5x5 Rubik S Cube Solution Pdf 68 BEST
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There are two main versions of the Megaminx. The 6-color variant uses the 6 colors common to puzzle cubes - white, yellow, blue, green, red and orange - with opposite faces having the same color. The 12-color variant uses a unique color for each face of the puzzle. The most common 12-color scheme starts with white bordered by yellow, dark blue, red, dark green and purple. Directly opposite each of these faces, respectively, are gray, beige, light blue, orange, light green and pink. Black is a common alternate face color, either replacing gray which improves color contrast on what is typically the last layer solved, or replacing beige, which typically shares two neighbors (red and green) with the white face which can make piece identification challenging in certain lighting conditions. The 12-color Megaminx is the only type legal in official WCA competitions (color scheme variations are legal as long as each face is uniquely-colored), and is therefore much more popular than the 6-color version.
Regardless of color variant, the most common solution strategy shares qualities with common methods for solving a Rubik's Cube. The solution begins with one face (most often white), where the solver will reconstruct the \"star\" formed by the edge pieces adjacent to that face, each one properly paired with the neighboring center color (analogous to the \"white cross\" of the beginner and CFOP methods for a Rubik's Cube). Once this is done, the five white corner pieces are maneuvered into place, commonly paired with the appropriate edge piece on the opposing side of each corner from the starting face. The star, first five corners and first five edges are known as the \"first two layers\", similar to those of a 3x3 cube. From here, the solver will typically proceed to the \"second two layers\", which involves a similar process of joining and positioning corner and edge pieces around the equator of the puzzle, and then the corners and edges in the hemisphere opposite the starting face. This is all typically done \"intuitively\", turning faces of the puzzle based on where the solver wishes the pieces to move, with the result of each move being predictable given some experience manipulating the puzzle.
This leaves the solver with the \"last face\", most commonly grey (opposite the white starting face), which will require a series of more complicated move sequences to orient (flip/rotate) and permute (rearrange) the pieces of the last layer in predefined ways without disturbing the solved lower layers. These \"algorithms\" are similar to those used on a Rubik's Cube and thus familiar to those experienced in solving one, with the obvious allowance made for the extra side of each face (and thus the extra turn possible as part of a full revolution of the face). The \"beginner\" method requires only four or five algorithms of between three and six moves each to be memorized, which will be used to orient and then permute the last layer's edge pieces to form the last layer star, followed by orienting and permuting last layer corners. More complicated algorithm sets require memorization of a wider variety of distinct sequences of longer length, which nevertheless are more efficient solutions for each of a more specific set of possible permutations of the latest layer's pieces, performing orientation and permutation of both edges and corners in fewer sequences (and total moves) than the shorter but more repetitive beginner algorithms.
The 6-color Megaminx comes with an additional challenge which is not immediately obvious (and which does not occur on the 12-color puzzle). Its edge pieces come in visually identical pairs, because of the duplicated colors of opposite faces. However, although visually indistinguishable, they are nevertheless mathematically bound in a parity relationship. In any legal position (reachable from the solved state without disassembling the puzzle), there is always an even number of swapped pairs of edges. However, since swaps may be between visually identical edges, one may find that having solved almost the entire puzzle, one is left with a pair of swapped (distinct) edges that seems to defy all attempts to exchange them. The solution is to swap a single pair of 'identical' edges to resolve the parity issue, and then restore the rest of the puzzle.
There are many similar puzzles with different numbers of layers, most of which change the \"mega\" in the puzzle's name to another metric prefix. They are the Kilominx (2 layers), Master Kilominx (4 layers), Gigaminx (5 layers), Elite Kilominx (6 layers), Teraminx (7 layers), 88 Kilominx (8 layers), Petaminx (9 layers), Examinx (11 layers), Zettaminx (13 layers), Yottaminx (15 layers), and Atlasminx Or Quettaminx (19 layers).[3][4][5] The highest order mass-produced variant of the Megaminx is the Zettaminx, which was released by ShengShou in 2021, and the highest order variant of the Megaminx ever made to date is the Minx of Madness, created by Corenpuzzle using 3D printing. The Minx of Madness was revealed in May 2022. It is the dodecahedral equivalent to a 21x21x21 Rubik's cube.[6]
Tony Fisher has produced a shape modification of the Megaminx into a cube form which he called the Hexaminx.[7] Another variant is the Holey Megaminx, which has no center pieces, like the Void Cube. It is being produced by Mèffert as of July 2009. Other variants include the Flowerminx, Megaminx Ball, and Crazy Megaminx.
The Rubik's Cube can be very frustrating and may seem next to impossible to restore to its original configuration. However, once you know a few algorithms, it is very easy to solve. The method described in this article is the Layer by Layer method also known as the Beginner's Method: first, you solve one face of the cube (first layer), then the middle layer, and finally the last layer.
This ACM Paper describes several alternative ways that it has used to represent a rubik's cube and compares them against eachother. Sadly, I don't have an account to get the full text but the description states:
Seven alternative representations of Rubik's Cube are presented and compared: a 3-by-3-by-3 array of 3-digit integers; a 6-by-3-by-3 array of literals; a 5-by-12 literal matrix; an ll-by-ll sparse literal matrix; a 54-element vector; a 4-dimension array; and a 3-by-3-by-3 nested array. APL functions are given for orientation moves and quarter-turns plus several useful tools for solving the cube.
The short answer is that it depends on how you're going to solve the cube. If your solver is going to use a human method like the layer-by-layer approach or the Fridrich method then the underlying data structure won't make much of a difference. A computer can solve a cube using a human method in negligible time (well under a second) even in the slowest of programming languages. But if you are going to solve the cube using a more computationally intensive method such as Thistlethwaite's 52-move algorithm, Reid's 29-move algorithm, or Korf's 20-move algorithm, then the data structure and programming language are of utmost importance.
I implemented a Rubik's Cube program that renders the cube using OpenGL, and it has two different types of solvers built in (Thistlethwaite and Korf). The solver has to generate billions of moves and compare each cube state billions of times, so the underlying structure has to be fast. I tried the following structures:
Expanding on method (3) above, each face of the cube is made up of 9 stickers, but the center is stationary so only 8 need to be stored. And there are 6 colors, so each color fits in a byte. Given these color definitions:
Expanding on method (4) above, some of the algorithms for programmatically solving the Rubik's Cube use an iterative deepening depth-first search with A*, using pattern databases as a heuristic. For example, Korf's algorithm utilizes three pattern databases: one stores the index and orientation of the 8 corner cubies; one stores the index and orientation of 6 of the 12 edge pieces; the last stores the index and orientation of the other 6 edges. When using pattern databases, a fast approach is to store the cube as a set of indexes and orientations.
The orientation of each cubie needs to be kept as well. An edge piece can be in one of two orientations (oriented or flipped), while a corner piece can be in three different orientations (oriented, rotated once, or rotated twice). More details about the orientation of pieces can be found here: =eoline#eo_detection With this model, rotating a face means updating indexes and orientations. This representation is the most difficult because it's hard for a human (for me at least) to look at a big blob of index and orientation numbers and verify their correctness. That being said, this model is significantly faster than dynamically calculating indexes and orientations using one of the other models described above, and so it's the best choice when using pattern databases. You can see an implementation of this model here: -cube-cracker/tree/4.0.0/Model (see RubiksCubeIndexModel.{h,cpp}).
As mentioned, the program also renders the cube. I used a different structure for that part. I defined a \"cubie\" class, which is six squares with 1, 2, or 3 colored faces for center, edge, and corner pieces, respectively. The Rubik's Cube is then composed of 26 cubies. The faces are rotated using quaternions. The code for the cubies and cube is here: -cube-cracker/tree/4.0.0/Model/WorldObject
An interesting method to represent the cube is used by the software \"Cube Explorer\". Using a lot of clever maths that method can represent the cube using only 5 integers. The author explains the maths behind his program on his website. According to the author the representation is suited to implement fast solvers.
Probably a bigger concern is how to represent the various transforms. Rotating a single face of a physical cube (all cube moves are essentially rotations of a single face) would have to be represented by swapping around a lot of cuboid objects. 153554b96e