corners Rotated in Same Direction                               |
|                                                                                             |
| At this time, you should be able to pick up your Cube and rotate all eight corner cubies    |
| into their proper orientation. You should be able to accomplish Goal Two. Basically, there  |
| are three ways to do this.                                                                  |
|                                                                                             |
| The first way is, "everything all at once". You have tools to make lots of diagrams of      |
| patterns of rotation for the corner cubies. You should be able to make a diagram of your    |
| partially unscrambled Cube. You should be able to determine what pattern will rotate the    |
| corner cubies to their proper orientation.                                                  |
|                                                                                             |
| Here's an example of what I mean by this. Suppose your Cube has this pattern:               |
|                                                                                             |
|   0 0 + + +   What pattern will     0 0 - - -         For each "0" in the problem,          |
|   + - 0 - +   solve this problem?   - + 0 + -            write "0" in the solution.         |
|   A possible                        The pattern       For each "+" in the problem,          |
|   problem.                          that solves it.      write "-" in the solution.         |
|                                                       For each "-" in the problem,          |
|                                                          write "+" in the solution.         |
|                                                                                             |
|   DIAGRAM 7-7. Find a Solution For a Rotation Problem                                       |
|                                                                                             |
| Now, all you need to do is find this solution in your notebook (You did make a notebook,    |
| didn't you?) and apply it to your Cube (You did record the moves to make your patterns,     |
| didn't you?). The advantage of "everything all at once" is that it gives you an immediate   |
| sequence of moves to complete fixing all eight corner cubies. The disadvantages are (1) you |
| must have lots of good, accurate notes -- there are (3 to the 8th power) / 3 = 2,187        |
| possible patterns, and (2) I can't remember 2,187 moves, or even what the next              |
| disadvantage was.                                                                           |
|                                                                                             |
| The second method is, "one or two cubies at a time". This is easy to memorize, but slow --  |
| you may have to use this method up to seven times. Diagrams 7-2, 7-3, and 7-4 show how to   |
| rotate two corner cubies in opposite directions. So, find two corner cubies which are not   |
| oriented properly, move them into position (step 1 of customizing), rotate the two corners, |
| and undo step 1 (step 3 of customizing). One more, or possibly two more, corner cubies      |
| should now be correctly oriented. Keep doing this until all eight corners are correctly     |
| oriented.                                                                                   |
|                                                                                             |
| The third way is to use a combination of methods -- find a rotation pattern that matches    |
| several corners of your required solution, apply it, then clean up any remaining problems   |
| using the "one or two cubies at a time" method.                                             |
|                                                                                             |
| The second and third methods are something fairly easy to remember and use. If you can      |
| finish this phase of the solution by yourself, that's great! If you need the help of a      |
| recipe to do this, here it comes.                                                           |
|                                                                                             |
| But first, I need to show you another diagram. You may use this diagram to find the         |
| "customization" moves you need.                                                             |
|                                                                                             |
|      Case 1: ?? #1 #2 ?? ??  No customization is needed.                                    |
|              ?? ?? ?? ?? ??                                                                 |
|                  [F]   [K]                                                                  |
|                                                                                             |
|      Case 2: ?? #1 ?? #2 ??  Customize by doing "R^".                                       |
|              ?? ?? ?? ?? ??                                                                 |
|                  [F]   [K]                                                                  |
|                                                                                             |
|      Case 3: #2 #1 ?? ?? #2  Customize by doing "3T^".                                      |
|              ?? ?? ?? ?? ??                                                                 |
|                  [F]   [K]                                                                  |
|                                                                                             |
|      Case 4: ?? #1 ?? ?? ??  Customize by doing "K^ 3T^".                                   |
|              #2 ?? ?? ?? #2                                                                 |
|                  [F]   [K]                                                                  |
|                                                                                             |
|      Case 5: ?? #1 ?? ?? ??  Customize by doing "3Fv".                                      |
|              ?? #2 ?? ?? ??                                                                 |
|                  [F]   [K]                                                                  |
|                                                                                             |
|      Case 6: ?? #1 ?? ?? ??  Customize by doing "Rv".                                       |
|              ?? ?? #2 ?? ??                                                                 |
|                  [F]   [K]                                                                  |
|                                                                                             |
|      Case 7: ?? #1 ?? ?? ??  Customize by doing "R2".                                       |
|              ?? ?? ?? #2 ??                                                                 |
|                  [F]   [K]                                                                  |
|                                                                                             |
|      DIAGRAM 7-8. Find Second Corner Cubie to Rotate, Then Apply These Moves                |
|                                                                                             |
| While looking at diagram 7-8, you may have thought that there are often several different   |
| ways to "customize". You are correct. Pick one way -- whatever you feel comfortable with -- |
| and stick with it. Just don't change your mind part way through the three-step process.     |
|                                                                                             |
| Paragraph A:                                                                                |
| Find a corner cubie which is properly located, but not properly oriented. Call this cubie   |
| "#1". Rotate the entire Cube until this cubie is located at the FRONT LEFT TOP corner. Find |
| another corner cubie which is also not properly oriented. There are seven possibilities, as |
| shown in diagram 7-8. Follow the directions to "customize" your moves, then perform the     |
| sequence,                                                                                   |
|                       Fv R^ Fv R^ Fv R^                                                     |
|                       Fv R^ Fv R^ Fv R^                                                     |
|                    Fv Fv R^ Fv R^ Fv R^ F^.                                                 |
|                                                                                             |
| Paragraph B:                                                                                |
| It is possible that no improvement resulted from performing paragraph A, because both       |
| corner cubies were rotated in the wrong direction. If the corner cubie in the FRONT LEFT    |
| TOP position is still not correcxtly oriented, simply repeat this sequence again,           |
|                       Fv R^ Fv R^ Fv R^                                                     |
|                       Fv R^ Fv R^ Fv R^                                                     |
|                    Fv Fv R^ Fv R^ Fv R^ F^.                                                 |
|                                                                                             |
|                          _ * _                                        _ * _                 |
|                      _ * _   _ * _                                _ * _   _ * _             |
|                  _ * _   _ * _   _ * _                        _ * _   _ * _   _ * _         |
|                * _ T _ * _ T _ * _   _ *                    * _ l _ * _ T _ * _   _ *       |
|          (L)...|   * _   _ * _   _ *   |              (f)...|   * _   _ * _   _ *   |       |
|                | F |   * _#2 _ *   |   |                    | t |   * _#2 _ *   |   |       |
|      The       * _ |   |   *   |   | _ *          The       * _ |   |   *   |   | _ *       |
|      FRONT     |   * _ |#2 |#2 | _ *   |          FRONT     |   * _ |#2 |#2 | _ *   |       |
|      LEFT      |   |   * _ | _ *   |   |          LEFT      |   |   * _ | _ *   |   |       |
|      TOP       * _ | F |   *   | R | _ *          TOP       * _ | F |   *   | R | _ *       |
|      cubie     |   * _ |   |   | _ *   |          cubie     |   * _ |   |   | _ *   |       |
|      is        |   |   * _ | _ *   |   |          needs     |   |   * _ | _ *   |   |       |
|      correctly * _ |   |   *   |   | _ *          more      * _ |   |   *   |   | _ *       |
|      oriented;     * _ |   |   | _ *              work;         * _ |   |   | _ *           |
|      go on to          * _ | _ *                  repeat the        * _ | _ *               |
|      paragraph C.          *                      sequence of moves.    *                   |
|                                                                                             |
|      DIAGRAM 7-9A.                                DIAGRAM 7-9B.                             |
|                                                                                             |
|      DIAGRAM 7-9. Is the FRONT LEFT TOP Corner Cubie Properly Oriented?                     |
|                                                                                             |
| Paragraph C:                                                                                |
| Undo the customization you used in paragraph A. (You do remember what you did, don't you?)  |
|                                                                                             |
| Paragraph D:                                                                                |
| If all eight corner cubies are properly positioned and properly oriented, you are done!     |
| Otherwise, repeat this process (starting at paragraph A) until all eight corner cubies are  |
| properly oriented.                                                                          |
|                                                                                             |
# How much progress have we made at the end of chapter seven? There are now only              #
# ( ( 12 factorial) * (2 to the 12th power) / 4 ) = 479,001,600 * 4,096 / 4 =                 #
# 490,497,638,400 ways to arrange the cubies of your Cube.                                    #
|                                                                                             |
| By the way, you should now be able to see a pretty "X" pattern of matching cubies on all    |
| six sides of your Cube.                                                                     |
|                                                                                             |
|                             +---+---+---+                                                   |
|                             |TOP| ? |TOP|                                                   |
|                             +---+---+---+                                                   |
|                             | ? |TOP| ? |                                                   |
|                             +---+---+---+                                                   |
|                             |TOP| ? |TOP|                                                   |
|                 +---+---+---+---+---+---+---+---+---+---+---+---+                           |
|                 | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                           |
|                 +---+---+---+---+---+---+---+---+---+---+---+---+                           |
|                 | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |                           |
|                 +---+---+---+---+---+---+---+---+---+---+---+---+                           |
|                 | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                           |
|                 +---+---+---+---+---+---+---+---+---+---+---+---+                           |
|                             | B | ? | B |                                                   |
|                             +---+---+---+                                                   |
|                             | ? | B | ? |                                                   |
|                             +---+---+---+                                                   |
|                             | B | ? | B |                                                   |
|                             +---+---+---+                                                   |
|                                                                                             |
|                 DIAGRAM 7-10. X Marks Our Progress                                          |
|                                                                                             |
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