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 |
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| Chapter Seven - - - - - - - - - Finishing the Orientation of Corner Cubies |
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| 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. |
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| Can you find these patterns? |
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| + 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] |
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| DIAGRAM 7-1. Typical Coded Rotation Problems |
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| 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? |
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| 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. |
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| 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]. |
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| 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). |
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| 0 0 0 0 0 Fv R^ Fv R^ Fv R^ 0 + - - 0 |
| 0 0 0 0 0 -> -> -> -> -> -> 0 + + - 0 |
| [F] [K] [F] [K] |
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| DIAGRAM 7-2. A Sequence of Moves Which Changes Orientations od Corner Cubies |
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| If we do that again, we have this. |
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| 0 + + - 0 Fv R^ Fv R^ Fv R^ 0 - + + 0 |
| 0 + - - 0 -> -> -> -> -> -> 0 - - + 0 |
| [F] [K] [F] [K] |
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| DIAGRAM 7-3. Repeating the Sequence of Moves |
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| 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. |
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| 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] |
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| DIAGRAM 7-4. A Sucessful Experiment |
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| 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.) |
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| How do you solve this problem? |
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| 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! |
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| DIAGRAM 7-5. Solving a Typical Problem |
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| 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! |
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| 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. |
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| 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] |
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| DIAGRAM 7-6. Solving Three Corners Rotated in Same Direction |
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| 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. |
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| 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. |
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| Here's an example of what I mean by this. Suppose your Cube has this pattern: |
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| 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. |
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| DIAGRAM 7-7. Find a Solution For a Rotation Problem |
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| 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. |
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| 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. |
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| 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. |
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| 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. |
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| But first, I need to show you another diagram. You may use this diagram to find the |
| "customization" moves you need. |
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| Case 1: ?? #1 #2 ?? ?? No customization is needed. |
| ?? ?? ?? ?? ?? |
| [F] [K] |
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| Case 2: ?? #1 ?? #2 ?? Customize by doing "R^". |
| ?? ?? ?? ?? ?? |
| [F] [K] |
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| Case 3: #2 #1 ?? ?? #2 Customize by doing "3T^". |
| ?? ?? ?? ?? ?? |
| [F] [K] |
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| Case 4: ?? #1 ?? ?? ?? Customize by doing "K^ 3T^". |
| #2 ?? ?? ?? #2 |
| [F] [K] |
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| Case 5: ?? #1 ?? ?? ?? Customize by doing "3Fv". |
| ?? #2 ?? ?? ?? |
| [F] [K] |
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| Case 6: ?? #1 ?? ?? ?? Customize by doing "Rv". |
| ?? ?? #2 ?? ?? |
| [F] [K] |
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| Case 7: ?? #1 ?? ?? ?? Customize by doing "R2". |
| ?? ?? ?? #2 ?? |
| [F] [K] |
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| DIAGRAM 7-8. Find Second Corner Cubie to Rotate, Then Apply These Moves |
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| 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. |
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| 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^. |
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| 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^. |
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| _ * _ _ * _ |
| _ * _ _ * _ _ * _ _ * _ |
| _ * _ _ * _ _ * _ _ * _ _ * _ _ * _ |
| * _ 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. * |
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| DIAGRAM 7-9A. DIAGRAM 7-9B. |
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| DIAGRAM 7-9. Is the FRONT LEFT TOP Corner Cubie Properly Oriented? |
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| Paragraph C: |
| Undo the customization you used in paragraph A. (You do remember what you did, don't you?) |
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| 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. |
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# 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. #
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| By the way, you should now be able to see a pretty "X" pattern of matching cubies on all |
| six sides of your Cube. |
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| +---+---+---+ |
| |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 | |
| +---+---+---+ |
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| DIAGRAM 7-10. X Marks Our Progress |
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