Rubik's Cube/Keeping Track of Details
This is where Chapter 05 has been installed. Some diagrams may have been messed up in translation. I will try to fix them as I convert from Top, Bottom, and bacK to Up, Down, and Back.
Ray Calvin Baker 03:43, 10 November 2011 (UTC)
+---------------------------------------------------------------------------------------------+ | 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 resources | | | | Chapter Five - - - - - - - - - - Keeping Track of Lots of Details -- Rotating Corner Cubies | | | | Goal Two is to get all of the corner cubies properly oriented in their proper corners of | | the Cube. | | | | We will need to consider more details when we try to orient the corner cubies, without | | messing up where they are placed. For now, we are still ignoring the edge cubies. So, let's | | start with a diagram of the corner cubies of Rubik's Cube. (Yes, I have still left out some | | details which are not essential for what I am trying to do.) I am using arbitrary letters | | and numbers to identify the labels on the visible faces of each corner cubie. And yes, this | | is another chapter where we use pencil and paper more than Cube turning. | | | | This diagram is like an isometric drawing, with some lines left out. I tried to emphasize | | the three sides of the visible corner cubies, and I added some notes, "(1)", "(C)", "(5)", | | "(G)", "(e)", "(f)", and "(g)" to indicate the hidden labels on several of the corner | | cubies. Later on, I will show you several other types of diagrams that I used; I hope a few | | of them will also help you. | | | | d a, b, c, and d are on the TOP side. | | a TOP c | | (1) b (C) A, B, E, and F are on the FRONT side. | | A 3 | | B 2 2, 3, 6, and 7 are on the RIGHT side. | | (5) FRONT RIGHT (G) | | E 7 (Items indicated in parentheses are | | F 6 actually hidden, around the corner.) | | (e) (g) | | (f) Position 0 | | | | DIAGRAM 5-1. Diagram for Orienting Corner Cubies -- Position Zero | | | | (There is a cubie in the back, which would bear the labels "H", "h", and "8".) | | | | If we are not yet cubemeisters, we will need to do SOMETHING to get started. I propose that | | we explore what happens with two simple moves. As you will see, we may be able to find LOTS | | of clever operators with just a little bit of VERY CAREFUL analysis. | | | | So, here is our first (arbitary) move, Fv. Rotate the FRONT face clockwise one quarter | | turn. | | | | d | | 5 TOP c | | (e) 1 (C) | | E 3 | | A a | | (f) FRONT RIGHT (G) | | F 7 | | B b | | (6) (g) | | (2) Position 1 | | | | DIAGRAM 5-2. What FRONT Clockwise Did | | | | Our second arbitrary move is R^. Rotate the RIGHT face counterclockwise one quarter turn. | | | | d | | 5 TOP G | | (e) C (g) | | E 7 | | c 3 | | (f) FRONT RIGHT (2) | | F b | | 1 a | | (6) (B) | | (A) Position 2 | | | | DIAGRAM 5-3. What RIGHT Counterclockwise Did -- Position Two | | | | Now let's compare position 0 with position 2. | | | | d d | | a TOP c 5 TOP G | | (1) b (C) (e) C (g) | | A 3 E 7 | | B 2 c 3 | | (5) FRONT RIGHT (G) Fv R^ (f) FRONT RIGHT (2) | | E 7 ----> F b | | F 6 1 a | | (e) (g) (6) (B) | | (f) (A) | | Position 0 Position 2 | | | | DIAGRAM 5-4. Comparing Position Zero With Position Two | | | | "A" started on the front side of the TOP LEFT FRONT cubie. "A" ended on the bottom side of | | the BOTTOM RIGHT FRONT cubie. | | | | "f" started on the bottom side of the BOTTOM RIGHT FRONT cubie. "f" ended on the left side | | of the BOTTOM LEFT FRONT cubie. | | | | "5" started on the left side of the BOTTOM LEFT FRONT cubie. "5" ended on the top side of | | the TOP LEFT FRONT cubie. | | | | We can summarize these facts like this: A -> f -> 5 -> a -> etc. The entire series is: | | A -> f -> 5 -> a -> 6 -> e -> 1 -> F -> E -> A. | | | | "a" started on the top side of the TOP LEFT FRONT cubie. But this means that if we repeat | | these two moves three times, the TOP LEFT FRONT cube will be rotated counterclockwise by | | 120 degrees. Other things will happen also, so we need to think things through. | | | | "B" started on the front side of the TOP RIGHT FRONT cubie. "B" ended on the bottom side of | | the BOTTOM RIGHT BACK cubie. | | | | "g" started on the bottom side of the BOTTOM RIGHT BACK cubie. "g" ended on the back side | | of the TOP RIGHT BACK cubie. | | | | "C" started on the back side of the TOP RIGHT BACK cubie. "C" ended on the top side of the | | TOP RIGHT FRONT cubie. | | | | "b" started on the top side of the TOP RIGHT FRONT cubie. We can summarize these facts like | | this: B -> g -> C -> b -> etc. The entire series is: | | B -> g -> C -> b -> 7 -> 3 -> 2 -> G -> c -> B. | | Repeating the two moves three times will also rotate the TOP RIGHT FROMT cube clockwise by | | 120 degrees. | | | | You should be able to verify these sequences: | | 3 -> 2 -> G -> c -> B -> g -> C -> b -> 7 -> 3 -> 2, and | | E -> A -> f -> 5, F -> E -> A -> f, and 7 -> 3 -> 2 -> G. | | Notice that three repetitions of the two moves rotates the TOP RIGHT BACK cubie clockwise | | by 120 degrees. | | | | d d | | a TOP c 5 TOP G | | (1) b (C) (e) C (g) | | A 3 E 7 | | B 2 c 3 | | (5) FRONT RIGHT (G) Fv R^ (f) FRONT RIGHT (2) Fv R^ | | E 7 ----> F b ----> | | F 6 1 a | | (e) (g) (6) (B) | | (f) (A) | | Position 0 Position 2 | | | | d d | | f TOP 2 A TOP 3 | | (6) g (B) (a) + B - (c) | | F b 1 - C | | G 7 2 b | | (A) FRONT RIGHT (3) Fv R^ (E) FRONT RIGHT (7) | | 1 C ----> e g | | e 5 + 6 f - | | (a) (c) (5) + (G) | | (E) (F) | | "After Four Turns" "After Six Turns" | | | | DIAGRAM 5-5. After Six Turns | | | | In this last picture, "After Six Turns", I have marked clockwise rotations of corner cubies | | with "-" signs, and counterclockwise rotations with "+" signs. (The two cubies in the BACK | | LEFT column are not rotated at all. I can use the "0" symbol to indicate this, if I need | | to.) | | | | Hmmm. Three cubies are rotated clockwise, three are rotated counterclockwise, and two are | | not rotated at all. Because we noticed that all cubies are returned to their stating | | locations, we will, hopefully, not have to worry about messing up Goal One. Can we do | | something useful with the information we have learned so far? Yes! | | | | Let's try to ignore "irrelevant" details, and concentrate on the patterns of rotation of | | the corner cubies. I make a copy of the picture "After Six Turns" (with slightly different | | perspective), unroll it, then remove some details. NOTE: These operations are NOT something | | you can actually do on a real Cube -- these are conceptual drawings only, intended to help | | us visualize more of the surface of the Cube. We can NOT take a Cube apart in this fashion. | | but we CAN make drawings of all six sides. | | | | TOP | | explodes; | | _ *-------* _ | | * |corner | * | | _ * _ _ * _ /| ? |cubies | ? |\\ | | _ * _ d _ * _ _ * _ d _ * _ / * _ *stretch* _ * \\ | | _ * _ ? _ * _ ? _ * _ _ * _ ? _ * _ ? _ * _ /A | *-------* | 3\\ | | * _ a _ * _TOP_ * _ c _ * * _ A _ * _TOP_ * _ 3 _ * * _ | ? | | ? | _ * | | | * _ ? _ * _ ? _ * | | * _ ? _ * _ ? _ * | | * _ | B | _ * | | | | A | * _ b _ * | 3 | | 1 | * _ B _ * | C | | 1 | * _ _ * | C | | | * _ | ? | * | ? | _ * * _ | ? | * | ? | _ * * _ | ? | * | ? | _ * | | | * _ | B | 2 | _ * | | * _ | 2 | b | _ * | | * _ | 2 | b | _ * | | | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | ? | * _ | _ * | ? | | | * _ FRONT * RIGHT _ * * _ FRONT * RIGHT _ * * _ FRONT * RIGHT _ * | | | * _ | ? | ? | _ * | | * _ | ? | ? | _ * | | * _ | ? | ? | _ * | | | | E | * _ | _ * | 7 | | e | * _ | _ * | g | | e | * _ | _ * | g | | | * _ | ? | * | ? | _ * * _ | ? | * | ? | _ * * _ | ? | * | ? | _ * | | * _ | F | 6 | _ * * _ | 6 | f | _ * \\ * _ | 6 | f | _ * / | | * _ | _ * * _ | _ * \\ | * _ | _ * | / | | * * BOTTOM \\| ? | * | ? |/ | | explodes * _ | F | _ * | | *-------* | | Position 0 After Six Moves Exploding the Cube | | | | DIAGRAM 5-6A. DIAGRAM 5-6B. DIAGRAM 5-6C. | | | | * * * * | | /| |\ |\\ /| | | RIP! * | Opening | * | * * | | | _ _ /| * the * |\ *d|\\ /|d* | | _ * \\ / * _ * |/| Cube's |\| * |\\| * * |/| | | _ * | V | * _ /|?* | surface | *?|\ | *?|\\ /|?* | | | ( _ | ? | _ - _ | ? | _ ) ( _/| * * |\\_) * |\\| \\_ _/ |/| * | | | * _ * | * _ * | | *-| |-* | |\\| * * _ _ * * |/| | | | | * _ | _ * | | | | * _ _ * | | | *?|\\ A | * _ _ * | 3 /|?* | | | | * _ | ? | - | ? | _ * * _ | ? | - | ? | _ * *?|\\| * _ | ? | - | ? | _* |/|?* | | | | * _ | | _ * | | * _ | | _ * | |\\| *a| * _ | B | _ * |c* |/| | | | | * _ _ * | | | | * _ _ * | | | *L|\\| 1 | * _ _ * | C |/|K* | | | * _ | ? | * | ? | _ * * _ | ? | * | ? | _ * * |\\|?* _ | ? | * | ? | _ * |/| * | | | * _ | | | _ * | | * _ | | | _ * | |\\| * | * _ | 2 | b | _ * |?* |/| | | | ? | * _ | _ * | ? | | | * _ | _ * | | | *?|\\| ? | * _ | _ * | ? |/|?* | | | * _ FRONT * RIGHT _ * * _ FRONT * RIGHT _ * * |\\| * _ FRONT * RIGHT _ * |/| * | | | * _ | ? | ? | _ * | | * _ | | | _ * | \\| *E| * _ | ? | ? | _ * |7* |/ | | | | * _ | _ * | | | | * _ | _ * | | *?|\\| e | * _ | _ * | g |/|?* | | * _ | ? | * | ? | _ * * _ | | * | | _ * \\| * _ | ? | * | ? | _ * |/ | | | * _ | | | _ * | | * _ | | | _ * | * * _ | 6 | f | _ * * | | | | * _ | _ * | | | | * _ | _ * | | \\ 5 | * _ | _ * | G / | | ( _ | ? | * | ? | _ ) ( _ | | * | | _ ) \\_ | ? | * | ? | _/ | | * _ | | _ * * _ | | _ * * _ | F | _ * | | * _ _ * * _ _ * * _ _ * | | - - - | | Starting to Rip Apart Separating the LEFT We Are Unwrapping the Cube! | | the Surface of the Side From the BACK | | Cube Side | | | | DIAGRAM 5-6D. DIAGRAM 5-6E. DIAGRAM 5-6F. | | | | DIAGRAM 5-6. Unwrapping the Cube | | | | There are several ways to show all six sides of the Cube on one diagram. Each way | | introduces some kind of distortion, but sometimes we can live with that if we can gain more | | insight into how the Cube works. | | | | Examples: | | | | Why not choose a way that emphasizes the details we are interested in? For now, we are | | interested primarily in how the corner cubies can be rotated. So, let's take a diagram of | | an "unwrapped" Cube and ignore the parts we don't need now. Later, we may want to emphasize | | some other details, instead. | | | | +---+---+---+ A completely unwrapped Cube | | --> |TOP|TOP|TOP| | | +---+---+---+ The distortions in this kind of | | |TOP|TOP|TOP| diagram separate parts of some | | +---+---+---+ cubies, often by a large distance. | | |TOP|TOP|TOP| | | +---+---+---+---+---+---+---+---+---+---+---+---+ For example, | | --> | L | L | L | F | F | F | R | R | R | K | K | K | <--- the three arrows | | +---+---+---+---+---+---+---+---+---+---+---+---+ point to separated | | | L | L | L | F | F | F | R | R | R | K | K | K | parts of the | | +---+---+---+---+---+---+---+---+---+---+---+---+ BACK LEFT TOP | | | L | L | L | F | F | F | R | R | R | K | K | K | corner cubie. | | +---+---+---+---+---+---+---+---+---+---+---+---+ | | | B | B | B | | | +---+---+---+ This is a diagram of the Cube | | | B | B | B | as we would like to see it | | +---+---+---+ when we are finished working | | | B | B | B | with it. | | +---+---+---+ | | | | DIAGRAM 5-7. A Completely Unwrapped Cube | | | | (0)--+---+--(-) A completely unwrapped Cube | | --> | 4 | ? | 3 | | | +---+---+---+ The distortions in this kind of | | | ? |TOP| ? | diagram separate parts of some | | +---+---+---+ cubies, often by a large distance. | | | A | ? | B | | | (0)--+---+--(+)--+---+--(-)--+---+--(-)--+---+--(0) For example, | | --> | D | ? | a | 1 | ? | 2 | b | ? | C | c | ? | d | <--- the three arrows | | +---+---+---+---+---+---+---+---+---+---+---+---+ point to separated | | | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? | parts of the | | +---+---+---+---+---+---+---+---+---+---+---+---+ BACK LEFT TOP | | | | ? | E | e | ? | 6 | f | ? | g | 7 | ? | | corner cubie. | | (0)--+---+--(+)--+---+--(+)--+---+--(-)--+---+--(0) | | | 5 | ? | F | | | +---+---+---+ This is what diagram 5-6F woulf look like | | | ? | B | ? | if we finished unwrapping it. | | +---+---+---+ | | | | ? | G | I emphasize the rotation of each corner | | (0)--+---+--(-) cubie with "(+)", "(-)", and "(0)" signs. | | | | DIAGRAM 5-7. Unwrapped Cube After Six Moves | | | | If we ignore portions of diagram 5-7 which do not pertain to the rotations of corner | | cubies, we can make some simplified diagrams, which may be easier to work with. | | | | | A | ? | B | | | (0)--+---+--(+)--+---+--(-)--+---+--(-)--+---+--(0) First, we ignore portions | | | D | ? | a | 1 | ? | 2 | b | ? | C | c | ? | d | of the TOP and BOTTOM. | | +---+---+---+---+---+---+---+---+---+---+---+---+ | | | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? | Then, we ignore the lines | | +---+---+---+---+---+---+---+---+---+---+---+---+ and the edge cubies. | | | | ? | E | e | ? | 6 | f | ? | g | 7 | ? | | | | (0)--+---+--(+)--+---+--(+)--+---+--(-)--+---+--(0) | | | 5 | ? | F | | | | | DIAGRAM 5-8A. | | | | TOP For short, we can draw it like this: | | 0 + - - 0 | | D a 1 2 b C c d 0 + - - 0 | | LEFT FRONT RIGHT BACK 0 + + - 0 | | H E e 6 f g 7 h [F] [K] <-- These notes serve only to remind | | 0 + + - 0 us which are the FRONT and BACK | | BOTTOM sides of the Cube. | | | | DIAGRAM 5-8B. DIAGRAM 5-8C. | | | | DIAGRAM 5-8. Developing a New Type of Diagram | | | | See if you can figure out the rather peculiar arithmetic of "rotate by 120 degrees". | | | | "0" & "0" = "0" "0" & "+" = "+" "0" & "-" = "-" | | "+" & "0" = "+" "+" & "+" = "-" "+" & "-" = "0" | | "-" & "0" = "-" "-" & "+" = "0" "-" & "-" = "+" | | | | DIAGRAM 5-9. The Peculiar Arithmetic of Rotate by 120 Degrees | | | | Now I remove even more details from the stuff between "TOP" and "BOTTOM" above, and make | | four copies. (By the way, what we are doing here is "paper and pencil" work, organized in a | | systematic fashion. It does not yet require that you do anything with a Cube.) | | | | 0 + - - 0 0 + - - 0 0 + - - 0 0 + - - 0 (I used "[F]" and "[K]" to | | 0 + + - 0 0 + + - 0 0 + + - 0 0 + + - 0 emphasize how the Cube | | [F] [K] [F] [K] [F] [K] [F] [K] is oriented.) | | | | Now combine these (using "&" arithmetic) with rotated copies. (There is some important | | information concerning "how to do this" as it is related to working with an actual Cube in | | Chapter Six, "Customize Your Moves -- Commutation". Please don't rush into things just | | yet -- leave your Cube on the table just a little while longer.) | | | | 0 + - - 0 - 0 + - - - - 0 + - + - - 0 + | | 0 + + - 0 - 0 + + - + - 0 + + + + - 0 + | | [F] [K] [F] [K] [F] [K] [F] [K] | | | | ... and the results are: | | | | 0 - + + 0 - + 0 + - - 0 - 0 - + + + + + | | 0 - - + 0 - + - 0 - + 0 + 0 + - + - 0 - | | [F] [K] [F] [k] [F] [K] [F] [K] | | | | These strange-looking diagrams really are coded solutions for problems involving the | | orientation of the corner cubies of the Cube. Chapter Seven, "Finishing the Orientation of | | Corner Cubies", will explain how we can use these patterns, and many others, to get all | | eight corners of the Cube perfectly oriented. But first, let me explain in Chapter Six, | | "Customize Your Moves -- Commutation", how to "customize" any series of moves you can learn | | to solve lots and lots and lots of Cube problems. | | | | Diagram 5-5 shows us that we can rotate several corner cubies without changing their | | locations. In Chapter Seven, "Finishing the Orientation of Corner Cubies", we will find | | some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, | | we need to understand a very helpful principle that will guide all of our work. This will | | be explained in Chapter Six, "Customize Your Moves -- Commutation". | | | +---------------------------------------------------------------------------------------------+