Latest revision as of 00:34, 19 May 2022

2x2 Pocket Cube

Main article: Pocket Cube

The 2x2 Rubiks Cube is a great introduction to the Rubik's Cube, as it provides a simple introduction to the step-by-step method of solving cubes using algorithms.

Solving

A basic 2x2 solution technique: Pocket Cube/Solution Method

Mathematical Representation

See the Rubiks Cube section below for a lot more detail on mathematical representations of Rubiks Cubes.

We should write up a tuple representation of the 2x2 pocket cube, and how to represent permutations on the pocket cube.

3x3 Rubiks Cube

The standard Rubiks Cube size is a 3x3.

Pages on the 3x3 cube:

Rubiks Cube/Layer Method (algorithm for solving)
Rubiks Cube/Tuple (matheamtical representation of the cube)
Rubiks Cube/Permutations (permutation algebra applied to the cube)
Rubiks Cube/Numbers (counting states)
Rubiks Cube/Patterns (group theory and algorithm patterns)

Solving the 3x3

A basic, layer-by-layer Rubiks Cube solution technique: Rubiks Cube/Layer Method

There are other, more advanced techniques:

Fridrich Method (CFOP) [1] - also see Rubiks Cube/Fridrich Method
2x2 Method (blockbuilding)
Blindfolded [2]
Blindfolded (another) [3]

Mathematical Representation of the 3x3

The representation of the Rubik's Cube for use in mathematics (or computer programs) is an important topic. To turn a particular Rubik's Cube configuration into a mathematical representation agnostic to the particulars of the cube, we can label each face with an integer 1 to 54 (from 9 squares per face and 6 total faces), and pick a particular order for each face. Then we can write a sorted cube as (1 2 3 4 5 ... 52 53 54). This also enables enumerating combinatoric properties of the cube.

See Rubiks Cube/Tuple for more on this tuple representation.

See Rubiks Cube/Permutations for more on representing permutations and uncovering their properties.

Counting Permutations for 3x3

Counting the number of possible configurations of a Rubik's Cube requires thinking about how the puzzle works and the unique pieces that compose any Rubik's Cube puzzle. We can enumerate the state of each of the three types of pieces: corner pieces, edge pieces, and center pieces.

See Rubiks Cube/Numbers for the procedure and rundown of how we count the number of permutations of a 3x3 or 4x4 Rubik's Cube.

Patterns and Sequences of 3x3

The symmetry properties of the Rubik's Cube lead to some interesting properties.

Notes on group theory and Rubiks Cube patterns: Rubiks Cube/Patterns

4x4 Rubiks Revenge

Main article: Rubiks Revenge

Pages on the 4x4 cube:

Rubiks Revenge/Layer Method (algorithm for solving)
Rubiks Revenge/Tuple (mathematical representation of the 4x4 cube)
Rubiks Revenge/Permutations (permutation algebra applied to 4x4 cube)
Rubiks Revenge/Numbers (counting states)
Rubiks Revenge/Patterns (group theory and algorithm patterns)

Solving the 4x4

Main article: Rubiks Revenge/Layer Method

For the most part, solving the 4x4 is a lot like solving the 3x3, but complicated by two initial steps that need to happen first - arranging the center 2x2 cubies of each face, then orienting the two edge pairs of cubies of each of the twelve sides of the cube to match up.

Then you can think of these double edge pairs as sticking together to form a single edge cubie, and the four center face cubies as a single fixed face cubie, and the whole thing becomes a 3x3 cube, but with 2 rows/columns instead of a single middle row/column.

However, the parity case complicates things. Whereas the 3x3 cube has a few ending scenarios with regard to the parity of the top layer and how the squares are oriented (memorizing these becomes crucial to fast solves), the 4x4 cube has some situations that are impossible on a 3x3 cube (such as two corner cubies being swapped or a cube that's entirely solved except for two inside-out cubies on one of the top edges).

See Rubiks Revenge page, in particular the Rubiks Revenge#Algorithms Cheat Sheet section, for algorithms that apply only to the 4x4 case.

Parity on the 4x4

The 4x4 has an even number of cubies per side, which means that it has more flexibility than the 3x3 or 5x5 (or any other cube with an odd number of cubies per side) because there is no fixed center face.

This difference means the 4x4 can reach any possible cube permutations (arrangements of the cubies), unlike the 3x3 or 5x5 cubes, which can only reach half of the possible permutations (the cubes would need to be mechanically dismantled to reach the other half).

This shows up on the very first step of solving the 4x4, which requires lining up the center faces.

(Image: faces ok and oriented correctly)

(Image: faces ok but oriented incorrectly)

Mathematical Representation of the 4x4

To represent the Rubik's Revenge as a mathematical object, we can use a 96-tuple (that is, a permutation of the integers from 1 to 96, non-repeating). We have 8 corner pieces, 24 double-edge pieces (12 left-hand double edge pieces, 12 right-hand double edge pieces), and 24 center pieces, for a total of 56 pieces. However, some of these pieces have multiple faces, and we specify the state of the cube by enumerating the faces.

See Rubiks Cube/Tuple for information about how to turn a Rubik's Cube permutation, or arrangement, using a tuple of integers.

5x5 Professors Cube

Main article: Professors Cube

Solving the 5x5

The basic algorithm for solving the 5x5 is to use the layer method, by first reducing the 5x5 to a 3x3, then solving the 3x3 with the layer method.

Main: Professors Cube/Layer Method

Parity on the 5x5

If you consider the solved cube, with each face correctly oriented, a cube with even parity (the 4x4) can have two corner cubies, three cubies, or four corner cubies swapped; it can also have two neighboring edge cubies oriented inside-out. But a cube with odd parity (like the 3x3 or 5x5) can only have three or four corner cubies swapped; it is not possible to swap two corner cubies on a cube with odd parity.

The end solution for the 5x5 cube is actually simpler than the 4x4 cube, due to the odd parity. While the 4x4 cube could end up with two corner cubies swapped or two edge cubies inverted, both of which require additional algorithms to fix, those states cannot be reached on a 5x5 cube, so there are no extra algorithms to learn for the end step.

Another step where even parity creates a more complicated solution step is the very first step - when there are four center pieces that are all interchangeable, the center pieces can move anywhere - so when you are solving the 4x4 and you aren't paying close attention to the orientation of the colors of the faces, you might end up inverting red and orange, or white and yellow, which you may not uncover until you've gone several steps ahead in the algorithm - which is when you'll discover that you need to swap two face colors.

With the 5x5 cube, as with the 3x3 cube, there is a single center piece that is fixed on each face, meaning the extra cognitive burden of correctly orienting each face is already done for you.

References

Flags

@@ Line 18: / Line 18: @@
 The standard Rubiks Cube size is a 3x3.
+[[Image:RubiksCube LayerMethod 8Solved.jpg|200px]]
 Pages on the 3x3 cube:
@@ Line 31: / Line 33: @@
 There are other, more advanced techniques:
-* Friedrichs Method (CFOP) [https://ruwix.com/the-rubiks-cube/advanced-cfop-fridrich/]
+* Fridrich Method (CFOP) [https://ruwix.com/the-rubiks-cube/advanced-cfop-fridrich/] - also see [[Rubiks Cube/Fridrich Method]]
-* 2x2 Method
+* 2x2 Method (blockbuilding)
 * Blindfolded [https://ruwix.com/the-rubiks-cube/how-to-solve-the-rubiks-cube-blindfolded-tutorial/]
-* Blildfolded (another) [http://www.ws.binghamton.edu/fridrich/Richard/BlindfoldRevenge.pdf]
+* Blindfolded (another) [http://www.ws.binghamton.edu/fridrich/Richard/BlindfoldRevenge.pdf]
-==Mathematical Representation==
+==Mathematical Representation of the 3x3==
 The representation of the Rubik's Cube for use in mathematics (or computer programs) is an important topic. To turn a particular Rubik's Cube configuration into a mathematical representation agnostic to the particulars of the cube, we can label each face with an integer 1 to 54 (from 9 squares per face and 6 total faces), and pick a particular order for each face. Then we can write a sorted cube as (1 2 3 4 5 ... 52 53 54). This also enables enumerating combinatoric properties of the cube.
@@ Line 44: / Line 46: @@
 See [[Rubiks Cube/Permutations]] for more on representing permutations and uncovering their properties.
-==Counting Permutations==
+==Counting Permutations for 3x3==
 Counting the number of possible configurations of a Rubik's Cube requires thinking about how the puzzle works and the unique pieces that compose any Rubik's Cube puzzle. We can enumerate the state of each of the three types of pieces: corner pieces, edge pieces, and center pieces.
@@ Line 50: / Line 52: @@
 See [[Rubiks Cube/Numbers]] for the procedure and rundown of how we count the number of permutations of a 3x3 or 4x4 Rubik's Cube.
-==Patterns and Sequences==
+==Patterns and Sequences of 3x3==
 The symmetry properties of the Rubik's Cube lead to some interesting properties.
@@ Line 70: / Line 72: @@
 * [[Rubiks Revenge/Patterns]] (group theory and algorithm patterns)
-==Solving==
+==Solving the 4x4==
 {{Main|Rubiks Revenge/Layer Method}}
@@ Line 76: / Line 78: @@
 For the most part, solving the 4x4 is a lot like solving the 3x3, but complicated by two initial steps that need to happen first - arranging the center 2x2 cubies of each face, then orienting the two edge pairs of cubies of each of the twelve sides of the cube to match up.
-Then you can think of these edge pairs as sticking together, and the whole thing becomes a 3x3 cube, but with 2 rows/columns instead of a single middle row/column.
+Then you can think of these double edge pairs as sticking together to form a single edge cubie, and the four center face cubies as a single fixed face cubie, and the whole thing becomes a 3x3 cube, but with 2 rows/columns instead of a single middle row/column.
-However, the parity case complicates things. Whereas the 3x3 cube has a few ending scenarios with regard to the parity of the top layer and how the squares are oriented (memorizing these becomes crucial to fast solves), the 4x4 cube has some situations that are impossible on a 3x3 cube (such as two corner cubies being swapped) or a cube that's entirely solved except for two inside-out cubies on one of the top edges.
+However, the parity case complicates things. Whereas the 3x3 cube has a few ending scenarios with regard to the parity of the top layer and how the squares are oriented (memorizing these becomes crucial to fast solves), the 4x4 cube has some situations that are impossible on a 3x3 cube (such as two corner cubies being swapped or a cube that's entirely solved except for two inside-out cubies on one of the top edges).
-See [[Rubiks Revenge]] page for pictures.
+See [[Rubiks Revenge]] page, in particular the [[Rubiks Revenge#Algorithms Cheat Sheet]] section, for algorithms that apply only to the 4x4 case.
-==Mathematical Representation==
+==Parity on the 4x4==
-To represent the Rubik's Revenge as a mathematical object, we can use a 96-tuple (that is, a permutation of the integers from 1 to 96, non-repeating). We have 8 corner pieces, 24 double-edge pieces (12 left-hand double edge pieces, 12 right-hand double edge pieces), and 24 center pieces, for a total of 56 pieces. However, some of these pieces have multiple faces, and we specify the state of the cube by enumerating the faces.
+The 4x4 has an even number of cubies per side, which means that it has more flexibility than the 3x3 or 5x5 (or any other cube with an odd number of cubies per side) because there is no fixed center face.
-See [[Rubiks Cube/Tuple]] for information about how to turn a Rubik's Cube permutation, or arrangement, using a tuple of integers.
+This difference means the 4x4 can reach any possible cube permutations (arrangements of the cubies), unlike the 3x3 or 5x5 cubes, which can only reach half of the possible permutations (the cubes would need to be mechanically dismantled to reach the other half).
-=5x5 Professors Cube=
+This shows up on the very first step of solving the 4x4, which requires lining up the center faces.
-{{Main|Professors Cube}}
+(Image: faces ok and oriented correctly)
-[[Image:ProfessorsCube_5x5.jpg|200px]]
+(Image: faces ok but oriented incorrectly)
-==Solving==
+==Mathematical Representation of the 4x4==
-===Solution Algorithm===
+To represent the Rubik's Revenge as a mathematical object, we can use a 96-tuple (that is, a permutation of the integers from 1 to 96, non-repeating). We have 8 corner pieces, 24 double-edge pieces (12 left-hand double edge pieces, 12 right-hand double edge pieces), and 24 center pieces, for a total of 56 pieces. However, some of these pieces have multiple faces, and we specify the state of the cube by enumerating the faces.
-The solution template for the 5x5 cube follows the solution template for the 4x4 fairly closely, although there are some important differences. This is due to the fact that the 5x5 has an odd number of cubies per side (odd parity), meaning half of all possible permutations of cubies are not possible to reach. The 4x4 cube, on the other hand, has an even number of cubies per side (even parity), which means that all permutations of cubies are permitted.
+See [[Rubiks Cube/Tuple]] for information about how to turn a Rubik's Cube permutation, or arrangement, using a tuple of integers.
-The solution algorithm, at a high level, proceeds like this:
+=5x5 Professors Cube=
-. Form the crosses on each respective cube face. (Start with yellow cross, then do white cross on opposite face. Do red-then-orange and blue-then-green. Like the 3x3 step of forming a cross, this utilizes a few algorithms that were all worked out through trial and error.)
+[[Category:5x5 Cube]]
-. Use an algorithm to exchange cubies on the corners of the center of the face (non-cross portion) until each face of the 5x5 cube has 9 cubies of a single color.
+{{Main|Professors Cube}}
-. Use algorithms to line up triple edges so that they are all of a single color combination. Proceed as follows: line up the three cubies in a triple edge and "store" it on the top of the cube. Once all four triple edges on the top of the cube are correctly oriented, flip the cube over and repeat the process for the next four triple edges of the cube.
-. With 8 of the triple edges already solved, use algorithms to solve two of the remaining four edges (so there are 10 triple edges solved). Repeat to solve the last two edges. If the cube has all edges oriented correctly in the end, skip the next step. If the final cube has one edge with a center piece that is inside-out, continue to the next step.
-. (Only applies if Step 4 results in one triple edge with an inside-out center cubie) Apply the Holy Cube Algorithm:
+[[Image:ProfessorsCube_5x5_Hipsta.jpg|200px]]
-<pre>
+==Solving the 5x5==
-(Rr)2  B2   U2
-(Ll)   U2
-(Rr)'  U2
-(Rr)   U2
-F2     (Rr)
-F2     (Ll)'
-B2     (Rr)2
-</pre>
-. All center pieces are a solid square of nine colors, and all triple edges are aligned, so the cube is now in a state that is exactly equivalent to a 3x3 Rubik's Cube. You should now be able to tear through the remainder of the solve in the classical manner (yellow cross - yellow face - second layer - top cross - cycle corner cubies - orient corner cubies).
+The basic algorithm for solving the 5x5 is to use the layer method, by first reducing the 5x5 to a 3x3, then solving the 3x3 with the layer method.
-For more detail about each step, see the [[Professors Cube]] page.
+* Main: [[Professors Cube/Layer Method]]
-===Parity===
+==Parity on the 5x5==
 If you consider the solved cube, with each face correctly oriented, a cube with even parity (the 4x4) can have two corner cubies, three cubies, or four corner cubies swapped; it can also have two neighboring edge cubies oriented inside-out. But a cube with odd parity (like the 3x3 or 5x5) can only have three or four corner cubies swapped; it is not possible to swap two corner cubies on a cube with odd parity.

Rubiks Cube: Difference between revisions

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