# Rubik's Cube/Finishing the Orientation of Corner Cubies

Jump to navigation
Jump to search

+---------------------------------------------------------------------------------------------+ | 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 | | | +---------------------------------------------------------------------------------------------+