Rubiks Cube/Tuple - Revision history

Admin: /* 4x4 Rubiks Cube Representation */

2019-04-22T22:01:25Z

4x4 Rubiks Cube Representation

← Older revision		Revision as of 22:01, 22 April 2019
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	=4x4 Rubiks Cube Representation=		=4x4 Rubiks Cube Representation=

	==Numbering System==		==Numbering System for 4x4==

	Start with a numbering system for the cube. The nxnxn rubiks cube solver library I'm using (https://github.com/dwalton76/rubiks-cube-NxNxN-solver) implements the following numbering system:		Start with a numbering system for the cube. The nxnxn rubiks cube solver library I'm using (https://github.com/dwalton76/rubiks-cube-NxNxN-solver) implements the following numbering system:
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	</pre>		</pre>

	==Moves as Permutations==		==4x4 Moves Mapped to Permutations==

	Each move of a face - U, D, R, L, F, B, and other two-layer or second-layer moves, as well as sequences of moves - can now be thought of as a permutation of these 96 integers.		Each move of a face - U, D, R, L, F, B, and other two-layer or second-layer moves, as well as sequences of moves - can now be thought of as a permutation of these 96 integers.

Admin: /* 3x3 Rubiks Cube Representation */

2019-04-22T22:00:45Z

3x3 Rubiks Cube Representation

← Older revision		Revision as of 22:00, 22 April 2019
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	=3x3 Rubiks Cube Representation=		=3x3 Rubiks Cube Representation=

			==Numbering System for 3x3 Cube==

			==Mapping Moves to Permutations for 3x3==

			==3x3 Permutations and Properties==

			==3x3 Cube Code==

	=4x4 Rubiks Cube Representation=		=4x4 Rubiks Cube Representation=

Admin: /* 4x4 Rubiks Cube Representation */

2019-04-22T21:59:59Z

4x4 Rubiks Cube Representation

← Older revision		Revision as of 21:59, 22 April 2019
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	To skip straight to the notes on permutations, see [[Rubiks Cube/Permutations]]		To skip straight to the notes on permutations, see [[Rubiks Cube/Permutations]]

			=3x3 Rubiks Cube Representation=

	=4x4 Rubiks Cube Representation=		=4x4 Rubiks Cube Representation=

Admin: /* Tuple Representation */

2019-04-22T21:50:59Z

Tuple Representation

← Older revision		Revision as of 21:50, 22 April 2019
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	* 8 corner cubies		* 8 corner cubies
	* 12 edge cubies		* 12 edge cubies
	* 6 center cubies		* 6 center cubies (fixed)

	Rubik's Revenge (4x4 cube): 56 total cubies (mechanical pieces)		Rubik's Revenge (4x4 cube): 56 total cubies (mechanical pieces)

	* 8 corner cubies		* 8 corner cubies
	* 24 double-edge cubies (12 left~~-hand~~, 12 right~~-hand~~)		* 24 double-edge (dedge) wing cubies (12 left wing, 12 right wing)
	* 24 center cubies		* 24 center cubies

	~~However, it is important to note that we are ''not'' trying to find the ''minimal'' representation of the~~ cube~~, we are simply trying to find a ''unique'' representation of the cube. Listing the state of every single face requires more information - there are more faces than pieces, because corners have~~ 3 ~~faces and double edge~~ pieces ~~have 2 faces - but it is much simpler to accomplish:~~		Professors Cube (5x5 cube): 252 + 163 = 98 total cubies (mechanical pieces)

	* The 3x3 Rubik's Cube has 9 squares on each face, and 6 faces, for a total of 54 squares.		* 8 corner cubies
	* The 4x4 Rubik's Revenge has 16 squares on each face, and 6 faces, for 96 total squares.		* 24 triple-edge (tredge) wing cubies (12 left wing, 12 right wing)
			* 12 triple-edge (tredge) center cubies
			* 6 center (of face) cubies (fixed)
			* 48 face cubies (mobile)

			However, it is important to note that we are ''not'' trying to find the ''minimal'' representation of the cube, we are simply trying to find a ''unique'' representation of the cube. Listing the state of every single face requires more information - there are more faces than pieces, because corners have 3 faces and double edge pieces have 2 faces - but it is much simpler to accomplish, by thinking about "unfolding"the cube into a map of colored squares.

			* The 3x3 Rubik's Cube has 9 squares on each face, and 6 faces, for a total of 54 colored squares.
			* The 4x4 Rubik's Revenge has 16 squares on each face, and 6 faces, for 96 total colored squares.
			* The 5x5 Professor's Cube has 25 squares on each face, and 6 faces, for 150 total colored squares.

	Now, if we were looking for a ''minimal'' representation, we would utilize the fact that some of these squares are innately linked (for example, the three faces representing a corner piece are always positioned in the same way relative to one another, even though they may move relative to the rest of the pieces on the cube).		Now, if we were looking for a ''minimal'' representation, we would utilize the fact that some of these squares are innately linked (for example, the three faces representing a corner piece are always positioned in the same way relative to one another, even though they may move relative to the rest of the pieces on the cube).

	However, we simply want a ''unique'' representation, so we can represent the state of any 3x3 Rubik's Cube using a 54-~~tuple~~, or the state of any 4x4 Rubik's Revenge using a 96-~~tuple~~.		However, we simply want a ''unique'' representation, so we can represent the state of any 3x3 Rubik's Cube using a 54-uple, or the state of any 4x4 Rubik's Revenge cube using a 96-uple, or the state of any 5x5 Professor's cube using a 150-uple.

	==Why A Tuple Representation==		==Why A Tuple Representation==

Admin: /* Tuple Representation */

2019-04-21T18:32:09Z

Tuple Representation

← Older revision		Revision as of 18:32, 21 April 2019
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	Now, if we think about how a 3x3 Rubik's Cube or 4x4 Rubik's Revenge is mechanically constructed, we see that the cube consists of:		Now, if we think about how a 3x3 Rubik's Cube or 4x4 Rubik's Revenge is mechanically constructed, we see that the cube consists of:

	Rubik's Cube (3x3 cube): 26 total (mechanical) ~~pieces~~		Rubik's Cube (3x3 cube): 26 total cubies (mechanical pieces)

	* 8 corner ~~pieces~~		* 8 corner cubies
	* 12 edge ~~pieces~~		* 12 edge cubies
	* 6 center ~~pieces~~		* 6 center cubies

	Rubik's Revenge (4x4 cube): 56 total (mechanical) ~~pieces~~		Rubik's Revenge (4x4 cube): 56 total cubies (mechanical pieces)

	* 8 corner ~~pieces~~		* 8 corner cubies
	* 24 double-edge ~~pieces~~ (12 left-hand, 12 right-hand)		* 24 double-edge cubies (12 left-hand, 12 right-hand)
	* 24 center ~~pieces~~		* 24 center cubies

	However, it is important to note that we are ''not'' trying to find the ''minimal'' representation of the cube, we are simply trying to find a ''unique'' representation of the cube. Listing the state of every single face requires more information - there are more faces than pieces, because corners have 3 faces and double edge pieces have 2 faces - but it is much simpler to accomplish:		However, it is important to note that we are ''not'' trying to find the ''minimal'' representation of the cube, we are simply trying to find a ''unique'' representation of the cube. Listing the state of every single face requires more information - there are more faces than pieces, because corners have 3 faces and double edge pieces have 2 faces - but it is much simpler to accomplish:

	* The 3x3 Rubik's Cube has 9 squares on each face, and 6 faces, for a total of 36 squares.		* The 3x3 Rubik's Cube has 9 squares on each face, and 6 faces, for a total of 54 squares.
	* The 4x4 Rubik's Revenge has 16 squares on each face, and 6 faces, for 96 total squares.		* The 4x4 Rubik's Revenge has 16 squares on each face, and 6 faces, for 96 total squares.

	Now, if we were looking for a ''minimal'' representation, we would utilize the fact that some of these squares are innately linked (for example, the three faces representing a corner piece are always positioned in the same way relative to one another, even though they may move relative to the rest of the pieces on the cube).		Now, if we were looking for a ''minimal'' representation, we would utilize the fact that some of these squares are innately linked (for example, the three faces representing a corner piece are always positioned in the same way relative to one another, even though they may move relative to the rest of the pieces on the cube).

	However, we simply want a ''unique'' representation, so we can represent the state of any 3x3 Rubik's Cube using a 36-tuple, or the state of any 4x4 Rubik's Revenge using a 96-tuple.		However, we simply want a ''unique'' representation, so we can represent the state of any 3x3 Rubik's Cube using a 54-tuple, or the state of any 4x4 Rubik's Revenge using a 96-tuple.

	==Why A Tuple Representation==		==Why A Tuple Representation==

Admin at 02:46, 11 January 2018

2018-01-11T02:46:12Z

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	[[Category:Rubiks Revenge]]		[[Category:Rubiks Revenge]]
	[[Category:Rubiks Cube]]		[[Category:Rubiks Cube]]
	~~[[Category:Rubiks]]~~

Admin: /* Flags */

2018-01-11T02:38:38Z

Flags

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	=Flags=		=Flags=

	{{~~RubiksCubeFlag~~}}		{{RubiksFlag}}

	[[Category:Rubiks Revenge]]		[[Category:Rubiks Revenge]]
	[[Category:Rubiks Cube]]		[[Category:Rubiks Cube]]
	[[Category:Rubiks]]		[[Category:Rubiks]]

Admin: /* Flags */

2018-01-11T02:38:31Z

Flags

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	=Flags=		=Flags=

	{{~~RubiksFlag~~}}		{{RubiksCubeFlag}}

	[[Category:Rubiks Revenge]]		[[Category:Rubiks Revenge]]
	[[Category:Rubiks Cube]]		[[Category:Rubiks Cube]]
	[[Category:Rubiks]]		[[Category:Rubiks]]

Admin: /* Moves as Permutations */

2018-01-11T00:22:15Z

Moves as Permutations

← Older revision		Revision as of 00:22, 11 January 2018
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	Once we can write a sequence of moves as a permutation of 96 integers, we can start to dig deeper into the effect that it has on the cube state.		Once we can write a sequence of moves as a permutation of 96 integers, we can start to dig deeper into the effect that it has on the cube state.

			==Permutations and Their Properties==

			Now that we have a tuple representation of the cube, we can start to use it to characterize sequences of moves, the permutations they lead to, and their properties.

			See [[Rubiks Cube/Permutations]]

	=Flags=		=Flags=

Admin: /* Permutation Algebra */

2018-01-11T00:21:26Z

Permutation Algebra

← Older revision		Revision as of 00:21, 11 January 2018
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	Once we can write a sequence of moves as a permutation of 96 integers, we can start to dig deeper into the effect that it has on the cube state.		Once we can write a sequence of moves as a permutation of 96 integers, we can start to dig deeper into the effect that it has on the cube state.

	~~==Permutation Algebra==~~

	=Flags=		=Flags=