Presentation on theme: "Rubiks Cube Demystified. Rubiks Cube Basics Much is made of the number of permutations in the cube - some 43,252,003,274,489,900,000. But really, the."— Presentation transcript:
Rubiks Cube Demystified
Rubiks Cube Basics Much is made of the number of permutations in the cube - some 43,252,003,274,489,900,000. But really, the number of permutations is not important. If I put cards with letters A-Z on a table and said put them in order, youd be able to solve it in a minute. Yet the cards have 403,291,461,126,606,000,000,000,000 permutations. There is a way of moving beyond fixed formulas and patterns and actually begin to understand the cube. When people see me doing it fast they always make a joke and say I used to be able to solve it too – I used to peel off the stickers, LOL… Or they say, I used to be able to solve it too – I used to take the pieces apart and put them back together, LOL… Think about these comments for a second. What they are saying is that its not the number of pieces or combinations thats the problem – if the pieces could be moved INDIVIDUALLY, solving the problem is easy. The problem with the cube is that you cant move just one piece or swap two pieces. Each turn moves 9 pieces. Make 2 turns and half the pieces have moved. So this is the real problem. The pieces are intertwined and cannot move independently. If you tried to sort the A-Z cards, and each time you moved one card, several others moved, youd have a real puzzle on your hands there too. The problem with the cube is that, compared to the cards, treating the pieces independently is not so easy. However, there is a way to separate out the pieces. There is a method for moving just a few pieces at a time. Once you master these basic skills, you can literally move any piece to any position that you want at any time. So lets begin. The solution in my book, Jeff Conquers the Cube was designed for speed. It had very easy to spot patterns, and then you performed a specified short sequence of moves and then looked for another easy to spot pattern. Although there is logic to that solution, its primary focus is speed and easy pattern recognition. So that solution is easier to learn. The solution in this document is very different. The purpose of this solution is to give you an understanding and some control over the cube. It is to demystify it. This is not the fastest solution. In fact, its probably one of the slowest. But it is the clearest and most easily understood method around. Once you understand how to move the pieces individually, then in theory you should be able to make any pattern. One thing I do occasionally is to scramble one cube and then, using a solution like this one, take a solved cube and match it exactly to the scramble. In other words, this type of solution should give you total control and mastery over the puzzle.
Notation I will not be using the notation that I used in my book or that many use. Instead, I will use these little pictures. As you will see, these are really more useful than the alphabetic notation and I wish now that my book had used these pictures. One of the guiding ideas behind this particular solution is the idea of symmetry. Its a lot easier to see the symmetries this way, because the little diagrams can be seen going one way, and then the opposite way. I hope these are pretty self explanatory. Double arrows indicate 2 moves Individual pieces on the cube are referred to using the older notation, that I used in my book. Sides are given a letter: U – Up D – Down L – Left R – Right F – Front B – Back So an edge, which sits along 2 sides, is referred to by 2 letters. UR indicates the edge between the Up & Right sides. Corners, which sit on 3 sides, uses 3 letters, such as URF for Up, Right & Front
Counting moves As we proceed, I will speak of a sequence being an odd number of moves or an even number of moves. So I want to clarify what constitutes one move of the cube. This is 1 move This is really 2 moves. It consists of this double move: This counts as 0 moves. The only thing thats changed is the orientation, or really your own personal perspective This is 2 moves: Plus an orientation: This is 2 moves Therefore, this is 4 moves.
The Centers Never Move Lets set up some basics. Many people think of the individual colors and how they relate to one another. Its more useful to think of the puzzle in terms of its pieces. There are 8 corners, 12 edges and 6 centers. The best way to think of it is to realize that the centers never really move. So this move: Is Really the same as these 2 moves: Plus this reposition: Consider this simple sequence. It looks like its job is to move the centers (perform this on a solved cube) But you can do the same thing with this sequence, which never moves the centers. Can you see that the centers never move and really the edges and corners move around the centers? Heres another way of thinking about it. Starting with a solve cube turn any one side. Have the centers moved relative to each other? No. If one move cant move the centers relative to each other then no combination of moves can.
4 Tasks to Completion Since the centers never move, we have only to deal with the edges and the corners. On a standard 3x3 cube, our job is to line these up relative to the immobile centers. Each piece has 2 attributes: –Its Position –Its Orientation Therefore our solution needs 4 basic moves. –Edge Reposition –Corner Reposition –Edge Orientation –Corner Orientation I am going to show to show you 4 basic moves to perform each of these 4 tasks. However all 4 moves are really based on a single idea. That is the beauty of this particular method. The whole thing is based, really, on just one idea and one basic move format.
Basic Move Structure Action: Start with a solved cube and focus your eyes on the UR slot. The UR slot is the Action Target Slot. Whatever piece is in that slot will be the target of the action. Now follow the first 5 moves. Can you see that –The piece in the UR slot flips –The rest of the Upper side is untouched –The lower 2/3 of the puzzle are essentially scrambled. Heres the trick to the whole thing: How can you unscramble the lower 2/3 of the cube? Its kind of a trick question. The answer is that to unscramble the lower 2/3, you must perform the 5 moves in REVERSE. Its pretty obvious that thats guaranteed to work. But how does that help you? Wont that also undo the Action that we wanted (we flipped the UR piece)? The reverse move will unscramble the bottom, while flipping the piece in the UR slot in the OPPOSITE direction. But thats where we perform one key step. Since the 5 moves in the Action Block never touch any of the other 8 pieces on the upper side, you can safely turn that side between the forward and reverse Action Blocks, without causing much effect. The only effect is that the piece in the UR slot changes. So go ahead and perform the move labeled M. This moves a new piece into the UR slot by moving the piece in the UF slot into the UR position. This changes the Action Target. When you reverse the 5 moves in the Action Block, THAT NEW PIECE in the UR slot ( the piece that started in the UF position but was moved into the UR slot by the Manipulation Block) will be the one flipped in reverse, not the piece that was originally in the UR Slot. The move works on whatever piece is in the Action TargetSlot Go ahead and perform the next 5 moves. Can you see that this is the exact reverse of the first 5. Move 5 is undone, then move 4 is undone, etc. Finally reverse move M M12345M 54321M54321M Now the whole cube is restored except for the 2 pieces that occupied the UR slot (the original and the one that was move there by the Manipulation Block). Can you see that this HAS to work, logically. Its the principle of forward action, switch target, reverse sequence, reverse switch target. This will work regardless of the actual move in the Action Block. The goal is to develop moves that act on just a few pieces at a time. This way we can move the pieces at will and avoid the problem of the pieces being intertwined. So each move will perform some action on just a few pieces. The moves have a specific structure. Each move has 3 blocks: –The Setup Block. This block can be empty and will be explained in later pages. Ignore it for now. –The Action block is where the actual task occurs. The Action is designed to reposition or reoriented whatever piece sits in the Action Target Slot. –The Manipulation block positions pieces into the Action Target Slot so that the Action Block can work on them. Although all 4 moves are essentially the same, the easiest one to see is Edge Orientation, so Lets begin with that one.
Variations on the Basic Move Exercise: The Action block below does basically the same thing as the Action block on the previous page did. It reorients an Edge piece (in this case the UF piece). Work out the moves for Action and Manipulation yourself M M M See how easy it is! Manipulation Block: The basic concept is that you pick a side that will isolate the action. Lets call this the Manipulation Side. In the previous example, the Upper side is chosen. The Manipulation Block will consist of one move of that side. So Realistically, M can be any turn of that side. Possible moves for the Manipulation Block: Action Block: Will act on one piece on the Manipulation Side Lets call this the Action Target. All other pieces on the Manipulation Side are off limits to the Action Block. Will perform the needed action: It Positions or Orients a piece. Can really be any sequence of moves, except for a turn of the Manipulation Side. The Action block is almost anything you want. The beauty of this method is that it doesnt really matter how good or bad or long or short the Action Block is. It has the one very minor restriction, which is that it cant turn the Manipulation Side or affect any of its pieces (except Action Target). But other than that it s pretty wide open. So the Action Block positions or orients the Action Target and the Manipulation Block moves a new piece from the Manipulation Side into the Action Target Slot. We will call this other piece from the Manipulation Side the Reverse Action Target. At the end of the whole move, Action is fully reversed. Any random scramble is restored. The only effect is that Action Target and the Reverse Action Target have been affected in opposite ways. For example, if Action rotates an Action Target corner clockwise, then Reverse Action Target will be rotated counterclockwise.
Block Sequence The Manipulation Block can come AFTER or BEFORE the Action Block. So this is acceptable: Action 1-NM Action N`-1M Action 1-NM Action N`-1M In either Case, one move the Manipulation Block is between Action and Action. This is what is important. And So is this
The Setup Block The Setup Block: Purpose of the Setup Block is to move all desired Action Targets onto a single Manipulation Side prior to beginning the Action Block. For example. Lets say that I wanted to flip the UR and LF pieces. These pieces do not share a side. Therefore there is no clear Manipulation Side. So we must preface the Action Block with a move in order to Setup a viable Manipulation Side. In this example, Either of these moves would work: Moves the LF to the UL position Moves the LF to the UF position The Setup Block has no real restrictions, but in practice there will be no moves of the Manipulation Side. Any Position or Orientation move can be Setup in 0, 1 or 2 moves. However there are cases when you are trying to do both Position and Orientation at the same time and these may involve more complex Setups.
Final Block Sequence Either of these Sequences is acceptable. The Setup and its reverse wrap the move. As long as the Action and Manipulation interleave, the sequence is unimportant. Action 1-NM Action N`-1M Action 1-NM Action N`-1M Setup 1-N Setup N`-1 Setup 1-N Setup N`-1 Or
Inside the Action Blocks Look at how easy these Action Blocks are to construct. Not only is there symmetry between the blocks, there is a lot of symmetry within the Action Block itself. Regardless of which of the 4 moves you are trying to achieve, the same rules apply: Choosing a Manipulation Side is just a matter of orienting the cube. So you can always have this be the Upper Side if thats easier for you to focus on. You can always rotate the Cube so that one of the Action Targets is either the UR or URF pieces. The Action block will work on the Action Target but nothing else on the Manipulation Side. Therefore you must move the Action Target off the Manipulation Side. Therefore A move like this, is always a valid first move: The next move will continue to move the Action Target Piece. The first move moved it off the Upper side to the Front. Therefore the next move will have to touch the front. Since you dont want to move the right side again, nor break anything else on the Manipulation Side, you most likely second moves are some thing in this group: So one of these for Edge Moves: Or one of these for Corner Moves: Action blocks can be run either Forward or reverse. In other words, Action can be run first and then Action. The results are not necessarily identical, but are always still valid. They are still Action blocks and always within the same class of the 4 moves (position/corner orient/position). For example. Action might rotate a corner clockwise. Then Action rotates it counter-clockwise. Action can be run before Action, as needed.
4 Sample Action Blocks 3 - Orient Edges – Action Target = UR: 4 - Orient Corners – Action Target = URF: 1 -Position Edges – Action Target = UR: 2 -Position Corners – Action Target = URF: Notice that these are practically the Same. Its just that you move the middle slice when dealing with the Edges, and the bottom slice when dealing with a corner: When Reorienting, we start out the same as with Positioning, since we must move the Action Target off the Manipulation Side before we can do anything with it. But there are a few more moves since we must also put the piece back. These are opposites: Since the first turn moves multiple pieces off the Manipulation Side, you must put all but the Action Target Back. These are the same repeated Notice that all we are doing is alternating between these types of moves:
Position Blocks Orientation blocks are pretty easy to understand. Whatever you do is exactly reversed. Clockwise is followed by counterclockwise. Everything is done in pairs. The only affected pieces are the Action Target and the Reverse Action Target. Position blocks are shorter, but a bit harder to understand theoretically. Everything is done in 3s. You cannot swap just two pieces on the cube (more on this later). In addition to Action Target and Reverse Action Target a third piece called Piece C is involved. Action moves the Action Target off the Manipulation Side. Since no other pieces on the Manipulation Side can be moved by Action, and since some piece must replace the Action Target, then that introduces Piece C into the equation. Piece C is simply some other piece from anywhere else on the other 2/3 of the cube. Action does this: Moves Action Target off the Manipulation Side Moves Piece C into Action Targets position After the Manipulation Block, Action will have the reverse affect on Reverse Action Target: Moves Reverse Action Target into Piece Cs position Moves Action Target back onto the Manipulation Side but into Reverse Action Targets position. This is a bit confusing at first. Follow it visually. Start with A, B & C. B is the Action Target and A is the Reverse Action Target. Look at the trivial case first, where M is skipped. A B C Action moves B off Manipulation Side, replaced by C AB C A B C Action Simply undoes Action. Nothing has happened. C is pushed back off Manipulation Side But when you introduce M, the situation Changes. M swaps A & C, so Action pushes A off the Manipulation Side back into Piece Cs position. AB CC B A CB A BC AA B C Action moves B off Manipulation Side, replaced by C Action A is pushed into Cs original position. Meanwhile B is moved into As Slot M swaps A & C M swaps B & C
Symmetry Notice that all of the moves so far have been an even number, since they are all done as a forward/reverse pair. There are certain restrictions with the cube. Not every combination is reachable without taking the puzzle apart. For example, you cant have just one edge or one corner flipped. These can only be done in pairs. This is why all of the orientation moves work on pairs of pieces. Regarding positioning, you cannot have just 2 pieces switched. The closest you can come is: 3 edges switched 3 corners switched 2 corners switched + 2 edges switched Its this last move that is interesting. With the first 2 (where 3 pieces are switched) you are just one Action/Action sequence away from being putting all pieces in position. But the last situation is different. Notice that all the moves Ive outlined in this solution are an even number of moves because they are done in forward/reverse pairs. But what if the cube is an odd number of moves away from solution? What if you start with a solved cube, make a single turn and then use Action/Action sequences as outlined here to solve the puzzle? If you did that, the closest you could come would be to end up with 2 corners switched/2 edges switched. You cannot get to the solved state from here using an even number of turns. The only way out of this is to make a single turn (anywhere). You can now proceed with Action/Action sequences and the puzzle will come into place. Usually, when I do this method I still like to solve all the corner positioning first. This is not a requirement, but just a preference. If I notice that I have 6 corners in position, and 2 out of position, I know that Ive got an odd number of turns to go. So I have to make a single turn and then proceed. This will temporarily knock some pieces out of place and I may end up with 3 or 4 out of position. But this is the way to go.
Summary This presentation outlines a powerful method for understanding the cube and complexity in general. Complexity arises when factors become intertwined and variables are allowed to permutate and grow out of control. Power over the situation can be achieved by developing methods which allow isolated action, thus circumventing the permutation problem. Using this method you can take small action and solve the pieces of the Cube a few at a time, without those moves affected the puzzle as a whole. The problem of exponentially expanding permutations is solved – we now have a method that grows only arithmetically, rather than geometrically. Using the same basic move format, all aspects of the cube can be controlled. Although other Action Blocks can be developed, the entire cube can be solved using just this small sampling of moves: Action 1-NM Action N`-1M Action 1-NM Action N`-1M Setup 1-N Setup N`-1 Setup 1-N Setup N`-1 Or Orient Edges – Action Target = UR: Orient Corners – Action Target = URF: Position Edges – Action Target = UR: Position Corners – Action Target = URF: Manipulation Block Or Move Formats Sample Moves
6 Sided Cube Heres the key move to do the 6 sided Cube, starting with a solved cube: First you have to position the corners using the move that was done a few pages back for positioning corners Obviously, you then perform M and Action. Then reorient the cube by doing this Action 1-NM The URF corner is now the basis for the 2x2 cube that will sit inside the big cube. The pattern cannot be made on the opposite side because youd have to switch the UBL and BLD corners and, of course, you cant switch just 2 corners. This move will reposition the centers. This is a weird sequence that I cant explain without a whole long explanation. But basically, this puts the centers into position relative to the corners: From this point out, you can rely exclusively on Positioning and Orienting Edge moves the way that Ive described. I know Im leaving you with a bit of work, But see what you can do from here using the lessons in this method.