CFOP method: Difference between revisions

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{{short description|Method in speedcubing}}

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[[File:A speedcube midway through being solved.jpg|thumb|Cube mid-solve on the OLL step.]]

The '''CFOP method''' (Cross – F2L (first 2 layers) – OLL (orientate last layer) – PLL (permutate last layer)), also known as the '''Fridrich method''', is one of the most commonly used methods in [[speedcubing|speedsolving]] a 3×3×3 [[Rubik's Cube]]. It is one of the fastest methods with the other most notable ones being [[Roux method|Roux]] and [[ZZ method|ZZ]]. This method was first developed in the early 1980s, combining innovations by a number of speed cubersspeedcubers. [[Jessica Fridrich]], a Czech speed cuberspeedcuber and the namesake of the method, is generally credited for popularizing it by publishing it online in 1997.<ref>{{cite web

|author=Shotaro "Macky" Makisumi

|title=Speedcubing

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}}</ref>

The method works by first solving a cross typically on the bottom, continuing to solve the first two layers together (F2L), orienting the last layer (OLL), and finally permuting the last layer (PLL). There are 119 algorithms in total to learn the full method, with 41 for F2L (these being mostly intuitive), 57 for full OLL, and 21 for full PLL,. andOn top of that, there are other algorithm sets like ZBLL and COLL (corners of the last layer) that can be learned in addition to CFOP to improve solving efficiency even further. F2L can be improved using special algorithms to reduce the need to rotate or change grip on the cube; this is known as ''advanced F2L.'' This method of F2L has far more algorithms than the basic 41, and the fastest speedsolvers can memorize hundreds of algorithms for this step, including learning multiple algorithms for the same case.

However, the F2L step can also be done with ''intuitive F2L'', where the solver intuitively solves the step through basic rules, requiring no memorisation of notated algorithms, at the expense of efficiency. By doing F2L intuitively, and by splitting OLL and PLL into two sections each (leaving 10 algorithms for OLL and 6 for PLL), the method can be done with as few as 16 algorithms.

==History==

Basic [[Layer by Layer|layer-by-layer]] (LBL) methods were among the first to arise during the early 1980s craze, such as James Nourse's ''[[The Simple Solution to Rubik's Cube]]'', which proposed the use of a cross, and worked its way down. [[David Singmaster]] published a faster layer-based solution in 1980.<ref>{{cite web|title=Beginner's Rubik's Cube Solution|url=http://www.ryanheise.com/cube/beginner.html|access-date=15 June 2012|url-status=dead|archive-url=https://web.archive.org/web/20150926083418/http://www.ryanheise.com/cube/beginner.html|archive-date=26 September 2015}}</ref>

The major innovation of CFOP over the simpler LBL methods is its use of F2L, which solves the first two layers simultaneously by solving top-corners and vertical edges together after the Cross is established. Guus Razoux Schultz used this method as part of his CFCE method during the [[1982 World Rubik's Cube Championship]], but he did not invent this F2L method. [[Jessica Fridrich]], also competing at this championship, was then using a LBL method. The first official publication of CFOP was done by Anneke Treep and Kurt Dockhorn in 1981 in the Netherlands, based on the F2L-pairing idea of the Dutch professor René Schoof.

Another difference between the CFOP method over Nourse's simple solution is that it orients the Last Layer first, and then positions the pieces. In the simple solution, the pieces are positioned first, and then oriented.

The last layer steps OLL (orient last layer) and PLL (position last layer) involve first orienting the last layer pieces, then permuting them into their correct positions. First publishingThis was first published in Dutch by Hans Dockhorn and Anneke Treep in 1981. Jessica Fridrich developed OLL-PLL parallel in the Czech Republic.

Fridrich switched to F2L later in 1982 after she got the idea from Guus Razoux Schultz. Her main contribution to the method was developing the OLL and PLL algorithms, which together allowed any last layer position to be solved with two algorithms and was significantly faster than the previous last layer systems.<ref name=fridhist>{{cite web|last=Fridrich|first=Jessica|title=20 years of speedcubing|url=http://ws2.binghamton.edu/fridrich/history.html|access-date=15 June 2012}}</ref>

CFOP, with small tweaks, is by far the most popular method that top cubers use. Users include [[Mats Valk]], [[Feliks Zemdegs]], {{ill|[[Tymon Kolasiński|fi}}]], [[Yiheng Wang]], and [[Max Park]].

==Method==

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Speedcubers are generally given 15 seconds to inspect the cube. They are allowed to rotate the cube, but not to make any moves.

When using the CFOP method, cubers generally use this time to look at how to solve the cross. More advanced cubers can also look ahead into their first pair of F2L ("Cross + 1"), and can even set up theirthat first pair to be solved faster by altering their cross solution.

===Cross===

This first stage of solving involves solving the four edge pieces around one center, matching the colors of that center and each of the adjacent centers, forming the eponymous cross shape on the first layer. Most beginner methods start with the daisy, which is putting the white edge pieces required to solve the cross around the yellow center, then matching them with the same colored center, and moving them down to match them with the white center. However, while the beginner method typically recommends doing the daisy, most CFOP tutorials recommend solving the cross on the bottom side to avoid cube rotations and to get an overall better view of the important pieces needed for the next step (F2L). If the solver is particularly advanced, they could skip the first pair and solve an "X-cross" (the cross and a first pair solved at the same time).<ref>{{Cite web |title=XCross - Speedsolving.com Wiki |url=https://www.speedsolving.com/wiki/index.php?title=Extended_cross |access-date=2024-06-17 |website=www.speedsolving.com}}</ref> This is usually done using a technique called Keyhole, which solves one piece of the first pair while ignoring the other one.<ref>{{Cite web |title=Keyhole F2L - Speedsolving.com Wiki |url=https://www.speedsolving.com/wiki/index.php?title=Keyhole |access-date=2024-06-17 |website=www.speedsolving.com}}</ref>

This first stage of solving involves solving the four edge pieces around one center piece, matching the colors of that center and each of the centers of the adjacent centerssides, forming the eponymous cross shape on the first layer. Most beginner methods start withsolve the daisy,cross whichby isfirst putting the white edge pieces requiredaround tothe solveyellow thecenter cross aroundon the yellow centertop, then matching them with the same colored center, and finally moving them down to match them with the white center. However, while the beginner method typically recommends doing the daisy, mostMost CFOP tutorials instead recommend solving the cross on the bottom side to avoid cube rotations and to get an overall better view of the important pieces needed for the next step (F2L). If the solver is particularly advanced, they couldcan skip separately solving the first F2L pair andafter solvethe cross by solving an "''X-cross"'' (solving the cross and athe first pairF2L solvedpair at the same time).<ref>{{Cite web |title=XCross - Speedsolving.com Wiki |url=https://www.speedsolving.com/wiki/index.php?title=Extended_cross |access-date=2024-06-17 |website=www.speedsolving.com}}</ref> This is usually done using a technique called ''Keyhole'', which solves one piece of the first F2L pair while ignoring the other onepiece of that pair.<ref>{{Cite web |title=Keyhole F2L - Speedsolving.com Wiki |url=https://www.speedsolving.com/wiki/index.php?title=Keyhole |access-date=2024-06-17 |website=www.speedsolving.com}}</ref>

This step is usually performed intuitively, although some techniques, such as replacement, and edge orientation. The white cross is most commonly used for demonstration and by beginner and intermediate speedsolvers, though more advanced speedcubers can use any of the six colors to form the cross (choosing the one that requires the fewest/easiest moves), a practice known as "color neutrality".<ref>{{cite web | url=https://www.cubeskills.com/blog/colour-neutrality | title=Colour Neutrality }}</ref> Cross can always be solved in 8 moves or less.<ref>{{Cite web |title=CubeZone - Cross study |url=http://www.cubezone.be/crossstudy.html |access-date=2023-08-05 |website=www.cubezone.be}}</ref>

This stepcross is usually performedsolved intuitively, although some techniques, such as replacement, and edge orientation, are used. The white cross is most commonly used for demonstration and by beginner and intermediate speedsolvers, though more advanced speedcubers can use any of the six colors to form the cross (choosing the one that requires the fewest/easiest moves), a practice known as "''color neutrality"''.<ref>{{cite web | url=https://www.cubeskills.com/blog/colour-neutrality | title=Colour Neutrality }}</ref> CrossThe cross can always be solved in 8 moves or lessfewer.<ref>{{Cite web |title=CubeZone - Cross study |url=http://www.cubezone.be/crossstudy.html |access-date=2023-08-05 |website=www.cubezone.be}}</ref>

[[File:CfopF2L.svg|alt=|thumb|First Two Layers (F2L) solved|210x210px]]

===First Two Layers (F2L)===

While the beginner methods focusescontinues on not onlyby solving the four white corners butof the first layer and then matching the vertical edges to the corners to solve the second layer, the CFOP method solves each corner along with its vertical edge at the same time. There are 4142 unique cases for the permutations of a corner and its matching edge on the cube (one of which corresponds to the solved pair), and the most efficient algorithm to solve eachany other case without "breaking" any already-solved pair is known and can be memorized. There are 3 main casescategories of the 41 uniquethese cases: white on top, same color on top, and different color on top. All these algorithms are based on a simple sequence which brings the pieces to the top layer, aligns them with the color faces showing, and then inserting them into the pair's "slot"correct position between the matching centers (called a ''slot''). This sequence can be intuitivelydone followedintuitively, andbut thereknowing arethe different special cases, thatand canmaking improveuse on the general-case solution for a pair if other conditions are met (such asof another slot being still unsolved or "open"), can improve on the general-case solution.

[[File:CfopOLL.svg|alt=|thumb|Orientation of the Last Layer (OLL) complete|210x210px]]

===Orient Last Layer (OLL)===

This stage involves manipulating the top layer (yellow, if the cross is solved on white) so that all the pieces have the correct color on top, while largely ignoring the sides of these pieces. This stage involves a total of 57 algorithms, each solving a unique permutation of the top layer in a single sequence. A simpler version, called "two-look OLL", orients the edges first to produce a cross, then uses a second algorithm to orient the corners. This reduces the 57 algorithms down to 10: 3 for edge orientation and 7 for orientating the corners. The three algorithms - Dot, L and Line - are used for edge orientation, and seven - Sune, Antisune, Pi, H, Bowtie, Headlights and T - for corner orientation. Edge orientation in two-look is commonly taught as two algorithms, one of which is a simple variation of the other; the Dot case is solved by performing both algorithms consecutively. Additionally, the required algorithms for corner orientation can be reduced to just two, the Sune and Antisune, as all other permutations can be solved either by performing two Sunes or a Sune followed by an Antisune. Additional algorithms, more efficient than the Sune-Antisune sequences, can be learned at the solver's own pace.

==== Full OLL ====

This stage involves manipulating the top layer (yellow, if the cross is solved on white) so that all the pieces have the correct color on top, while largely ignoring the sides of these pieces. Doing this in one step is called ''Full OLL''. There are 58 possible combinations of piece orientations, so once again ignoring the solved case, this stage involves learning a total of 57 algorithms. Learning this stage in full is typically the last step chosen in learning CFOP, due to the large number of algorithms.

==== Two-look OLL ====

A simpler version, called ''two-look OLL'', orients the top layer in two stages: Edge Orientation (EO) orients the edges first to produce a cross, then uses a second algorithm for Corner Orientation (CO). This reduces the 57 algorithms down to 3 for EO and 7 for CO, totalling 10. EO commonly begins by teaching only algorithms for two of the three possible cases (''Line'' and ''L)'' because the last case (''Dot'') can be split up into performing the other two algorithms consecutively.

==== Minimizing algorithms learnt ====

This step can be made even more beginner friendly by dividing it into multiple smaller steps, which require learning less algorithms. This would be at the cost of more “looks” at the cube to determine which algorithm is needed to solve the current sub-step, slowing down the solver, but can be used as a stepping stone into two-look OLL. For example, here is one way OLL can be learnt in just two algorithms, by fully breaking down EO and CO.

EO: Note the following: Performing the ''Line'' algorithm (used to create a cross from the center piece end two edge pieces oriented in a line) on a ''Dot'' case (none of the edge pieces are oriented correctly) will yield an ''L'' case (the center piece and two neighboring edge pieces are oriented correctly). Doing the same on an ''L'' case (in a correct orientation) will yield a ''Line'' case, which can then be solved. With this knowledge, all EO cases can be solved in repetitions of only one algorithm.

CO: Note that executing the Sune algorithm on an Antisune case (again in the correct rotation) will yield a Sune case. Furthermore, any non (Sune/Antisune) case can be transformed into a Sune/Antisune case by executing a Sune algorithm at a correct orientation. So this step also only needs only one algorithm.

This method above, while only requiring knowledge of two algorithms, would constitute a “6-look OLL”, since at worst the solver needs to identify a case 6 different times, which is highly impractical. However, these minimal-algorithm methods are a very beginner-friendly way to introduce beginners into more advanced methods, and can easily be built upon by learning more algorithms at the solver's own pace.

===Permute Last Layer (PLL)===

The final stage involves moving the pieces of the top layer while preserving their orientation. There are a total of 21 algorithms for this stage. They are distinguished by letter names, often based on what they look like with arrows representing what pieces are swapped around (e.g., A-perm, F-perm, T-perm, etc.). "''Two-look" PLL'' solves the corners first, followed by the edges, and requires learning just six algorithms of the full PLL set. The most common subset uses the AT-perm and EY-perm to solve corners (as these algorithms only permute the corners), then the U-perm (in clockwise and counter-clockwise variants), H-perm and Z-perm for edges. However, as corners are solved first in two-look, the relative position of edges is unimportant, and so algorithms that permute both corners and edges can be used to solve corners. The J, T, F, and R-perms are all valid substitutes for the A-perm, while the N, V and Y-perm can do the same job as the E-perm. Even fewer algorithms can be used to solve PLL (as few as two, such as the A-perm and U-perm) at the expense of having to repeat these algorithms to solve other cases, with additional "looks" to identify the next step.<ref>{{cite web |last1=Zemdegs |first1=Feliks |title=2-look last layer |url=https://www.cubeskills.com/tutorials/2-look-last-layer |website=Cubeskills}}</ref>

can do the same job as the E-perm. Even fewer algorithms can be used to solve PLL - as few as two, such as the A-perm and U-perm - at the expense of having to repeat these algorithms to solve other cases, with additional "looks" to identify the next step.<ref>{{cite web |last1=Zemdegs |first1=Feliks |title=2-look last layer |url=https://www.cubeskills.com/tutorials/2-look-last-layer |website=Cubeskills}}</ref>

===Skipped stages===

Depending on the initial state of the cube and the exact moves made in previous stages, it is possible to complete one stage in such a way that the next stage is also already complete. This is known as a "''skip"'', commonly referred to specifically by the stage that isn't required in the solve. A "''PLL skip"'' is the most common, occurring (when "unforced") approximately once in 72 solves, followed by an ''OLL skip'' with a 1 in 216 chance to occur. A combination of the two, a full "Last''last Layerlayer Skip"skip'', occurs approximately once in 15,552 solves. The Cross and F2L stages of a competition-legal scramble are almost certainly not skippable, though a scramble may present the solver with "free" cross pieces or F2L pairs that are already solved or matched. As speedsolving time is closely related to the number of moves required, any opportunity to make fewer moves presents a significant advantage to the solver. Many speedsolvers have the ability, falling under the general skillset of "lookahead", to identify the likely permutation they will see for the next stage based on the progress of the current stage, and they can vary their solution to avoid permutations that require more moves or an algorithm they are slower to perform. This same ability can allow the solver, in specific known scenarios, to "force" a stage skip with a particular sequence of moves to solve the remainder of the current stage; for instance, by recognizing a particular OLL permutation and performing a specific OLL algorithm, the solver can simultaneously solve PLL, effectively obtaining a PLL skip.<ref>{{Cite web|title=PLL Skip Cases - Sarah's Cubing Site|url=https://sarah.cubing.net/3x3x3/pll-skip-cases|access-date=2022-12-16}}</ref>

There also exist many advanced extension algorithm sets to be used alongside CFOP, such as COLL,<ref>{{Cite web |title=COLL |url=https://jperm.net/algs/coll |access-date=2022-09-18 |website=jperm.net}}</ref> Winter Variation,<ref>{{Cite web |title=Winter Variation |url=https://jperm.net/algs/wv |access-date=2022-09-18 |website=jperm.net}}</ref> VLS, ZBLL, and more. However, it is not necessary to learn them in order to solve the cube or to use the CFOP method. These sets usually have extremea large numbers of algorithms; ZBLL has a total of 472 of them. Therefore, most solvers do not learn these sets and instead focus on improving their skills within regular CFOP.

==Competition use==

CFOP is heavily used and relied upon by many [[Speedcubing|speedcubers]], including [[Max Park]], [[Feliks Zemdegs]], [[Yiheng Wang]] and [[Tymon Kolasiński]] (Though [[Tymon Kolasiński|Kolasiński]] often uses a more complicated method, forZB) its. This is because of the method's heavy reliance on algorithms, [[pattern recognition]], and [[muscle memory]], as opposed to more intuitive methods such as the [[Speedcubing#Roux method|Roux]], [[Speedcubing#Petrus method|Petrus]], and [[Speedcubing#ZZ|ZZ]] methods. The vast majority of top speedcubers on the WCA ranking list are CFOP solvers, including the current 3x3x3 singleaverage world record holder [[MaxYiheng ParkWang]] with a time of 3.1390 seconds.<ref>{{Cite web |title=WCARankings {{!}} World Cube LiveAssociation |url=https://livewww.worldcubeassociation.org/competitionsresults/2758rankings/competitors333/269213average |access-date=20232025-06-1210 |website=livewww.worldcubeassociation.org |language=en}}</ref>

==References==