Rubiks Cubes

Thursday, December 15, 2011

Algorithms

In Rubik's cubists' parlance, a memorised sequence of moves that has a desired effect on the cube is called an algorithm. This terminology is derived from the mathematical use of algorithm, meaning a list of well-defined instructions for performing a task from a given initial state, through well-defined successive states, to a desired end-state. Each method of solving the Rubik's Cube employs its own set of algorithms, together with descriptions of what the effect of the algorithm is, and when it can be used to bring the cube closer to being solved.

Most algorithms are designed to transform only a small part of the cube without scrambling other parts that have already been solved, so that they can be applied repeatedly to different parts of the cube until the whole is solved. For example, there are well-known algorithms for cycling three corners without changing the rest of the puzzle, or flipping the orientation of a pair of edges while leaving the others intact.

Some algorithms have a certain desired effect on the cube (for example, swapping two corners) but may also have the side-effect of changing other parts of the cube (such as permuting some edges). Such algorithms are often simpler than the ones without side-effects, and are employed early on in the solution when most of the puzzle has not yet been solved and the side-effects are not important. Towards the end of the solution, the more specific (and usually more complicated) algorithms are used instead, to prevent scrambling parts of the puzzle that have already been solved.

Center Faces

The original Rubik's Cube had no orientation markings on the centre faces (although some carried the words "Rubik's Cube" on the centre square of the white face), and therefore solving it does not require any attention to orienting those faces correctly. However, with marker pens, one could, for example, mark the central squares of an unscrambled Cube with four coloured marks on each edge, each corresponding to the colour of the adjacent face. Some Cubes have also been produced commercially with markings on all of the squares, such as the Lo Shu magic square or playing card suits. Thus one can nominally solve a Cube yet have the markings on the centres rotated; it then becomes an additional test to solve the centres as well.

Marking the Rubik's Cube increases its difficulty because this expands its set of distinguishable possible configurations. When the Cube is unscrambled apart from the orientations of the central squares, there will always be an even number of squares requiring a quarter turn. Thus there are 4⁶/2 = 2,048 possible orientations of the centre squares in the otherwise unscrambled position, increasing the total number of possible Cube permutations from 43,252,003,274,489,856,000 (4.3×10¹⁹) to 88,580,102,706,155,225,088,000 (8.9×10²²).^[22]

When turning a cube over is considered to be a change in permutation then we must also count arrangements of the centre faces. Nominally there are 6! ways to arrange the six centre faces of the cube, but only 24 of these are achievable without disassembly of the cube. When the orientations of centres are also counted, as above, this increases the total number of possible Cube permutations from 88,580,102,706,155,225,088,000 (8.9×10²²) to 2,125,922,464,947,725,402,112,000 (2.1×10²⁴).

Algorithms

Rubik's Cube Patterns

Pons asinorum

F² B² R² L² U² D²

(12q*, 6f*)

Checkerboards of order 3

F B² R' D² B R U D' R L' D' F' R² D F² B'

(20q*, 16f*)

Checkerboards of order 6

R' D' F' D L F U² B' L U D' R' D' L F L² U F'	(20q*, 18f)
R² L² U B L² D' F B² R L' F' B R D F² L' U'	(17f*, 22q)

Stripes

F U F R L² B D' R D² L D' B R² L F U F

(20q*, 17f*)

Cube in a cube

F L F U' R U F² L² U' L' B D' B' L² U

(18q*, 15f*)

Cube in a cube in a cube

U' L' U' F' R² B' R F U B² U B' L U' F U R F'	(20q*, 18f)
F' U B' R' U F² U² F' U' F U² D B' D' R² B² U'	(17f*, 22q)

Christman's cross

U F B' L² U² L² F' B U² L² U

(16q*, 11f*)

Plummer's cross

R² L' D F² R' D' R' L U' D R D B² R' U D²

(20q*, 16f*)

Anaconda

L U B' U' R L' B R' F B' D R D' F'

(14q*, 14f*)

Python

F² R' B' U R' L F' L F' B D' R B L²

(16q*, 14f*)

Black Mamba

R D L F' R L' D R' U D' B U' R' D'

(14q*, 14f*)

Green Mamba

R D R F R' F' B D R' U' B' U D²

(14q*, 13f*)

Female Rattlesnake

U² D' L² D B U B' R' L² U² F U' F R

(18q*, 14f*)

Male Rattlesnake

R' F' U F' U² R L² B U' B' D' L² U² D

(18q*, 14f*)

Female Boa

R U' R² U² F D² R² U' D' R D' F'

(16q*, 12f*)

Male Boa

F D R' U D R² D² F' U² R² U R'

(16q*, 12f*)

Four spot

F² B² U D' R² L² U D'

(12q*, 8f*)

Six spot

U D' R L' F B' U D'

(8q*, 8f*)

Orthogonal bars

F R' U L F' L' F U' R U L' U' L F'

(14q*, 14f*)

Six T's

F² R² U² F' B D² L² F B

(14q*, 9f*)

Six-two-one

U B² D² L B' L' U' L' B D² B²

(15q*, 11f*)

Exchanged peaks

F U² L F L' B L U B' R' L' U R' D' F' B R²	(19q*, 17f)
F² R² D R² U D F² D' R' D' F L² F' D R U'	(16f*, 21q)

Two twisted peaks

F B' U F U F U L B L² B' U F' L U L' B	(18q*, 17f)
F D² B R B' L' F D' L² F² R F' R' F² L' F'	(16f*, 20q)

Four twisted peaks

U' D B R' F R B' L' F' B L F R' B' R F' U' D

(18q*, 18f*)

Exchanged chicken feet

F L' D' B' L F U F' D' F L² B' R' U L² D' F

(19q*, 17f*)

Twisted chicken feet

F L' D F' U' B U F U' F R' F² L U' R' D²

(18q*, 16f*)

Exchanged rings

B' U' B' L' D B U D² B U L D' L' U' L² D	(18q*, 16f)
F U D' L' B² L U' D F U R² L² U' L² F²	(15f*, 20q)

Twisted rings

F D F' D² L' B' U L D R U L' F' U L U²

(18q*, 16f*)

Edge hexagon of order 2

U B² U' F' U' D L' D² L U D' F D' L² B² D'

(20q*, 16f*)

Edge hexagon of order 3

D L' U R' B' R B U² D B D' B' L U D'	(16q*, 15f)
F L B U L F² B² R' F² B² U' B' L' F'	(14f*, 18q)

Tom Parks' pattern

L U F² R L' U² B' U D B² L F B' R' L F' R

(20q*, 17f*)

Ron's cube in a cube

F D' F' R D F' R' D R D L' F L D R' F D'	(17q*, 17f)
L² D² L' D² B² L² B² L' D² L² B² L' B²	(13f*, 23q)

Twisted duck feet

F R' B R U F' L' F' U² L' U' D² B D' F B' U²

(20q*, 17f*)

Exchanged duck feet

U F R² F' D' R U B² U² F' R² F D B² R B'

(21q*, 16f*)

Math Behing the Rubik's Cube

The original (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 3⁷ (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 2¹¹ (2,048) possibilities.^[20]

${8! \times 3^7 \times 12!/2 \times 2^{11}} = 43,252,003,274,489,856,000$

which is approximately forty-three quintillion.^[21]

The puzzle is often advertised as having only "billions" of positions, as the larger numbers are unfamiliar to many. To put this into perspective, if one had as many 57-millimeter Rubik's Cubes as there are permutations, they could cover the Earth's surface 275 times.

The preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If one considers permutations reached through disassembly of the cube, the number becomes twelve times as large:

${8! \times 3^8 \times 12! \times 2^{12}} = 519,024,039,293,878,272,000.$

which is approximately five hundred and nineteen quintillion^[21] possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge cube. Thus there are twelve possible sets of reachable configurations, sometimes called "universes" or "orbits", into which the Cube can be placed by dismantling and reassembling it.

Centre

Other Cubes

There are different variations of Rubik's Cubes with up to seven layers: the 2×2×2 (Pocket/Mini Cube), the standard 3×3×3 cube, the 4×4×4 (Rubik's Revenge/Master Cube), and the 5×5×5 (Professor's Cube), the 6×6×6 (V-Cube 6), and 7×7×7 (V-Cube 7).

CESailor Tech's E-cube is an electronic variant of the 3×3×3 cube, made with RGB LEDs and switches.^[46] There are two switches on each row and column. Pressing the switches indicates the direction of rotation, which causes the LED display to change colours, simulating real rotations. The product was demonstrated at the Taiwan government show of college designs on October 30, 2008.

Another electronic variation of the 3×3×3 Cube is the Rubik's TouchCube. Sliding a finger across its faces causes its patterns of coloured lights to rotate the same way they would on a mechanical cube. The TouchCube was introduced at theAmerican International Toy Fair in New York on February 15, 2009.^[47]^[48]

The Cube has inspired an entire category of similar puzzles, commonly referred to as twisty puzzles, which includes the cubes of different sizes mentioned above as well as various other geometric shapes. Some such shapes include thetetrahedron (Pyraminx), the octahedron (Skewb Diamond), the dodecahedron (Megaminx), the icosahedron (Dogic). There are also puzzles that change shape such as Rubik's Snake and the Square One.

Rubik's Cube World Record Times

Type	Result	Person	Citizen of	Competition	Result Details



Rubik's Cube
Single	5.66	Feliks Zemdegs	Australia	Melbourne Winter Open 2011
Average	7.64	Feliks Zemdegs	Australia	Melbourne Winter Open 2011	7.03 8.11 8.36 5.66 7.78

4x4 Cube
Single	30.28	Feliks Zemdegs	Australia	Australian Nationals 2011
Average	35.22	Feliks Zemdegs	Australia	World Championship 2011	33.33 38.71 33.28 33.63 39.33

5x5 Cube
Single	56.22	Feliks Zemdegs	Australia	World Championship 2011
Average	59.94	Feliks Zemdegs	Australia	World Championship 2011	59.59 58.41 1:01.81 1:05.40 56.22

2x2 Cube
Single	0.69	Christian Kaserer	Italy	Trentin Open 2011
Average	2.12	Feliks Zemdegs	Australia	Melbourne Cube Day 2010	2.38 1.77 1.75 2.21 2.46

Rubik's Cube: Blindfolded
Single	30.58	Yuhui Xu (许宇辉)	China	Suzhou Open 2011

Rubik's Cube: One-handed
Single	10.68	Piotr Tomczyk	Poland	Swierklany Open 2011
Average	13.57	Michał Pleskowicz	Poland	World Championship 2011	12.34 15.83 12.97 15.11 12.63

Rubik's Cube: Fewest moves
Single	22	Jimmy Coll	Belgium	Barcelona Open 2009
		István Kocza	Hungary	Czech Open 2010

Rubik's Cube: With feet
Single	31.56	Anssi Vanhala	Finland	Helsinki Open 2011
Average	39.98	Anssi Vanhala	Finland	Kotka Open 2011	37.81 39.30 42.84

Megaminx
Single	42.28	Simon Westlund	Sweden	Danish Open 2011
Average	49.90	Simon Westlund	Sweden	Danish Open 2011	49.46 49.30 48.61 52.44 50.94

Pyraminx
Single	1.93	Yohei Oka (岡要平)	Japan	Japan Open 2011
Average	3.39	Yohei Oka (岡要平)	Japan	Kyotanabe Open 2011	3.65 2.86 2.75 3.65 4.59

Square-1
Single	8.65	Bingliang Li (李炳良)	China	Guangdong Open 2010
Average	11.33	Bingliang Li (李炳良)	China	Changsha Open 2011	11.83 14.13 11.44 10.72 10.11

Rubik's Clock
Single	5.83	Javier Tirado Ortiz	Spain	World Championship 2011
Average	7.33	Sam Zhixiao Wang (王志骁)	China	Guildford Summer Open 2011	11.30 6.78 8.15 5.88 7.05

6x6 Cube
Single	1:54.81	Kevin Hays	USA	Vancouver Winter 2011
Average	2:02.13	Kevin Hays	USA	Vancouver Winter 2011	2:00.93 1:54.81 2:10.66

7x7 Cube
Single	3:13.19	Michał Halczuk	Poland	Polish Nationals 2011
Average	3:25.10	Michał Halczuk	Poland	Swierklany Open 2011	3:17.97 3:31.68 3:25.66

Rubik's Magic
Single	0.69	Yuxuan Wang (王宇轩)	China	Beijing Spring 2011
Average	0.76	Yuxuan Wang (王宇轩)	China	Beijing Summer Open 2011	0.72 0.77 0.77 0.75 DNF

Master Magic
Single	1.68	Ernie Pulchny	USA	Park Ridge Open 2011
Average	1.75	Ernie Pulchny	USA	US Nationals 2011	1.78 1.71 1.77 1.71 3.08

4x4 Cube: Blindfolded
Single	3:26.11	Daniel Sheppard	United Kingdom	Guildford Summer Open 2011

5x5 Cube: Blindfolded
Single	9:48.58	Ville Seppänen	Finland	Kirkkonummi Open 2011

Rubik's Cube: Multiple Blindfolded
Single	23/25 57:48	Zane Carney	Australia	Melbourne Cube Day 2011

Rubik's Cube Picture

Thursday, December 15, 2011

Algorithms

Center Faces

Algorithms

Rubik's Cube Patterns

Math Behing the Rubik's Cube

Centre

Other Cubes

Rubik's Cube World Record Times