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Okay, I'll give a group-theoretic answer, since that was requested, which goes most of the way to a proof. I'm assuming you mean a 3x3x3 Rubik's cube. For larger cubes, the proof is just a little bit less convenient, because these cubes don't form groups (or more precisely, the action of the group of all moves on the cube on the states of the cube is not ...

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Yes. Examples: Superflip "Checkerboard"

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You shouldn't think about the cube in terms of tiles, but on the mini-cubes it is split into, sometimes called cubies or cubelets. There are 26 of these (3*3*3 minus the center), and there are 3 types. The centers (1 color), the edges (2 colors), and the corners (3 colors). For instance, it is not just a blue tile, it is a blue-red edge piece, which there is ...

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The answer is Because I do hope the explanation and pictures suffice with no real need for a geometrical argument, because I'm not sure I can give a rigorous proof. This solution was found by Robert S. Holmes as part of a challenge called "a set of quickies" by Martin Gardner (who originally had found a solution of 20). The problem and Martin Gardner's 20-...

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There are three techniques that allow you to come up with useful move sequences for solving a cube. Conjugation This is where you already have a move sequence that does one thing and allows you to apply it more generally. Suppose for example that you know a move sequence X that twists the two adjacent corners at the front of the top layer (i.e. at UFR and ...

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If you're willing to take the time: The World Cube Association has a specific method for shuffling their cubes. They use a program called TNoodle to do the scrambling; the program generates a "scramble sequence" which can be followed to produce a scrambled cube. Their code actually features a nice interface to generate the scrambles: I have yet to figure ...

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My answer at Why is a single-corner twist not a valid position on a Rubik's cube? basically explains this, but in simpler terminology here are the facts presented without proof. If you randomly disassemble and reassemble a $3 \times 3 \times 3$ cube, the probability that it will be solvable with just face turns is exactly $1 \over 12$. Furthermore, this ...

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Given any position you know that has 20 moves (like the Superflip) just do one move towards the solution and you have a position that requires 19 moves.

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The "most scrambled cube" is any configuration that requires the greatest number of rotations to solve the cube using a perfect algorithm. Although unknown, this algorithm is hypothetically called God's algorithm and in fact the maximum number of rotations, called God's number has been found to be 20. The list of such 20-move cubes is given here. To ...

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The only way you can have a solve-all sequence is if you have a sequence of moves that goes through all 43 quintillion configurations of the Rubik's Cube. In order to do this, you need to draw a transition graph between all the states of the Rubik's Cube and find a Hamiltonian cycle through them. This sequence of moves doesn't necessarily have to be 43 ...

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Something like this?

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The key is that the four center cubules on each face of a Rubik's Revenge are indistinguishable. When you do the move sequence to swap those two edges in the "3x3x3" phase of solving a 4x4x4 cube (or to flip a single edge or swap two corners), a bunch of the center cubules get reoriented, but you don't notice because they're all the same colour and ...

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In a Rubik's cube, every legal move swaps a even number of dowels, so any legal configuration can be obtained only with a even number of swaps. In this configuration, the difference between a legal cube (the solved one) and the current status consists of 1 swap; since 1 is odd, this is a No Win Scenario.

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It seems like you are trying to find One such example that satisfies this is In particular, As mentioned by armb in the comments, there is a good answer here discussing the maximum orders for an $n \times n \times n$ Rubiks cube.

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Each set of moves is designed to correct a parity error in the cube's layout, but each group of parity it corrects is essentially independent and random. Each step has a chance of already having the right parity. For instance, the way I do it is solve the bottom two layers and then attack the parity errors on the top one at a time Make sure all the edges ...

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The easiest explanation would be that in a 3x3 cube, only one cube is out of position, but in a 4x4 cube two cubes are out of position. In a 15 puzzle (the sliding puzzle where you try to put the numbers in order) half of all possible initial positions are unsolvable. They call the solvable positions "even" and the unsolvable positions "odd". The "odd" ...

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I believe you can place You can do so by

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My answer is This is reached by

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A Rubik's cube has six faces, so there are six different colors to work with. This suggests to me that it's So I changed each of the dates into Thus the colors represent: Read the pattern from the 3rd block: So my answer is: I'm not sure the significance of this day, if there is one.

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The World Cubing Association actually has an entire event dedicated to solving the cube with the fewest move count (FMC). You are given a scramble, paper, pencil, up to 3 cubes and an hour. The methods used vary widely and involve much more "freestyle" than regular speedsolving. But I'll explain one of the more common "methods." (Which is more like a set of ...

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An observation is.. Using this observation, one of many possible solutions is (left image is front of cube, right image is inside view):

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I haven't solved the 3x3x3 Cube myself. I bought mine in 2010, and used a tutorial to solve it. In 2012 I started a twisty puzzles collection, and I currently have more than 350 puzzles in my collection. So even though I haven't solved the 3x3x3 Cube myself, I have solved quite a few other puzzles myself (after I gained knowledge, understanding and ideas ...

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I'm afraid this is not possible to solve using normal moves. Two things could have happened: 1. The cube was taken apart or some pieces popped out, and then it was reassembled incorrectly. 2. The cube is loose enough that during scrambling or solving a corner got caught on another piece and twisted in the middle of a move. In the latter case you can ...

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To start off, we can reduce the number of flags to look at with some simple deductions. The only national flags able to be made on a rubik's cube, would be ones of either 3 horizontal or vertical stripes, with no "extra" pieces (No stars, no crowns, whatever), and can only use the Rubik's cube colours, those being Blue, Yellow, Green, Red, Orange, and White ...

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