My speed cubing page - Brandeis Universitystorer/JimPuzzles/RUBIK/Rubik... · My speed cubing page...

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My speed cubing page

This is a copy of Ernö Rubik's signature as it appears in my notebook. He signed it at the World Championship in Budapest in 1982

This system for advanced cubers and is not appropriate for a beginner. It is intendedfor those of you who can already solve the cube in a few minutes and want to getreally fast. If you are a complete beginner, please, visit Jasmine's Beginner Solution.

My system for solving Rubik's cube Unique features The first two layers (additional useful hints and examples of how I

solve the first two layers) The last layer 20 years of speed cubing (a short historical narrative) Watch me solving the cube

Hints for speed cubing Customizing algorithms Multiple algorithms Finger shortcuts Move algorithms to your subconsciousness No delays between algorithms Faster twisting does not have to mean shorter times Preparing the cube for record times Hard work What are the limits of speed cubing?

Collections of various algorithms (by Mirek Goljan, mgoljan (AT)binghamton. edu)

Swapping two edges and two corners

11/25/11 10:51 AMRubik's cube

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Swaping two and two edges Twisting and moving corners and edges in one layer (by Mirek Goljan,

mgoljan (AT) binghamton. edu) Pretty patterns by Mirek Goljan, mgoljan (AT) binghamton. edu

Richard Carr is THE expert on solving large cubes with a list of his recordtimes. Richard can solve the cube blindfolded and willingly shares with us hismethod. I met Richard in April 2003 and he showed me his incredible skills inperson.

Guus Razoux Schultz on speed cubing

The World Championship, Budapest 1982

Hana Bizek's cube art

Dutch Cube Day, October 6, 2002

San Francisco Cubing, January 19, 2003

The 2nd World Championship in Rubik's Cube in Toronto, August 23-24,2003

Press&Sun Bulletin, Binghamton Sep 11, 2003

Cube links

11/25/11 10:52 AMSystem for solving Rubik's cube

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My system for solving Rubik's cube

Winter 1996/97: The system described here enabled me to win the First CzechoslovakChampionship in Rubik's Cube, which took place in April 1982. When I was at my best,I routinely solved the cube in an average time of 17 seconds. At that time, I wasactively using more than 100 algorithms, but the basic required minimum is 53algorithms. Before I go on and describe the details of my system, I would like toexpress my thanks to Mike Pugh who retyped the algorithms from my old notebook toHTML and added nice graphics. His enthusiasm helped me to find the cube no lessinteresting than some 15+ years ago when I met it for the first time. Special thanksbelong to Mirek Goljan, my 1982 finale rival, who kindly provided his enormouscollection of algorithms as it appears here today.

There are a number of diferent systems suitable for speed cubing, but all can beroughly divided into two main categories: corners-edges and by-layers. My systembelongs to the second category even though the first two layers are really formed atthe same time rather than in sequence. The basic set of algorithms consists of 53algorithms for the last layer and a couple of simple moves for the second layertogether with a lot of experience. Most of the algorithms were developed by myselfduring the time period between the summer 1981 and the spring of 1983. However,other speed cubists, most noticeably Mirek Goljan, have also significantly contributedwith important moves. Here is my system in a nut shell:

Action description

Averagenumber

ofmoves

Time Result

Place the four edges from the first layer 7 2 sec.

Place four blocks each consisting of onecorner from the first layer and acorrespondingedge from the second layer.

4 x 7 4 x 2sec.

Simultaneously orient the corners AND



edgesso that the last layer has the required color (one algorithm out of 40).

9 3 sec.

Simultaneously permute the 8 cubes in thelast layer without rotating corners or flippingedges (one algorithm out of 13).

12 4 sec.

TOTAL 56 17

Unique features

One of the unique features of this system is that the last layer is always solved usingtwo algorithms of an average length of 9 and 12, which is very efficient. The averagelengths are based on frequencies with which various orientations and permutationsoccur and on the length of algoritms for each position. Another interesting feature isthat for the first two layers no lengthy algorithms are needed and you can use yourintuition and utilize the specifics of the particular initial state and subsequent states ofthe cube.

The first two layers

In an attempt to make this description complete, I supplied several algorithms whichcan help you solve the first two layers. Although most of the algorithms will be obviousto an experienced speed cubist, some of them are less trivial and are in my opinionvery valuable. In addition to that, one should always try to use the specifics of anygiven state of the cube rather than blindly apply the algorithms. For example, whentwo or three corners are already correctly placed, it may be advantageous to keep thelast corner free and insert all middle cubes using the free corner. Actually, some speedcubists use this approach as their default. Alternatively, when accidentaly (orintentionally) two or more middle cubes happen to be positioned correctly, one canplace the corners from the first layer via the free middle edge(s). All these moves and



a lot of practice should enable you to solve the first two layers in about 10-12 sec. Ofcourse, this requires a lot of practicing, but let us say that 15-20 sec. will be realisticfor most folks.

Because I was receiving a lot of requests for "additional hints" and advice for themiddle layer, I decided to include another section with practical advice for solving themiddle layer. Here are a few examples of how I think out loud when doing the middlelayer. I hope this will help you to master the system faster!

The last layer

Some systems for solving Rubik's cube "by-layers" divide the solution of the last layerinto four stages: orient edges, place edges, orient corners, place corners. It is possibleto group together two and two stages to speed up the process. It seems natural toorient and place edges in one move and then orient and place corners in the secondone. However, this approach has one big disadvantage - it is very difficult to recognizevarious positions quickly. A better approach is to orient edges and corners at the sametime and place all of them simultaneously. Convince yourself that there are 41different orientations of the cubies in the last layer, and 14 different permutations ofthose 8 cubies. Here, we do not count symmetric positions or inverse (backwards)positions as different because they can be solved using one algorithm. Differentorientations are easily recognizable by patterns formed by the color of the last layerand a brief look at the sides of the cube. There are two patterns "C", four "I", two "T",etc. Most of the permutations are also easily recognizable. Given an average twistingspeed of three moves per second, one can solve the last layer in 3 + 4 seconds (basedon the average number of moves).

In theory, we could come up with a much larger system of algorithms which wouldenable us to solve the last layer in one algorithm. However, the number of algorithmsone would need to learn is 1211.

11/25/11 10:52 AM20 years of speedcubing

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20 years of speedcubingDear visitor, I am sure my narrative will occassionally bring a smile ofdisbelief to your faces today, but I want to truthfully describe theatmosphere of excitement and mystery as we, the old-timers, lived throughthe "good ol' days" of the early eighties when Rubik's cube was makingheadlines around the world.

The first time I met the cube "face-to-face" was when I was 16 years old inMarch 1981. I was hooked since the first moment I saw this absolutelyunique combination of simplicity and ingenuity. There was no need to explainwhat needs to be done with it - a self-explanatory, remarkably difficultpuzzle with a devilishly mysterious mechanism inside - a fascinating silentchallenge. The owner of the cube was a 14-year old boy who could solve thecube in about a minute. He lent it to me for a few minutes just enough toassemble one face.

Although in March 1981 the cube was being sold by thousands in othercountries and despite the fact that the invention took place in theneighboring country, it was impossible to buy the cube in Czech Republic. Aclassical example of how inefficient and impotent the Eastern Blockeconomies were. I got my hands on a primitive solving system from aRussian magazine Kvant long before I actually owned the cube. I wouldanalyze simple moves and their action on a piece of paper, trying to figureout algorithms based on the commutator principle. Then later in the spring,our local astronomy club leader bought the cube during his trip to Hungary.He was unable to solve it and could not find anybody who would put thecube back into its original state. With the help of the commutator principleand those "Russian" moves, I solved the cube for the first time. It took meseveral hours.

I desperately wanted to get my own cube but whoever was lucky enough toown it, would never sell it. So, I had to wait a little longer and finally got myfirst cube in July 1981. A French family was visiting my sister and their twoteenage boys brought the cube with them. When they saw how attached Iquickly became to the piece of plastic, they did not have the heart to take



the cube with them back to France. That meant that I could finally startworking on my system! During the Summer, I persuaded my parents to visitHungary, where I bought three more cubes. It was still a challenge to getthe cubes because they were not available in stores. I bought the cubes froman old lady who was selling magazines and souvenirs in the street. When Imentioned "Buvos Kocka" to her, she smiled, quickly looked left and rightand handed it to me in a brown lunch bag, put her index finger across hermouth, and said "Shhh, one hundred and fifty Forints". I know all this soundsfunny now, especially to those from Western countries where it was a no-brainer to buy the cube. But this is how I really started.

At first, I was using the layer-by-layer system that I learned from a Czechmagazine. It was actually already quite advanced. First the first layer, thenthe four middle edges (just one algorithm), then flipping the edges, movingthe edges, flipping the corners, moving the corners. These were my basicalgorithms that I started using and I quickly got my average to about 1minute. It was September 1981, three months after I started playing withthe cube.

My last year of high school was a strange and exciting time. A kid solving thecube in public was a head-turner. Cube was a good conversation starter,although later it was termed by many as a conversation killer . It was normalfor two cubers who met on a bus to start the conversation without looking ateach others eyes, saying: "How do you this?" "Hmmm, could you do itslowly?" "Thanks". This is how we built our first primitive systems. I had oneschoolmate from high school who had the same disease - an unconditionallove for the Rubik's cube. His name was Ludek Marek. He was using thesame system as me, but for some reason he was always trailing about 20seconds behind me. He once noted while I was solving the cube pointing tomy cube: "Oh, I like this "T" pattern, because when you turn the edges, thewhole last layer will actually flip correctly." It was the shortest 6-move thatinfluences only the last layer - the move that perhaps all cubers know. Andthat sentence stuck in my mind. It was the germ that later blossomed intothe current system. I realized that in the system I was using it was possibleto first flip the edges, then the corners, then position edges and positioncorners. This is because the moves commuted. So, what if I had analgorithm for all flipping patterns and all permutations? Then I could solvethe last layer always in just two algorithms. Also, the number of patternswas not that big and they were easy to recognize fast. But where to get the



algorithms? I already knew some portion of them and I gradually startedadding more. Whenever I encountered an orientation that I did not know, Idid it the old way - flip edges and then flip corners. And whenever Iencountered a permutation for which I did not have an algorithm, I wouldcombine the permutation from the algorithms I already knew. I beganimproving very steadily as I improved my system and my ability to recognizepositions quickly. In December 1981, 6 months since time zero, I wasaveraging about 35. Occasionally, I would read an article about a studentfrom Great Britan who solved the cube in 28 seconds, then about a guy fromUSA who did it in 24, etc. I was always chasing the world, trying to catch up.I often felt like it was not possible to squeeze my times anymore, as if I wasalready at the limits of what I can do with my system, but nevertheless, withtime, I was able to get to those magic numbers I previously read in thenewspaper. At that time, I was getting ready for my final examination tofinish my high school. I was combining test preparation with cubing. When Iwas messing up the cube, I was staring into the text, learning the subject,then pausing for a while and solving the cube. And I could go like this forhours and hours. Surprisingly, during the late Summer and Fall 1981, thecubes became finally available in Czech Republic as well. The cube crazy hasofficially began. Local championships popped up at high schools anduniversities. I always participated in them often with my friend Ludek. Bothof us always left the competition far behind. There was nobody I knew withwhom I could compete. I was thus chasing the clock and the world.

In the Winter 1981/82, the Czech magazine Mlady Svet called for a nationalchampionship and people started submitting their times. In February 1982,the magazine printed a preliminary table showing the best ten timessubmitted. And I could not believe my eyes to see my name on the firstplace. I also noticed the name Mirek Goljan who was literallily "breathing onmy back". That gave me more energy for my practicing and I went into thenational championship on May 11, 1982 averaging about 25 seconds with mypersonal best of 18. Ten months from time zero.

I won the semifinals and 5 best advanced into the finals. Among them, MirekGoljan and my friend Ludek Marek. The finals were in front of TV cameras.We were allowed to use our own cubes. The best-of-three time determinedthe winner. We all solved the cube at the same time. I won the first andsecond rounds and Mirek won the last third round. My second time of 23.55got me the first place, Mirek was the second, trailing about 2 seconds, and



Ludek ended on the third place. The first prize was a plane ticket toBudapest to the first World Championship.

I became a "celebrity" for a few weeks receiving a lot of letters all asking forone thing - the description of my system. The letters actually did not havemy proper address, just the name and city, no street address or zip code.They all were delivered. I decided to publish my system in Mlady Svet. Itcontained all algorithms for permutations and orientations and a few movesfor the F2L. Most people were disappointed to learn that the method isactually quite "complex" requiring a lot of practicing and memorization. Mostexpected a simple trick that one can explain in a few minutes. What did yousay about the free lunch? I remember one really funny story that happenedto me on a train when I commuted to college from my home town. A guywas sitting next to me playing with the cube. I asked him about his system.He said: "I am using the Fridrich method." I asked with a surprise in myvoice: "You actually memorized ALL algorithms?" His answer was: "No, that'stoo much. I know only some of them." I replied with: "Well, you need tomemorize all of them otherwise you are not really utilizing its strength." Helooked at me frawning and said with his mouth half open: "Yeah, so what'syour system?" I answered with a big smile: "I use the Fridrich method, too,because I am Fridrich." He did not blink an eye, did not say anything andhanded me his messed-up cube. I solved the cube in about 20 seconds toprove my words and we both laughed at the coincidence.

I was acepted to college and still kept on improving. Later in 1982, Ichanged my F2L system to the current system. Before, I would do the firstlayer and then insert two cubies from the last layer into the middle layer. Ideveloped the algorithms and also algorithms that moved / flipped thecubies in the middle layer. When I switched to the current system for theF2L, I instantly improved by several seconds and got my average to around20 (15 months from time zero). By 1983, I was consistently averaging 17seconds. I knew three more cubers capable of achieving sub-20 averagesconsistently. We practiced together. As the cube rage cooled down, Istopped working on my system. The second Czech Championship took placein March 1983. Robert Pergl won all three rounds (if I remember correctly)with a best of 17.04. He was using basically my system but he knew morethan 600 algorithms (I was actively using about 120-150) and one could saythat he was using a "multi-system". From time to time, he was able to solvethe last layer in just one algorithm, perhaps due to preparing the LL a little



before finishing the F2L. And he stayed cool and psychologically stableduring the whole event. Psyche is a very important factor in championships.There is little value in being able to solve the cube in 16 seconds on averageif the nerves slow you down to 20 during the competition. You can't win a bigevent unless you work on the psychological factor as well. And Robert indeedwas consulting with a psychologist, preparing very carefully for the wholeyear. What can I say - it paid off.

I would dare to say that nothing important happened in speed cubing andcubing in general over the next decade. Then, in 1992 Herbert Kociembadeveloped a computer algorithm with a performance very close to the God'sAlgorithm (the shortest moves from any position). It was, in my opinion, thebiggest event in cubing in general. Suddenly, we could obtain the shortestmoves for any position and any pattern. Surprisingly, Kociemba's algorithmalways seemed to find a solution within 20 face moves. The famous cube-in-cube pattern turned out to have an elegant short solution L F L D'B D L! F!D'F'R U'R'F! D as we suspected for a long time but never found it. Progresshas been made in identifying the farthest positions on the cube (superflipand supertwist). To give you an idea how revolutionary Kociemba's discoverywas, the previous best computer solution was always able to solve the cubewithin 38 moves, but could not guarantee better (Thistlewaite's algorithm).Even though Kociemba's algorithm did not provide a proof that the diameterof the cube group is indeed 20 in face counting, it has been an impressivepiece of work, indeed. As the computer speed and memory increased,optimal solvers came to life, and suddenly, to my opinion, the cube lostmuch of its enigma.

I put my system in electronic form on the Internet in January 1997 after Ihad discussions with Mike Pugh on the Cube Lovers mailing list (one of theoldest mailing lists ever, established in 1980). He persuaded me that makingmy system available in electronic form would be useful for other cubers. Imade copies of my old, now yellowish, notebook and he made those smallpictures you now see on my pages. I included the patter and set up the site.I never put a counter on my page, so little did I know how popular thesystem became. Actually, to be completely honest, I was convinced thatnobody in their right mind will have the energy and will to learn the systemin its entirety. I thought that speedcubing was inactive and not popularenough for anybody to have the motivation to go through the pain ofmemorizing the algorithms. I know now how wrong I was. One should never



underestimate the power of the cube. I still admire those of you who enteredthe speedcubing now. Back in 1981, the cube was mysterious. We did nothave computers powerful enough to develop the shortest moves for us. Wedid not know if those algorithms we found by trial and error were the best orshortest. The unknown and unanswered questions were an importantingredient for many cubers. They were the engine that powered us forward.I do not intend to sound as an old lady complaining while recalling the oldgood days, but I am trying to convey what most of us, if not all, felt as wewere trying to uncover the curtain of secrecy of the cube.

At the end of 1996, I sent a postcard to Mirek Goljan and I typed the 14-move algorithm for cube-in-cube and nothing else on it. Mirek and I have notseen each other for at least 12 years and I was already pursuing my PhD inthe US by that time. We got in touch again via phone and later via e-mail. In1997 I visited Czech Republic and after almost 14 years, we started cubingtogether, admiring Kociemba's algorithm, and sharing our personal stories.Mirek later joined SUNY Binghamton and pursued his PhD degree in thesame field as me - steganography and digital watermarking. We becameprofessional colleagues and today we work together on puzzles of datahiding and discovering them in digital images. After 14 years, our journeysjoined again - two top Czech speed cubers uncovering the secrets of images.

That's all folks. Thanks for reading!

11/25/11 10:53 AMVideo Page

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Video page with finger tricksOn this page, you will find a few video clips of me solving the cube. You will need the(free) DivX codec, to view the movies. Also, if you want to view them frame by frame,I was told that you need VirtualDub under the GNU license. I have also put some ofmy favorite finger tricks because they are a lot of fun. In "Stick", I reach 10 moves persecond for three seconds. Overall, it was a lot of fun to watch myself on the screen. Itis a completely different experience!

This was my first solve in 16.74 sec. You can tell thatI am still stiff in it because of the delays betweenalgorithms. The time was not that bad considering myrough cubing style.

The F2L were in 9 seconds, which was not bad, butthen I got stuck in the permutation algorithm. Yeah, Ineed to work on my LL permutations, I know ... butthe final time was 15.98 sec.

OK, I will be critical to myself again. This one was in16.48 sec, again with lots of delays betweenalgorithms. I think I should rely less on speed and gofor a more reliable approach that is a bit slower butmore smooth without that many delays. Isn't thatwhat I advise everybody? :)

11/25/11 10:53 AMVideo Page

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This one was really really weird. 14.33 sec. - ablooper was cancelled out with a lucky permutation. Iinserted two corner-edge pieces in the F2L wrong andI had to switch them. Then I got an orientation that Ilike (four corners) and a trivial permutation. Withoutbeing lucky, I would have gotten probably around17.something.

My favorite finger trick - the Stick named aptly aftershifting gears in a car with manual transmission. No, Ido not drive that fast :) By counting frames, Idetermined 30 moves in 89 frames, which translatesto 10 moves per second sustained for 3 seconds. Iknow, this is just a repetitive move, so it is easier toachieve that speed. It will be very hard to beat LarsPetrus' Sune at 7 moves in 0.7 seconds. That lookslike magic, indeed!

Just a finger frenzy using a repetitive slice move.

I use those four moves when finishing thepermutation algorithm V (the second on mypermutations page).

11/25/11 10:53 AMHints for speed cubing

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Hints for speed cubing

Customizing algorithms

It is very important to customize each algorithm for your hands. Some of us are righthanded, some left handed, some may prefer algorithms which use only 2 or 3 faces sothat alternate twisting from left hand to right hand is avoided. Sometimes, it may bewise to perform an algorithm with the cube turned upside down, or twisted by 180degrees. This adjustment must be done by each individual separately becauseeveryone may have different views of which algorithms are user friendly and which arenot. This takes a lot of time, but it may cut an important chunk from your total time.

Multiple algorithms

As you may notice, some positions in the last layer have several algorithms associatedwith them. I alternate between them to minimize turning the cube as a whole, thuscutting on time.

Finger shortcuts

Most speed cubists have also developed special sequences ("shortcuts or macros") oftwo to four moves which can be performed astonishingly fast by pushing the faceswith your fingers. Yes, it does require some dexterity. On my video page you canwatch me solve the cube a few times. I also perform some finger tricks.

Move algorithms to your subconsciousness

It is also important that your brain automatizes the algorithms into inseparable units -elementary actions, because then you will not have to think about individual moves.The individual moves will be performed "by your hands" rather than making your brainbusy. At this stage, one can afford to think more about the next step rather thanabout the algorithm which is being performed. It is done for you automatically by yoursubconsciousness! I noticed that this automatization goes that far that if I aminterrupted while performing some longer algorithm, I will not be able to finish it! In asense, I do not know the sequence of moves and perceive the algorithm as one unit.This may sometimes create comical situatioins when somebody asks you about aspecific move, and you will not able to show it slowly - and will get stuck after severalmoves having to start over again to see the remainder of the algorithm.



No delays between algorithms

Another thing which is very important is to cut on delays between consecutivealgorithms. One should minimize the decision time to almost zero. This issue isstrongly connected with another one - the question of twisting speed.

Faster twisting does not have to mean shortertimes

Dogma: One needs to be especially dextered to be able to solve the cube that fast (in17 seconds). I would be lying to say that some dexterity is not important, but I insistthat an average person possesses the necessary dexterity to solve the cube in reallyshort times. I believe that almost everybody can achieve the twisting speed of 3 twistsper second. Remember, all you are required to do is to learn a finite set of algorithmsperform quickly. This relates to the important issue of adjusting algorithms for yourhands. So why is it possible that faster twisting speed may bring you longer times? Byperforming the moves really fast, one deprives him/herself of the [important]knowledge of what is actually happening to the cube. After performing an algorithm,one is then suddenly thrown into a new position and needs some time to decide whichmove to choose next. If you had turned the cube just little slower, you could actuallysee what is happening to the cube, and choose the best next move during the lastcouple of moves of the previous algorithm. If you compare the times: fast turns +delay between moves and slow turns + shorter delays, you will find out that thesecond summation may be shorter! Another argument for the second alternative isthat it is very hard to turn the cube really fast, and one often encounters "stuck"cubicles, or breaks the cube to its atoms. This can slow you down as well as frustrate.

Preparing the cube for record times.

I have heard people recommend a variety of different lubricants for the cube. Amongothers, sillicon oil, graphite, and soap were mentioned. From my experience, silliconoil worked best. Be careful before using other lubricants because some of them maybe pretty aggressive and speed up the aging process of your cube. Intense twistingcauses a fine dust to develope inside the cube. Some cubists say that this kind ofnatural lubricant is the best one. I recommend to grease the cube because a lubricatedcube will turn easier and you will be able to "cut corners" while speed cubing. But beaware of the fact that putting lubricant into a cube will make the cube more vulnerableto an accidental dismemberment.

Hard work

I would like to end with a couple of more remarks on the cube. First, the secret of



achieving amazingly short times is not just the algorithms themselves. After all, asystem will never solve the cube. Humans do! Probably the most important factor isdedication and a lot of practicing. As you may notice, some positions in the last layerhave several algorithms associated with them. I alternate between them to minimizeturning the cube as a whole, thus cutting on time.

So, what is the best system for speed cubing? I do not think that there is such thingas the best system. One system may better fit one person, other system may be morenatural for somebody else. I believe that any system which is worked out intosufficient perfection is good. We should not be comparing systems but cubists. Thosecertainly are comparable.

What are the limits of speed cubing?

Any algorithmic set which can be performed by a human must be limited to a couple ofhundreds at most thousands of algorithms. These algorithms need to be performed ina fast manner without too much thinking. This puts limits on the amount of timeneeded to solve the cube. If there was a hypothetical person who could see theshortest or the almost shortest algorithm right away in the beginning (which is quiteimprobable), he or she would need about 2 seconds, provided the farthest position isaround 20 face moves at the twist rate of 10 moves per second. Since the assumptionfor this estimate will probably be unrealistic for many years to come, I estimate thelimit for speed cubing at 5 seconds (the average time). One should totally abandon theconcept of a record time since it has very little informational value. If somebodymesses up the cube carelessly, one can take advantage of it and solve the cube in afew seconds. Therefore, for comparing purposes, I suggest to use an average of 10consecutive times. For my system, I defined the concept of a modified record: Idiscarded record times whenever more than one stage was skipped during the cubesolving. By skipping a stage, I mean: placing the four edges using less than 3 moves,too much luck for the four blocks (in the second layer), skipping the orientation of 8cubicles from the last layer, skipping the permutation part in the last layer. For thefirst two layers, it is hard to estimate the probabilities, but the last layer can becalculated exactly. The probability that after solving the second layer, the last layerwill have the correct color is 1/216, and the probablity that after orienting the cubes inthe last layer one will not need to permute them, is 1/72. So, for example, if the lastlayer got assembled by chance right after the second layer, I discarded the time sincethe probability of that happening is too small: 1/(216*72). So, what is my modifiedrecord? It is 11 seconds. My best average out of ten was often 17 in 1983. I keptmyself in a good shape for many years, and I can still get to an average of about 18after all those years. Going back to 17 or lower would require a lot of effort, goodcube, and a complete devotion that only a rookie can possess. So, good luckeverybody and do not give up!

11/25/11 10:54 AMTwo edges - two corners transpositions

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Two edges, two corners transpositions

These sequences are transpositions of two corners and two edges. The shortestalgorithm for a transposition of two edges and two corners has 13 quarter moves. Thelist below is an attempt to find all 13 and 15 quarter-move algorithms. The list wasfound by hands and is by no means considered to be complete. If you would like toadd some new algorithms, please contact Mirek Goljan mgoljan (AT) binghamton. eduJessica.

The two groups of the same length are sorted in alphabetical order. The alphabet is:A!s,A!,As,Aa,A,A'a,A',B!s,B!,Bs,Ba,B, B'a,B', ... ,F!s,F!,Fs,Fa,F,F'a,F'. FacesA,B,C,D,E,F represent U,R,D,L,B,F respectively in English notation and if you want youcan easily transfer algorithms to this or any other notation. Every algorithm is put inits minimal form (in alphabetical order) of all its cyclic shifts, inversions, mirrorings,and whole cube rotations. Number of a sequence means a number of quarter moves,[number] means number of used faces.

moves faces nameA²B²C E C'B²A F'A F (13) [5] CE1351A²BaA'B'A D'A²B A'B' (13) [3] CE1331A²B A B²C'F C B F'A F (13) [4] CE1341A²B A B'A D F D'A'F'A F (13) [4] CE1342A²B A B'A E'A B A B'A'E (13) [3] CE1332A²B A B'A E'B E A E'B'E (13) [3] CE1336A²B A B'A F'A E D E'A'F (13) [5] CE1352A²B A B'A F'A'B'F B A F (13) [3] CE1333A²B A B'A F'D C F C'D'F (13) [5] CE1353A²B A F'A E D'A'D F A E' (13) [5] CE1357A²B A'B'A'E B E'A'E B'E' (13) [3] CE1334A²B A'B'A'E D A'F'A D'E' (13) [5] CE1355AaB'A'B C'B A F B'F'A'B' (13) [4] CE1344A B A B'A D F A'F'A F A D' (13) [4] CE1343A B A B'A F'A E D A'F A E' (13) [5] CE1354A B A B'A'E'A'E B E'A E B' (13) [3] CE1335A B A B'A'E'A'E D A'F A D' (13) [5] CE1356 moves faces nameA²BsA E A'E'B'A D A D'A D (15) [4] CE1501A²BaA B²D'C'D F²D'C B (15) [5] CE1502A²B A²B'F'B'F B F'D F D'F (15) [4] CE1503A²B A²E A E'B'E'F'A F A'E (15) [4] CE1504A²B A²E D'E D E²B'E'B E (15) [4] CE1505A²B A B²A D A'B A B D'A B' (15) [3] CE1506A²B A B²A'E'B E B A F'A F (15) [4] CE1507A²B A B A'B'A F A'B'A B A F' (15) [3] CE1508

11/25/11 10:54 AMTwo edges - two corners transpositions

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A²B A B F B F'A'B²A F'A F (15) [3] CE1509A²B A B'A B A E'B'F'B E B'F (15) [4] CE1510A²B A B'A B F'A E A'F A E'B' (15) [4] CE1511A²B A B'A B F'D'A'D A F A B' (15) [4] CE1512A²B A B'A E'B E A D'E'B'E D (15) [4] CE1513A²B A B'A F'A D E D E'A'D'F (15) [5] CE1514A²B A B'A F'A D'A'D F D F D' (15) [4] CE1515A²B A B'A F'A D'A'F'D F D F' (15) [4] CE1516A²B A B'A F'D F²D'A'F A F' (15) [4] CE1517A²B A B'A F'D F B F'D'F A B' (15) [4] CE1518A²B A B'A F'D F D C'D'C F D' (15) [5] CE1519A²B A B'A F'D F D'F A D'A'D (15) [4] CE1520A²B A B'D'A B A'D A E A E'B' (15) [4] CE1521A²B A B'D'E D F'D'E'A D A F (15) [5] CE1522A²B A B'E F'A'D'A D A E'A F (15) [5] CE1523A²B A B'E F'A'F A E'A B A B' (15) [4] CE1524A²B A B'E F'D'E'A E D E'A F (15) [5] CE1525A²B A B'F'B'F B A F'D F D'F (15) [4] CE1526A²B A B'F'D'A D A E'A F A'E (15) [5] CE1527A²B A D'A E A²B'A'E'A'E D (15) [4] CE1528A²B A E A'B'A E'A E D'E D E' (15) [4] CE1529A²B A F B F'B'F'A F'D F D'F (15) [4] CE1530A²B A'B'A'B A'E D E'B'E D'E' (15) [4] CE1531A²B A'B'A'B E'C'E A'E'C E B' (15) [4] CE1532A²B A'B'A'D A'F'D F'D'F A D' (15) [4] CE1533A²B A'B'A'D A'F'D'A E D'E'D (15) [5] CE1534A²B A'D A B'A F A B A B'F'D' (15) [4] CE1535A²B A'E'B C'B'E A'B'F'B'F B (15) [5] CE1536A²B A'F'A B'A'F A'B'F'B'F B (15) [3] CE1537A²B A'F'A B'A'F A'F B F'B'F' (15) [3] CE1538A²B E A E'B'A F'A F B'F B F' (15) [4] CE1539A²B E B E'B'A B'A F B'C B F' (15) [5] CE1540A²B E D A'D'A'E'A'B E'B'E B' (15) [4] CE1541AsB A'B'AaB'A'B E'C'B'C E (15) [4] CE1542AsB C B'A'E'B E C'B'C E'B'E (15) [4] CE1543A B A B'A D C'F A'F'A C F A D' (15) [5] CE1544A B A B'A E'A B A B'F A'E A F' (15) [4] CE1545A B A'E D'A D E'A B'F B A'B'F' (15) [5] CE1546

11/25/11 10:54 AMTwo edges - two edges transpositions

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Two edges - two edges transpositionsApostrophe means 'turn counterclockwise'. Uppercase letter S means center slide, Sr (or only S) is the slideadjoining face R.Us,Rs,Ds,Ls,Bs,Fs are abbreviations for UD',RL',DU',LR',BF',FB', Ua,Ra,Da,La,Ba,Fa are abbreviations forUD,RL,DU,LR,BF,FB. The groups of the same length are sorted in alphabetical order. The alphabet is:U!s,U!,Us,Ua,U,U'a,U',R!s,R!,Rs,Ra,R,R'a,R',D!s,D!,Ds,Da,D,D'a,D',L!s,L!,Ls,La,L,L'a,L',B!s,B!,Bs,Ba,B,B'a,B',F!s,F!,Fs,Fa,F,F'a,F'.Every algorithm is put in its minimum (in alphabetical order) of all its cyclic shifts, inversions, mirroringsand cube rotations.

[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'

The first column each sequence denotes the number of face moves, the second column contains the numberof used faces.

Positions of order 2

U!sR!U!sL! 12 4U!sRsB!sLs = [S!u;Sr] 12 6U!R!U!R!U!R! 12 2U!R!U!RaF!R'a 12 4U!R!U!L!D!L! 12 4U!RsB LsBsL Fs 12 5U!RaBaU!B'aR'a = [U!RaBa] 12 5U!RaB'aD!BaR'a = U![RaB'a*D!] 12 6U!R B R'B'U!L'B'L B 12 4UsRsUsLsDsLs 12 4UsR UsF'DsR DsB' = SdB [Sd;B']S'cB' 12 5UsR U R'F'DsR B U'B' = [Sd;B U B'R'] 12 5UsR U F'R'DsB R U'B' = [Sd;B U R'B'] 12 5UaR U R'F'U'aB'U'B L 12 6UaR'U'R B U'aB U B'R' 12 4UaR'B'R B U'aF L F'L' 12 6U R U'R'B R B'U'B U R'B' 12 3U R U'R'B L U'F'U F L'B' 12 5U R U'L'U B R'U'L F U'F' 12 5

11/25/11 10:54 AMTwo edges - two edges transpositions

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U R U'B'R F'R B R'U'F R' 12 4U!R D R'UsR'UaR'D'R F 14 4UsR DsR B R'B'R'B'R'B R 14 4UsR B D'B'R'DsB L D L'B' 14 5

Positions of order 4

U!R UsF'U!F'DsR 12 4U!R U R U RaF'R'F'L' 12 4U!R U R'B'R!U'R'U B 12 3U!R U B R B!R'U'B'R' 12 3U!R U B L U!L'U'B'R' 12 4UsR U R DsB'U'B' 12 4UsR U'R DsB'U B' 12 4UsR DsB UsR'DsB' 12 4UsR B R B DsL'B'L'B' 12 5UsR B R B'DsL B'L'B' 12 5UsR B R'B DsL'B L'B' 12 5UsR B'R B'DsL B'L B' 12 5U!R U R B'R'D'R!D B U'R' 14 4

11/25/11 10:58 AMTwisting and moving corners and edges in one layer

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Twisting and moving corners and edges in onelayer

At most four corners are not correctCorner rotationsCorner trianglesCorner cross-transpositionsCorner parallel-transpositionsCorner triangle and one corner rotation

At most four edges are not correctEdge flipsEdge trianglesEdge cross-transpositionsEdge parallel-transpositionsEdge triangle and one edge flip

Twisting or flipping cubes in placeOne corner and two adjoining edges are correct

11/25/11 10:54 AMLast layer

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Corner rotationsThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).

The numbers in the second column denote the number of quarter moves and face moves.

[R B'R'U'B'U;F] 14L F U L U'R U L'U'F'L'F R'F' 14[U;F'D R'D!R F] 16,14[U;F'D L'F!L F] 16,14R'F!U F U'L'U F'U'F!R F'L F 16,14U'B U'B'U!BaU F'U F U!B'a 16,14U'F!U L!D'L U L!D L!U'L F! 18,13R(R F'D!F R'U!)!R' 18,13

Inversions of previous

D'F U F'D F L D L'U'L D'L'F' 14(D L'B!L D'F!)! 16,12[L'B L U B U';F!] 16,14R F'L'FaU!B'R!F!L F'R F' 16,13R!D!L D'R D'R D'L'UaF!U' 16,13L'B'RaF'L'FaR'F'RaB'R'Fa 16D'F D F!D'F L!F'U'F U L!D F! 18,14U'(R'F'R U'R U R'F)!U 18

R'(D'F L'F'L F'D F)!R 18

U'(R'F'R F'U F U'F)!U 18U'(F'R'F R'D R D'R)!U 18

11/25/11 10:54 AMLast layer

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R F U F'U'F U F'U'F R'D'L'U'F'U L D 18[U B D'B!U'B D;F!] 20,16R'F'D'F D F'D'F D!R UsF'D'F Ds 18,17[U'L B!U L!B'L;F!] 22,16

11/25/11 10:55 AMCorner triangles

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Corner trianglesThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


U!R!U'L'U R!U'L U' 12,9R'D R'U!R D'R'U!R! 12,9UaB!U'F'U B!U'F D' 12,10

D F!R'B'R F'R'B R F'D' 12,11

R'F!R D'R'D F!D'R D 12,10L'U!F D F'U!F D'F'L 12,10

L!D!L U!L'D!L U!L 14,9

R!B'R F!R'B R F!R 12,9

R'aD'R U!R'D R U!L 12,10

L U L'F!L U'L'U F!U' 12,10D F'L'F R!F'L F R!D' 12,10

D R!D L!D'R!D L!D! 14,9

11/25/11 10:55 AMCorner triangles

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U L D'L'U'L D L' 8

R'D'R U'R'D R U 8

U'R'D'R U R'D R 8

L D'L'U L D L'U' 8

R'F!R'B'R F!R'B R! 12,9L'U!R'D'R U!R'D R L 12,10

L'U'L B'L'U!L B L'U'L 12,11

L D'F!D L'D'L F!L'D 12,10

[R;F'L'F] 8

[F D F';U'] 8

11/25/11 10:55 AMCorner cross-transpositions

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Corner cross-transpositionsThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B,U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


R'aF!RaUaF!U'aF! 14,11[R'aB!Ra;F] 14,11

U!R!sD!B U!R!sD!F 18,10

R!D!R'U'R D!R!D R U R'D' 16,12

U L D'L'U'L!U!L'D L U!L! 16,12

R'F'R F'R'F R F'R'F R F'R'F!R 16,15L'U!R U R'U'R U R'U'R U R'U L 16,15(D'F U F D F U'F')! 16D'F'L D!B!R'B'R B'D!L!F L D 18,14

U'R'F D!L!B'L'B L'D!F!R F U 18,14

11/25/11 10:55 AMCorner cross-transpositions

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D'L'U'L D R'aU L U'R U 12L D L'U L U'aR U L'U'R' 12D'F!D R!U'R'U'L'U'L U'R' 14,12L'F'L F'L F L!F L!F!L'F! 16,12D'F!D!F D!F D F'D F'D'F! 16,12D'F!UaL D!L D L'D L'U F! 16,13D'F'aL!F L!B D F'D F'D'F! 16,13D'L'U'L D L'U F R F'L F R'F' 14L'F R F'L F U R'D R U'R'D'F' 14R!L D L B R'B'R!L'D'L'F'R F 16,14R BsL'FsR'F'R!B L B'R!F 16,14

R'D R F R!L B R D'R'B'R!L'F' 16,15

R D'B!D R'F'R D'B!D R'F 14,12

U F!U!F'LsF!RsF'U!F!U 18,13

The three-tuple associatedwith each algorithm denotes the number of quarter moves, face moves, and slicemoves.

11/25/11 10:55 AMCorner parallel-transpositions

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Corner parallel-transpositionsThe usual notation U, R, D, L, B, F is extended to slice and antislicemoves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B,U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


UaL U'R U L'U'aL D R'D'L' 14

R'F L'B!L F'RsF R'B!R F'L 16,14

U'R'F L'F!RaUaF!D'L F'L'F! 18,15D'L'D R'D'L D'L!D'R'D L!D'R D'R 18,16

D F!L'F!RaUaF!U'aF!R'F!D' 20,15

D'L'DsR'UsL DsR U 12L'U'R'F R U L D R F'R'D' 12

D'L'F'L F L D RsF'L'F Ls 14

U'F D F'U F UsL D L'U'L D'L'F' 16L'F R F'L F LsD L'U L D'L'U'F' 16R'D'L D'L'D!F!R U'R U R!F!R 18,14

U'F!D'B!D F!U F'U'F D'B!D F'U F 20,16



L'F!L'B'L F'D F'U'F D'F'U L'B L! 18,16

U!R!U'L'U R!U'L U!R U L'U'R'U L 20,16

U'R'D F'L!F R F'L!F U R'D'R 16,14

U'R'D'F!D R U!L'U'L!F!L'F! 18,13

U F!R B'R'F R B R'D'F U'F'D F 16,15U F!R'D R D'F!U'R'F!R!F!R' 18,13

R'F!R F R'F!D F'D'F'R D F!D'F' 18,15

R U L'U'R'U B'L F L'B L F'U' 14D R!U'R'U!F!U'F!R'D'L'F!L 18,13

D'L'U'F'R!F L F'R!F D L'U L 16,14

U L F L'F'(L F L'F')!U' 14[R'D!R!F!R'*D!]F! 20,12[R'D!L!B!R'*U!]F! 20,12

R'F!RaF!R'F!R F!R'aF!R F! 20,14

U'R U'R'U!L'U L U'L F!L'F! 16,13[U'R U'R'U R';F!] 16,14(R'F L F'R F L')!F! 16,15U'R'D R U R'D'R!U'R'D R U R'D' 16,15R'F!R U'R U R!F!D'L D'L'D!R 18,14U'F!D'B!D F!UsF!U'B!U F!D 20,14



D'B D!B L'U'L B'D!B L'U L B!D 18,15

R'F!R F!R U'R D R!U R'D'R F! 18,14

R U'R'F'R B U L U!L'U B'R'F U 16,15R F R'D'L F!L!F'L!F'L'F D F! 18,14D'L'U'L D L'U LsD'R U'R'D R U 16

R'F'D'R'aF'R'D R D'F RaF D R 16

11/25/11 10:55 AMCorner triangle and one corner rotation

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Corner triangle and one corner rotationThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


U!B'D B!U F'U'B'U FsD'B U 16,14

L B R'FsL B'L'F'L B!R B'L! 16,14

R'D'R U'R'UaL'U'RaD'L'U L D 16R'F'D F'U'F D'BsU F'U'B'U F'R 16R'D R D L'U'L D'R'D'R D'L'U L D 16

U'R'U L'U'R U!L'D!L U'L'D!L! 18,14

D'F!D F!D'F!L D R'D!L'D'R D' 18,14



D'F!D F!D'F D F'U F'D'F U'F D F 18,16

R'D R'U'R!D'R U F!R'F!R F!R' 18,14

U F D'F U'F'D F'U F U'F!U F!U'F 18,16

D R D'L D R'D!L'U L D L'U' 14,13

L'U'R U L U'R!D'R U R'D R 14,13

L D'L U!L!D R'D'L D RaU!L! 18,14

U!B U'F!U B'U'F D'F U'F'D F 16,14U F'D'L F'L'B'L F L'FaU'F'D F 16

D R D'L D R'UsL'F!L U'L'U F!U' 18,16

U F'D'F U'F D'B'D F!D'B D!F 16,14U F'D'FaL'F L B'L'F'L U'F'D F 16

L'F!L U'L'U F!U'LsD'R U R'D R 18,16



U'F D F'U!B!U'F U B!U'aF' 16,13U'aB!U F U'B!U!F'D F U'F' 16,13U'F'U!B'D B U B'UsF U'B U! 16,14R'F!RaF!R'F'R F!R'aF!R F' 18,14

R'F R F D'F!D F'R'F R D'F!D F! 18,15

U'R!D'R U L'U'R'D R!UsL D 16,14L'U'R U L U'R'U!L U'R U L'U'R' 16,15L'U B U'F!D'F U F'D FsU'F L 16,15R B L'U L B'R'U F'U!R'F'R F U 16,15

R'D F!D!L'D R D'L D!R'F!R D' 18,14

R U LsD!R D'L'U'L D R'D!L' 16,14D'L'U'L D L'U L!D'L'U L D L'U' 16,15U F L'FsR F'L F R'F!L'B L U' 16,15L F D F'D'L!F'L D'B'U L U'B D 16,15

R'D F!D'R!U R'D R U'R!F!R D' 18,14

U L U'R U L'U'R!D R U'R'D'R U 16,15R'D'F'UaL'U'R U L!D F D'L'U' 16,15L'U L U'F!L!D R'D R D!L!F! 18,13

R!U!R D!R'U!R D L D R D'L'D 18,14

D'L D R'D!L'D'R'D'R!U F!U' 16,13

U'R'D R'D'R'U R'F!R'F!R F! 16,13U'R'D R'D'R'U L'U!R'U!L F! 16,13U F!U'F!U'R U'L'U'L U'R'F! 16,13

D'F U F'D F U'R'F L F'R F L'F! 16,14



L D R'D'L'D R D!L'U'L D L'U L 16,15L'U'R'F R U!L D L'U'RaF'R'D' 16,15D!L!U L U'L D!F!R'D R D'F! 18,13

R U'R'D R U R D L!D'R!D L!D! 18,14

U'F'U F U L'U!F'U F U'F'U F L 16,15U F L F'L'F L F'L!U'L F L F'L' 16,15D R F R'F L D!B'L!F L!B L'F D' 18,15

R'F!R F U R'F!R F!R U'R!F R F! 20,15

L'F!L D!R'D'R'U'R!D'R U R' 16,13

11/25/11 10:55 AMEdge flips

Page 1 of 2http://ws2.binghamton.edu/fridrich/L1/ef.htm

Edge flipsThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


R U'B L F R U R'U F'L'B'R'U' 14R'D'L'U'F'U F'L D R U F U'F 14U F U!B'R'F'R FaU!F!U'F 16,13U'F'aR!F!R F'R'F'R!B U F 16,13U'aF!U!F U'F'U'F!D R U R' 16,13L!U!L U'L'U'L!D F U F'U'a 16,13UsF U'RsB U'FsR'F'LsD R 16,16S!F S F!S'F S'F S'F!S F S' 24,20U'F R'U F'LsD F'R D'F Rs 14RsU'F R'U F'LsD F'R D'F 14U'F'U F'L D R U F U'F R'D'L' 14U'B'R'F'R FaU!F!U'F U F U' 16,14RsU L DsF D'LsB D'FsL'F' 16[S B S'B S;F!] 20,18S F S F S F!S'F S'F S'F! 20,18R F U F'L F!L'U'F'R'F D'F!D 16,14*D L U(R Sf)!(R Sf)!U'L'D' 18U!LsF!R!sU'R!sF!RsU!F 22,14U!LsF!R!sU'F!RsU!R!sF 22,14R!U!R!F!RsU LsF!R!U!R!F 22,14[S'F!S F!S';F] 22,18

* -these algorithms have been added from 'Goodbook' supplied by Razoux Schulz

11/25/11 10:56 AMEdge triangles

Page 1 of 2http://ws2.binghamton.edu/fridrich/L1/et.htm

Edge trianglesThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


U!F RsU!LsF U! 12,9R F U F'U'R'aU'F'U F L 12

S!F'S F!S'F'S! 16,11

[R U'R';Sd] 10[Su;R F'R'] 10[Su;R'F'R ] 10R F R'D'L'U'F'U L D 10

F'S'F'S F'S'F S F S'F S 18

L D R F!R'D'L'U'F!U 12,10U'R'U!B!D!L'D!B!U' 14,9

S F S'F!S F S' 12,11

[Su;L'U L] 10[L F L';Sd] 10[L'F L ;Sd] 10R U F'U'R'D'L'F L D 10

11/25/11 10:56 AMEdge triangles

Page 2 of 2http://ws2.binghamton.edu/fridrich/L1/et.htm

S'F S F S'F S F'S'F'S F' 18

11/25/11 10:56 AMEdge cross-transpositions

Page 1 of 2http://ws2.binghamton.edu/fridrich/L1/ect.htm

Edge cross-transpositionsThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


R'aF!RaUaF!U'a 12,10[U!S!U!F] 18,10

S!F S!F!S!F S! 20,11

R'F'D'F!D R F U F!U' 12,10U F L F!L'U'F'R'F!R 12,10[S'B S B S';F] 18

S'F S F S'F S F S'F S F' 18

R U'R'F L'F R F'U'L F'U F R' 14U F'R FaU!F'aR!F!R F U' 16,13U S'U F S!F!S!F U'S U' 20,15

[S'B S B S';F] 18

U!D'B L'B'D U!R'D'F D R 14,12R'F R'D'F U'F D F'R'U F'R! 14,13RsB U B'LsF'DsR U R'D' 14RsF U F'U'F'U'F'U F LsF 14R'F'D R'D'R F R D'L'F'L F D 14R'D'L'F L F'D R D'L'F'L F D 14

11/25/11 10:56 AMEdge cross-transpositions

Page 2 of 2http://ws2.binghamton.edu/fridrich/L1/ect.htm

R'F'D'R U F'L'F L U'R'F D R 14R'F'D'L F D'L'D L F'L'F D R 14[U!S'U!S!F] 22,14[S'F!S'F!S';F] 22,18

11/25/11 10:56 AMEdge parallel-transpositions

Page 1 of 2http://ws2.binghamton.edu/fridrich/L1/eptr.htm

Edge parallel-transpositionsThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


R'F'R'D R D'F R D'F'L'F L D 14R'D'F L'F'L D R D'F'L'F L D 14D'F'D R!F U F'U'R!F'D'F!D 16,13R!F'R F R U!R'B'R B U!F'R 16,13S!F U!S!U!S!F'S! 22,12S!F S F!S!F!S F'S! 22,14S F S!F S!F S F!S!F' 22,14R'F'D'F D R!F U F'U'R' 12,11R!F U'R'D B R'B'R U R'D'R' 14,13

S F'S'F S F!S'F'S F'S'F S F!S'F 24,22

R'F'D R'D'R F R D F R F'R'D' 14R'D'L'F L F'D R D F R F'R'D' 14R'D'L'F L D F!R U L F'L'U'F! 16,14R'D'L'F!L D F R F!R'D'F'D R 16,14R'D!L D L'D R D F!D'F'D F'D' 16,14L'F L!R'D L'D!L D'R L!F'L F! 18,14L'D!U'R F D'R!D F'R'U D!L F! 18,14[S'F S F S';F!] 20,18U R U LsF'L'F'RsU LsU' 14R F U'R U R D R D L D L'F D 14

11/25/11 10:56 AMEdge parallel-transpositions

Page 2 of 2http://ws2.binghamton.edu/fridrich/L1/eptr.htm

R F'R'D'F!D F!R F!R'D'F'D 16,13U!S'F S F S'F'S F'U! 16,14S'F S F'S'F S F S'F!S 18,17

F S F S'F S F'S'F S F S' 18

U'aL FsLsF RsU!BsUa 16,15R'aU'R!F!R!F!R!F!U Ra 18,12(U R'S!fR U'F!)! 20,14

S F'S'F!S F'S!F S F!S'F S 24,20

11/25/11 10:57 AMEdge triangle and one edge flip

Page 1 of 2http://ws2.binghamton.edu/fridrich/L1/etr.htm

Edge triangle and one edge flipThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


D'F'D L B D!B'D'L'F D!F!D' 16,13*R F!R'F D'L'U'R'F'R F!U L D 16,14R F R'D'L'F'L DaL D F'D'F L'U' 16U'R'F R U L F'L'U F L'U L U'F'U' 16RsF'U!F!U!F!U!R'F'LsD R D' 20,15[S'F S F'S'F!S ;F] 24,22D L D LsB'L'B D'RsF D! 14,13F L B D'F D'F'D'F D!B'aL' 14,13*U L F'D F D'L'U'R'F'D'F D R 14U F U L'U'L F'U'R'F'D'F D R 14

F S'F'S'F!S F'S!F'S' 18,15

D'F!D!F R'D'B'D!B R D F'D' 16,13*D R U F!L F'L'U'R'D'F L'F!L 16,14U'R'F D'F'D R UaR F'R'D'L'F L 16U'F'U'R U R'F U R'F'R U L F L'U' 16D'L D RsF'L'U!F!U!F!U!F'Ls 20,15[S F!S'F'S F S';F] 24,22L F D'L D L'F'L'U'R'F'R F U 14L D R F'R'F D'L'U'R'F'R F U 14

11/25/11 10:57 AMEdge triangle and one edge flip

Page 2 of 2http://ws2.binghamton.edu/fridrich/L1/etr.htm

S F'S'F'S F'S'F'S F!S'F!S F'S'F 26,24


11/25/11 10:57 AMTwisting or flipping cubes in place

Page 1 of 4http://ws2.binghamton.edu/fridrich/L1/twfl.htm

Twisting or flipping cubes in placeThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


This is a set of algorithms for solving positions in the last layer when all the corners and edges are in placebut they are to be twisted or fliped.

Positions which appear in sections dealing solely with corners and edges are not included here.

R'D!L D L'F D R U!L'U'L F'U' 16,14U L!F!R'F L'F'RaF!L!F'U'F 18,14R F R'F'U!R!D R'U R U'aR!U! 18,14U'F'R'F R!B U!L'U L U B'R'U F! 18,15D'F'D F!D'F!R'F L'F'RaF D F! 18,15L F!L'F!D'F U'F'UaF L F!L'F' 18,15U F L'U L U!R'D'F'L D'L'D!R 16,14R F U!F!U'aF U F'D F!U!R'F' 18,14U!R!UaR'U'R D'R!U!F R F'R' 18,14U'F'R'aF R F'L F!U F!U'F U F! 18,15D'L B D'R'D'R D!B'L!F'L F D F! 18,15D F!D'F'L'aF L F'R F!D F!D'F 18,15D R!U!R'B'R B U!R!D'R U'R'U 18,14U'F!U!FaU'F'U B'U!F!R U R' 18,14U'F R D'F D F!U'B'R'D B'D'B!U! 18,15U'BsU F'aU!B U B'U!F!R U R' 18,15R'F D'R'aF'D'F D RaD!R'D'R!F' 18,16U'R U R'D R!U!B'R'B R U!R!D' 18,14R U'R'F!U!B U'F U F'aU!F!U 18,14



U!B!D B D'R B U F!D'F'D R'F'U 18,15R U'R'F!U!B U'B'U!FaU'FsU 18,15U!R U R!U'aR'F'R F UaR F'U F 18,16U L'U'L F'U'R'D!L D L'F D R U 16,15

U!R U R!F R D'F'D!R'D'R!F'R'U F 20,16

U'R'D'F'L D'L'D!R U F L'U L U' 16,15

R'D F D!L D L!F L D'F'D!R'D'R!F 20,16

U'F!L'F R'F'RaF U F!U'F'U F! 18,15

L'F!R F R'F!L!D F D'F'RsF R' 18,15

U'F U L D F D'F'D F'D'L'U'F'U F 16U F R'U'R F!R'F'D'F UaR F!U'F 18,16D'F'L'F L F'L D'L'D L D'L'D L'F L D 18R U!B L!F L'F'U'F!U L'B'U R U R! 20,16R U'R'U'B L U'F!U F L F'L!B'U! 18,15U'R'aF'R F U F!U'L U F'L'F'L F! 18,16U L F L'F'DsF U'aF!D F D'F!U 18,16U!B L!F L'F'U'F!U L'B'U R U R' 18,15R'F R F D'R'D F!D'F'L'F LaD F! 18,16U'F!D F'D'F!UaF'UsF L F'L'U' 18,16U L F L'F'L F'L'U'R'F'R F R'F R 16R F!R!F'D'F D R'B R'F'R B'R!F' 18,15R F!R!F'R F!L F'R F L'F U'R'U 18,15U!L'U'L!D R F U'R U R!D'L'F'U' 18,15D!R!U!R'B'R B U!R!D'R U'R'Us 20,15R'F'R F'R'F R U L F L'F L F'L'U' 16U!B U'F U B'U R'F'R F U!F!U'F 18,15U'R U F'L F'R'F L'F!R'F R!F!R' 18,15U F L D R!U'R'U F'R'D'L!U L U! 18,15DsR U R'D R!U!B'R'B R U!R!D! 20,15U'F'U!L'U'L!F'L'F! 12,9

U'R U'R'U!F'L F'L'F! 12,10



D'F!R'F'R!U'R'U!F'Ds 14,11R'D'F!R F U F'U'R'D F R F' 14,13R'D'F'D F R D'L'F'L F D F! 14,13UsF U!R U R!F R F!D 14,11U'F'R'U L F L'F'U'F!R U F 14,13U'F'R'F R U L'F'U'F U L F! 14,13L'B'U F'D R U R'D'F U'B L F 14

U L F!U'F'U!L'U'L!F'L!U' 16,12

U'B'R F'L D R'D'L'F R'B U F' 14

U L!F L!U L U!F U F!L'U' 16,12

R!L B'D!R D F D'F'R'B D R D'Rs 18,16

D F R UaF R F'R'U'aF'R'D' 14

U L F UaL F L'F'U'aL'F'U' 14

U F U!F'L F R'L'F R F'U!F U F! 18,15*

U F U!R U R!F R!F R!D R D!F D 20,15

L'U'R'F R U L F D F'D'F D F!D' 16,15

D F!D'F'D F D'F'L'U'R'F'R U L 16,15

U L F L'F'U'aL'F'L F D 12

D'F'L'F L UaF L F'L'U' 12



D'F'D F'D'R'aF'D'F D RaF!D 16,15

U'R'U L'U'R U'R!B!L'D L B!R!U!L F' 22,17

D'L!U!B!D'R'D B!U!L U'L D L'U L F 22,17

L'F'L!D'L'D!F'UsF U!R U R!F R 20,16

D'L!U!B!D'R B U B'D B U'B U!L!D F' 22,17

L'F'U'F U L U L F L'F'U'F! 14,13

U L F'D F D'F'L'U'F'R'F!R F! 16,14

(R'D'F D R U F'U')! 16

D B R'F L'U'R!U L F'R B'D'F! 16,14

RsB U B'R'B L F'L'B'L U'R U R'F! 18,17D F'U'F D'F'D B R B'UsF'R'D R D'F' 18(U'F'U!L'U'L!F'L')! 20,16


11/25/11 10:57 AMOne corner and two adjoining edges are correct

Page 1 of 11http://ws2.binghamton.edu/fridrich/L1/ece.htm

One corner and two adjoining edges are correctThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).


This is a set of algorithms for solving positions in the last layer when one corner and two adjoining edges arecorrect.

Some of these positions are already included in sections dealing with corners and edges and in the section ontwisting and flipping cubes in place. They are again included here for the completeness.

U!R!U'L'U R!U'L U' 12,9

R'D R'U!R D'R'U!R! 12,9

U!B'U'B U'L!D F'D'L! 13,10

L F'R F!L'F L F!R'aF 13,11

R'U'R'U R U!B U B'U F R F'R 15,14

R B'R'U'B'U F U'B U R B R'F' 14



U'F!U L!D'L U L!D L!U'L F! 18,13

R'F!R D'R'D F!D'R D 12,10

L!D!L U!L'D!L U!L 14,9

R!B'R F!R'B R F!R 12,9

L'B D!R'D R D B'L'D F D'L! 15,13L'F L F'D R'D R D!L D'L'D F 15,14

L!D F'D'F'D F L F L'D'L F'L 15,14

L!B R'D!R B'L'F!L'F' 13,10R'F!R F!D F'R'F R F D' 13,11

R'F L'F R F'L F!R'F R F 13,12

R'D'F'D F'R U L D F!D'L'U'F' 15,14



U B'U'L'B'L F'L'B L U B U'F 14

R'D R!U'R!D'R'U R!D'F!D F!

L U L'F!L U'L'U F!U' 12,10

D R!D L!D'R!D L!D! 14,9

U L D'L'U'L D L' 8

R'D'R U'R'D R U 8

L F'R F R'aF!R F R'F! 13,11

U'R!UsF'D F L'U'R!U L F' 15,12

U'R!D'R!U L F'D'F'D F L'D 15,13

U'R U F D'F D R D'R'F'R D F'R' 15U!F'U F U!L!D F'D'L!B'U B 17,13

U'R!B D'F!U'F U F'U F!D B'U 17,14



D'F U F'D F L D L'U'L D'L'F' 14D L'B!L D'F!D L'B!L D'F! 16,12

[L'B L U B U';F!] 16,14

R'F!R'B'R F!R'B R! 12,9

L D R!D'L'F U F'DsR!D'F' 15,13

L!B R'D!RsF'L B'L'F'L'F' 15,13

U'F D F D'F L!B'U B L!F U 15,13

U'R'D'R U R'D R 8

L D'L'U L D L'U' 8

L'F!L'B'R U!R'B L!F' 13,10L'F U F U'F'L F!U F!U' 13,11

L F L'F!R F'L F R'F L'F 13,12



R'D'F'D F D!L D L'D R F 13,12

U L D'L!B'L'B L'D L'U'F' 13,12

R F'L'FaU!B'R!F!L F'R F' 16,13

L D'F!D L'D'L F!L'D 12,10

R F'L'F R'F'L F 8

L'F L'F'D F!L F!L'D'F L! 15,12

D!F L'D'F!D F!L F'D'F D' 15,12



U'F!U!F U!F U L'F U F U'F'L 17,14

U'aF!U!F U'F'U'F!D R U R' 16,13U F U!B'R'F'R FaU!F!U'F 16,13

R'D'L'U'F'U F'L D R U F U'F 14

U!F'U'L'U'L!F'L!U'L U F! 16,12

U!F'U'R'U L'U'L U'F'U F R U'F 16,15

U L U L'U B!D'R D B!U 13,11

D!B R!U'R U R B'U'B R B'U D! 17,14

U'F'U!L'U'L!F'L'F! 12,9



R U'R'U F'L U L'F L U'L' 12R'D F D'R D F'D'R U'R'U 12R'F D!B'D F D'B D!F'R F' 14,12R F'U!B U'F U B'U!F R'F' 14,12R B'R!F R F'R!B R'U'R'U 14,12R B'R!F D'F D F'R!B R'F' 14,12R'B R!F'U F U'F R!B'R F' 14,12L U'F!U!F L F'L'U!F U F L' 16,13L F R'F!R!F U F'U'R!F Rs 16,13

U R'U!F D'F'U F!D F'U R U'F' 16,14

U R'D!L'D L D'L D!R U'F! 15,12

L'D R!U R'U R U'R!D'L F! 15,12

R B!D F'L!F D'B'U!B'U'R' 15,12



U F L'U L U!R'D'F'L D'L'D!R 16,14

R F U!F!U'aF U F'D F!U!R'F' 18,14

R'F L'D'L F L'D L F'R F' 12R'F D'L F L'D L F'RsF' 12R'F'U F!D'F D F!U'F R F' 14,12R'F!L F'R F L'F!R U'R'U 14,12R'F!L F'D'F D F L'F!R F' 14,12L'U!R B'R B R'U!L U'R'U 14,12L'U!R U R'U!L U'F'U'F U 14,12L F!L'F'U'F'U F L F!L'F' 14,12L'F'L F R F'U L'F L U'F R'F' 14L'F'L UaF U'R F R'UsF'U' 14U!B U!R U'R'B'R U R F R!F' 16,13

L D R F U F'U'F'R'F!D'F!L'F' 16,14

R!F R F'R'F!D'F'D F'U'R'U R! 17,14

R F U F'R'U F L F'U'R U L'U!R'F' 17,16

R B'U R U R'U'B U!R U!R! 15,12

R'F!R F'U'B'D R'D'B U F' 13,12



R U'R'F!U!B U'F U F'aU!F!U 18,14

U F'L D'F'D L'F U'F'D'F D F 14U F DsR F'R'U F'U'aL'F L 14U!F'U'L'U'B U L U'L!B'L!F 16,13

D F!R F!U F L F L'F'U'R'D'F 16,14

U R U'B'D'F R'F'D B R!F R F' 15,14

R F!D F D'F L!B'U B L!R' 15,12D R!U'aR U R'D R!U'R Us 15,13

D R B'R U'R U R!B R'U'R Us 15,14

U'F!U!FaU'F'U B'U!F!R U R' 18,14

D R F'U F DsF D'R'D F'D! 14,13R F'L'U'F'U B D L'D'FsL U R'F 16



L!F R'F'D!F!D F'D'F'D!L!R 18,13

U!R!U R!B R'U'R U R B'U 15,12

U'R!U R'U'R!UaR'U'R Us 15,13

DsR U F R F'aR F R'B R'F'D' 15

R'D!L D L'F D R U!L'U'L F'U' 16,14

U L!F!R'F L'F'RaF!L!F'U'F 18,14

R F'L U L'F L U'L'F R'F' 12RsF'L U L'F L U'F R'F' 12D'R D F'U F U'F D'R'D F' 12

R F D'F!U F U'F!D F'R'F' 14,12

D!F D'R D F'UsF'U'F R'D' 14,13

R!F R F'U'F'aR B R'F R!U 15,13



R F R!B U F'R'F U'B'D'R D F' 15,14

U'aR D'R'B R!D R!D'B'R U D! 17,14

11/25/11 10:58 AMPretty patterns

Page 1 of 23http://ws2.binghamton.edu/fridrich/ptrns.html

Pretty PatternsThis is the largest collection of pretty patterns that is known to me. It has been compiled by Mirek Goljan,who is also the author of most of the moves. Significant contributions were made by Peter Nanasy, MichaelReid, Mark Longridge, Hans Kloosterman, and others.

Contents: Notation and terminology Regular patterns

Semi-ornamentsLetters on all faces

U's ornamentsU's semi-ornamentsSupplements of the U's (Czech check problem)T's simple patternsH'sJ's ornamentsSupplements of the J's I's semi-ornamentsK'sSupplements of the K's L's

6 L's from Q26 L's from Q1

Regular ornamentsRegular ornaments -supplementsSpecials

Semi-regular patternsSemi-regular ornamentsSemi-regular ornaments - supplementsOrnaments composed from U2, U3 or U4Semi-regular semi-ornaments

Even positionsOdd positions

Cyclic patternsThree-color regular patternsThree-color regular ornamentsMore-color regular patternsMore-color nonregular patterns



Flips and twists

Patterns possible by disassemblyOrnamentsRegular semi-ornamentsSemi-regular semi-ornamentsCyclic patterns

Simple patternsOne little square of opposite colorTwo little corner squares of opposite color

Odd simple patterns

NOTATION AND TERMINOLOGYThe usual notation U, R, D, L, B, F is extended to slice and antislice moves.Us,Rs,Ds,Ls,Bs,Fs are slice moves UD',RL',DU',LR',BF',FB'.Ua,Ra,Da,La,Ba,Fa are antislice moves UD,RL,DU,LR,BF,FB.Sr, Sl, Sf, Sb, Su, Sd are the center slice moves adjoining faces R,L,F,B, U,D.[P*Q] means the same as P Q P'[P;Q] means the same as P Q P'Q'Lowercase letters u,r,d,l,b,f represent twists of the whole Cube, so that u = UsSu, r = RsSr and so on. Forexample dFR'U = LF'Ud = LF'U.Character ^ represents mirroring (usually R<->L).

Here is Mirek's classification of two-color patterns which he introduced in 1982.

1. OrnamentsA typical example of an ornament is "cube in cube". Colors of the faces are mixed: (U->R->F)(D->L->B). Axis of each pattern goes through the corners URF and DLB. In order to desribe ornaments in ashort, symbolic, form the following notation is introduced:D is an exchange of the centers (F,U,R) (L,D,B)V are twists of the corners FUR+ and LDB-S is a permutation of the edges (FU,UR,RF)S' is a permutation of the edges (LB,BD,DL)L is permutations of the edges (FU,FR,UR) (LB,LD,BD)O is a permutation of the corners (FUL,URB,RFD)O' is a permutation of the corners (LBU,BDR,DLF)H is a permutation of the edges (UL,RB,FD)H' is a permutation of the edges (UB,RD,FL)

2. Semi-ornamentsEvery face has at most two colors which are mixed: (F<->R)(B<->L)(U<->D)D is an exchange of the centers (FR)(BL)(UD)W are flips of the edges FR and BL.



O is an exchange of the edges FL and RB.R is an exchange of the corners FRU and RFDR' is an exchange of the corners BLU and LBDK is an exchange of the corners FLU and RBDK' is an exchange of the corners FLD and RBUS are permutations of the edges FU, FD, RU, RDS' are permutations of the edges BU, BD, LU, LDD is used to shorten the notation. [D] = [WORR'KK'S(14)(23)S'(14)(23)]The alphabetical order of letters is: D,W,O,R,K,S.Patterns containing all of R, R', K, K', S, S' are mostly denoted as [D...]. You should use mirroring (F<->R) to get K rather then K' and S(1.. rather then S(3.. or S(2.. .

3. Cyclic patternsEvery face has at most two colors which are mixed: (F->R->B->L)(U<->D)

4. Simple patternsEvery face is a simple two-color combination of colors from opposite faces of the Cube: (U<->D)(R<->L)(B<->F)

5. SpecialsOther permutations of colors, for ex. (F->L->U->B->R->D) or (F->B->L->R) (U<->D)

REGULAR PATTERNSRegular patterns are patterns which have all faces of the same look except for coloring and mirroring.

Semi-ornaments

F U L'B°L F!D'F D F L'D'L F L'D B°LU'F' 21,20 [OKS(14)S'(23)] 2 peaks

° can be replaced by the same but otherwise arbitrary exponent

U F'D'L B!L'D F D L!U'aF!D'F!L! 21,16 p117 Improve RCC3.9.36 (K)

D U!F!D!sR!sB!R!sD 20,11,8 [RR'KK'] crossesD R!sDaRaF!sRaU 16,12,10D RsFaL!F'aLsD'aL!U 16,14,12

U'R'F'U'R!sD B R'L!D!B!D!sB!U' 22,16,14 [ORR'KK'S'(12)] p147 ML's 6 ARMPattern (K)

R U'F R'B'D!R'U'BsLsD B L U'R'U'FU'Ls 23,22,19 [DK] MG.1982

RaB'D'aL'R!D!F!R F'aU'F!R!U D!L! 25,18 p68 Cherries (K)D'F!DsB!RsU RsB'R!B LsU RaB!U 24,20,16 [DOS(23)S'(14)](R!D L!)^4 Ds 22,14,10 [RR'KK'S(1342)S'(13)] p163 4 ARM Full



Letters on all faces

The U's - ornaments

BsU F'LsD F'U'BsR 12,12,9 U1 = [DH] P.NánásyF'R!D'RsF RaF D B LsU'F' 16,15,13 U2 = [DL] E.V.ChacønU B U'LsDsBsD R'UsB U'B' 16,16,12 U3 MG.1983B R D'R FsD'RsF UsL'F U'L' 16,16,13 MG.1982L D BsR'F L'F'DsR F R'D' 14,14,12 U4 MG.1982R B'LsD U!BsR'L!DsF D' 16,14,11R B!F UsL'FsU RsB'F!D' 16,14,11L'F'aR'DsB R F LsD'L D 14,14,12 U5 MG.1982

The U's - semi-ornaments

R'aB!RaD'aL BsD'B!D FsL'D! 20,17,15 U6 [0+0] MG.1982B LsD'LsB!RsD'LsF'aD!sFDs 22,19,13U'LsF'UsL!B!L'BsD B! 16,13,10 U7 [4+0] MG.1983F'L'FsR'F'D'F R BsL B L!BsD' 18,17,14 MG.1982B F!D!BsR'BsL'F DsR'F RsB'R 20,18,14 U8 [2+2] MG.1982R'F!LsD!B RsF L UsB'L'BsR'B 20,18,14 U9 [3+1] MG.1982L'B'LsF'L'UsB L FsR F U!FsR! 20,18,14 U10 [1+3]

Supplements of the 'U's

Here are some positions with exactly 8 squares correct on every face. The remaining positions are discussedin the article "A Czech Check Problem" by Michael Reid, Cubism For Fun 36, Feb 1995.

LsF L'FsU F'U'BsR 12,12,9 DU1 = [H]R D'LsF'RsD LsF L' 12,12,9F'R!D'RsF RaF D F UsL'U' 16,15,13 DU2 = [L] MG.1982F L F'RsD R'UsB U'B' 12,12,10 DU3 /E5/ MG.1983U L B'D BsR'FsD BsR'F L'U'=(ULB'DBsR'FL'U')! 16,16,13 MG.1982L U B'D BsR'FsD BsR'F U'L'=(LUB'DBsR'FU'L')! 16,16,13 MG.1982U L'DsF'UsL DsF D' 14,14,12 DU4 MG.1983UaL'DsF'UsL DsF D! 14,13,10 DU5



D'R'aD'RsF R F LsD'L D 14,14,12 MG.1982

The T's - simple patterns

R!U!R!U!R!FsU!Fs 16,10,8 T-ordinary MG.1982L!D!R!D!L!FsU!Fs 16,10,8 MG.1982L!D!B!D!F!L!B!U!B!U! 20,10 p97 6 T's AFaU!FaR!D!R!U!R!B! 18,11,10 T-orthogonal MG.1982F!D!R!U!B!D!B!R!B!D! 20,10

The H's

R!BsR!sBsL! 12,8,5 H1. - simple patternL!(U!F!s)!L!F!s 20,10,7 p174 6 H order 2 type 1FaR!F!sD!sL!F'a 16,10,8 H2. - simple patternRaDaRaD!sRaDaRa 16,14,13R!B!U!R!L!U!F!R! 16,8 p175 6 H order 2 type 2R'aUaR!sF!sUaRa 16,12,10 p175aR!F!sR!D F!sU!R!sD 20,11,8 H3. - semi-ornamentsU!R!sD F!sR!F!sR!U 20,11,8R!sU'F!sL!F!sL!D 18,10,7 H4. - cyclic patternD'L!F!sL!F!sD R!sDs 20,12,8R!sD'FsR!BsRsB!RsU 18,14,9 H5. - specialsD!R!F!U'F!sD R!sF!L!D! 22,12,10

The J's - ornaments

F'R!D'RsF RsF D'R!D F'L!F D'R!D!B LsU'F' 28,23,20 DLV MG.1982U'R B'R'U R'D!F U F U'F'D'U!BsLsD F 22,20,18 U4V MG.1997R'U'B L'B'L!U R U'L'U L D BsLsU'L F'L!F 24,23,21 U4V MG.1997R!F'D'F!L F'R F R'aF'D B LsDsF'U! 22,19,17 U5V MG.1997R!F'D'F!L F'L DsBsL D'aR'B R B'U! 22,19,17 U5V MG.1997R'B'L!U'F'U L'B R!B'L'B R DsBsL Da 22,20,18 U5V MG.1997R!F'D'R'D F L'F'aR F R'B RaU'BsL Da 22,21,20 U5V MG.1997

Supplements of the 'J's



R!U!B'U R B DaR B R'B'D'aR'B'U F!D'F!R! 26,21 LV MG.1997U'R B'R'U R'D!F U F U'F'D!U'R U 18,16 DU4V MG.1997

R!F'D'F!L F'R F R'aF'D F D'R! 18,15 DU5V MG.1997

The I's - semi-ornaments

U!R!D!U!R!D'a 12,7

The K's

There are 5 "K"-ornaments that can be obtained as a composition of "U"- ornaments and crosses-ornaments.Another 5 "K"-semi-ornaments can be obtained as a composition of "U"-semi-ornaments and crosses-semi-ornaments, and 2 "K"- simple patterns can be obtained from the one-edge-little-square simple patterns

D'aFsD!BsUsFsD!BsBsU!FsR!U!R!U!R!

composed with a chessboard simple pattern.

Supplements of the 'K's

D'L'DsF R D'LsB L'R!B'UsR!DsB'R!D!R UsLsD'F' 32,28,22 [DLHH']

The L's - simple patterns

The first 14 types are members of the square group Q2. I tried to get algorithms minimal in q-moves andusing a square group method too. Next 10 types are members of the group Q1. Q1 contains simple patternswhich are not achieveable by square moves. For example

U'F R F!R'F'U!R'F'R!F R U' 16,13U'F R F!R'F'U!F R F!R'F'U' 16,13U'R F'R'F R F'R'F R F'R'F U 14

Q0 is the group generated by [F!,R!,B!,L!,U!,D!,(FUL,FDR)(FL,FR)].Q0 consists of all positions whose colors of faces are two-color combinations of opposite faces of the Cube:(FB)(RL)(UD).Q2 is the group of positions generated by [F!,R!,B!,L!,U!,D!], It includes the three-cycle of edgesR!U!L!B!R!U!L!F! , E2+2 L!F!L!D!F!L!F!D!.Q1 is a group of positions generated by [F!,R!,B!,L!,U!,D!,(FUL,BUR,FDR)] - Q2,Q3 = Q0 - Q1 - Q2 are odd positions. For example



D'F!U L!U'F!D'R!U F!U'R!D 19,13 This pattern is not pretty one, of course

6 L's from Q2

There are four groups of relative 6L patterns (14=1+1+4+8). A short bridges of the type X!Y! exist betweenthe members of the group.

The first of them has the most interesting symmetries.

UsBsR!FaD'aF'aD!Fa 16,14,12 L1 MG.1982F!L!B!U!L!D!L!U!B!U!B!R! 24,12 p176 6 LFaL!F'aR!UsBsR!F'aUs 18,15,12 L2L!D!B!D!B!L!B!L!B!D!R!U! 24,12 MG.1997F'aD!BsLsD!Ra 12,10,8 L3 MG.1982R!D!R!F!R!F!R!F!D!F!R!s 24,12 MG.1997F!sL!D!L!B!L!F!L!F!D!B! 24,12 MG.1997F'aUaL!DsBsR! 12,10,8 L4 MG.1982R!D!R!F!R!F!R!F!D!F! 20,10 MG.1997R!U!L!B!L!B!L!B!U!F! 20,10 MG.1997FaD!BsLsU!R'a 12,10,8 L5 MG.1982R!D!R!F!R!F!R!F!D!B! 20,10 MG.1997R!U!L!B!L!B!L!B!U!B! 20,10 MG.1997FaUaR!DsBsL! 12,10,8 L6 MG.1982F!sL!D!L!B!L!F!L!F!D!F! 24,12 MG.1997R!D!R!F!R!F!R!F!D!B!R!s 24,12 MG.1997

{L3} + R!s = {L4}, {L3} + F!s = {L6}{L4} + F!s = {L5}, {L4} + R!s = {L3}{L5} + R!s = {L6}, {L5} + F!s = {L4}{L6} + F!s = {L3}, {L6} + R!s = {L5}

FaD!L!Fa 8,6 L7 MG.1982U!B!R!B!U!B!D!F! 16,8 MG.1997D!F!L!B!U!B!U!B! 16,8 MG.1997FaD!L!FaD!R! 12,8 L8 MG.1997U!B!R!B!U!B!D!F!D!R! 20,10 MG.1997B!L!U!L!BsD!Fa 14,9,8 L9 MG.1997



D!F!R!D!F!D!F!D!F!D! 20,10 MG.1997FaD!L!FaU!L! 12,8 L10 MG.1997

U!B!R!B!U!B!D!F!U!L! 20,10 MG.1997D!F!R!D!F!D!F!D!F!R! 20,10 MG.1997DsR!sDsF'aU!R!Fa 16,12,9 L11 MG.1997U!B!D!sR!F!D!F!D!F! 20,10,9 MG.1997DsL!UsFsR!F'aD! = L!UsB!RaF!R'aD'a 14,11,8 L12 MG.1997U!B!D!sR!F!D!F!D!F!D!R! 24,12,9 MG.1997U!B!R!B!U!B!D!F!D!R!U!s 24,12,9 MG.1997LsU!sLsF'aU!R!Fa = BsD!RsF!U!F!R'aFa 16,12,9 L13 MG.1997B!L!B!L!B!D!R!sF!R! 20,10,9 MG.1997DsR!UsBsR!F'aU! = R!UsF!RaF!R'aDa 14,11,8 L14 MG.1997= FaR!F'aR!UsF!Ua = FaR!F'aUaR!UsF! 14,11,10 MG.1997D!F!R!D!F!D!F!D!F!R!U!s 24,12,9 MG.1997

{L7} +D²R²={L8}, {L7} +U²L²={L10} {L8} +R²D²={L7}, {L8} +L²U²={L9}, {L8} +U²D²={L12}, {L8} +F²B²={L8}^ {L9} +D²R²={L10},{L9} +U²L²={L8} {L10}+R²D²={L9}, {L10}+L²U²={L7}, {L10}+U²D²={L14}, {L10}+F²B²={L10}^ {L11}+D²R²={L12},{L11}+U²L²={L14} {L12}+R²D²={L11},{L12}+L²U²={L13},{L12}+U²D²={L8}, {L12}+F²B²={L12}^ {L13}+D²R²={L14},{L13}+U²L²={L12} {L14}+R²D²={L13},{L14}+L²U²={L11},{L14}+U²D²={L10}, {L14}+F²B²={L14}^

In this group ^ means F<->B symmetry.

A schema of short bridges:

Connections between L3, L4, L5 and L6 are of the type F!B!.Connections between L7,L8,L9,L10,L11,L12 and L13 are of the type F!R! and F!B!.

L7 R²s / \ L3 --- L4 = L8 - L9 -L10 = F²s | | F²s U²D² | | U²D² L6 --- L5 =L12 -L11 -L14 = R²s \ / L13

= ... F²B² bridge to its own mirroring

Now let us consider bridges of the type F!R!sB!. In this way, we get connections between L1, L2.

{L1} + U²R²sD² = {L2}



{L1} + U²R²sD² = {L2} {L1} + F²U²sB² = ru{L2} {L1} + R²F²sL² = dl{L2} {L1} + D²R²sU² = {L2} {L1} + B²U²sF² = ru{L2} {L1} + L²F²sR² = dl{L2} ru.ru=dl, ru.ru.ru=identity

L1 / | \ L2 L2 L2 | \ | \ | \ L2^ L2^ L2^ \ | / L1°

° ... center of the cube symmetry (FB)(RL)(UD)

We also get new connections between L3,L4,L5,L6.

{L3} + D²R²sU² = {L5}^ (F<->B mirroring) {L4} + D²R²sU² = {L6}^ (F<->B mirroring) {L5} + D²R²sU² = {L3}^ (F<->B mirroring) {L6} + D²R²sU² = {L4}^ (F<->B mirroring) {L3} + F²R²sB² = {L6}^ (R<->L mirroring) L3 L4 {L4} + F²R²sB² = {L5}^ (R<->L mirroring) | X | {L5} + F²R²sB² = {L4}^ (R<->L mirroring) L6 L5 {L6} + F²R²sB² = {L3}^ (R<->L mirroring)

{L7} + L²F²sR² = {L9}^ (F<->B mirroring) {L11}+ L²F²sR² = {L13}^ (F<->B mirroring) {L7} + F²U²sB² = rr {L9} {L9} + F²U²sB² = rr {L7} {L8} + F²U²sB² = rr {L8} {L10}+ F²U²sB² = rr {L10} {L11}+ F²U²sB² = rr {L13} {L13}+ F²U²sB² = rr {L11} {L12}+ F²U²sB² = rr {L12} {L14}+ F²U²sB² = rr {L14}

About inversions: {L1}' = {L1}^ any mirroring along axis FRU--BLD {L2}' = {L2}^ (B<->U)(F<->D) {L4}' = {L4}^ (F<->R)(B<->L) {L3}' is not 6 L's {L5}' = f²u{L5} {L6}' is not 6 L's {L7}' = {L7}^ (R<->U)(L<->D) {L8}' is not 6 L's {L9}' = {L9}^ (R<->D)(L<->U) {L10}' is not 6 L's {L11}' = {L13}^ (F<->B)(R->U->L->D) {L12}' is not 6 L's {L13}' = {L11}^ (F<->B)(R->D->L->U) {L14}' is not 6 L's

U R!U R!D!F!D R!F!U!F!U!F!D' 24,14 L'9 MG.1997U R!U'B!D!L!U'F!U'F!D!B!D!F! 24,14 MG.1997D!F!L!U!D B R B!R'B'D!R'B'R!B R D 24,17 L'10 MG.1997

{L'4} = {L'3} + RsF²LsD² f²d{L'4} = {L'4} + UsB²DsL² {L'5} = {L'3} + UsB²DsL² f²d{L'5} = {L'4} + BsU²FsR² {L'6} = {L'3} + BsU²FsR² f²d{L'6} = {L'5} + BsU²FsR²



{L'9} = {L'8} + LsD²RsF² f²d{L'9} = {L'10} + RU²sL²U²sR {L'10} = {L'8}+ FsL²BsD² f²d{L'10} = {L'10} + DsL²UsB² f²d{L'10} = {L'6}+ R²F²R²F²R²F²

About inversions:

{L'1}' = {L'1}^ any mirroring along axis FRU--BLD {L'2}' = {L'2}^ (F<->R)(B<->L) {L'3}' = {L'3}^ (F<->R)(B<->L) {L'4}' = {L'6}^ (F<->R)(B<->L) {L'5}' = {L'5}^ (F<->R)(B<->L) {L'6}' = {L'4}^ (F<->R)(B<->L) {L'7}' = {L'7}^ (F<->R)(B<->L) {L'8}' = {L'9}^ (U<->D) {L'9}' = {L'8}^ (U<->D) {L'10}'= {L'10}^ (F<->R)(B<->L)

There are two cases for the 6L's with respect to the corners:

a) the two opposite corners are in place (L1,L2,L7,L9,L11,L13), (L'1,L'2,L'7).b) the two adjoining corners are in place (L3,L4,L5,L6,L8,L10,L12,L14),(L'3,L'4,L'5,L'6,L'8,L'9,L'10).

Only two types have the same orientation for all L's (L1,L6).

Regular ornaments

F D R D'L'D R'D'F'R'F L F'R 14 [V]L'D R'D'L D F L F'R F L'F'D' 14 [V]R'D!R B'U!B R'D!R B'U!B 16,12 [V]LsF L'FsU F'U'BsR 12,12,9 [H]R D'LsF'RsD LsF L' 12,12,9 [H]L'DsB D B'UsR D'Ls 12,12,9 [H]LsF!R!D'FsU'F!sDFsDR!F!Rs 24,18,13 [SS'] MG.1982R!F LsF RsU'F'U'F'RsU LsU R! 20,18,14 [HH'] MG.1983R'B L B D'B D FsL'B!F U F'U'F'R 18,17,16 [HH'] Nánásy.1983U'L'F L'F'R'F LsF'L F R F'R U 16,16,15 [OO'] MG.1982U'R F'R'F L F'RsF R'F'L F L'U 16,16,15 [OO'] Nánásy.1982U L D R!U R U'R B'D B'D'B!D'L'U' 18,16 [VH] MG.1997R U'R B'D B'D'B L B'U R'U R U!B!R!L'U 22,19 [VSS'] 2 peaks MG.+JF.R U'R B'D B'D'B L!U!F U F'U L'B R!L'U 22,19 [VSS'] 2 peaks MG.+JF.? [VHH']L!R'F D!L'F'RsFsL F U!L'B F! 20,16,14 [VOO'] crosses MG.1997= C.ful.C', C = L!R'F D!L'F'Rs derived from p1



R'F!sR!U!sL'BsL'F!sL!U!sR'Fs 28,18,12 [VOO'] MG.1983R'F!sR!U!sL'BsR'F!sR!U!sL'Fs 28,18,12 [VOO']B'L F'L F R'F'L F R B'R'B R F'LsB'R B!L F L' 24,23,22 [VOO'] MG.1982F R!sFaR!sF L B!sRaB!sL U!s 28,18,13 [VOO'] p101 (Plummer's Cross) (K)? [SS'H]U'F!U R U!R!U!R'F'R!F U'F!U 20,14 [HOO'] MG.1982B'L'D L'D'L'F LsF'L F R F'L B 16,16,15 [HOO'] MG.1982D L'B R D'R'D B'L B R'B'R D' 14 [HOO'] p87 Twisted Cube Edges

Regular ornaments - supplements

R'D!R B'U!B R'D!L DsBsR'F!L 20,16,14 [DV] MG.1982U L F R'DsBsR'D R'D'F'R'F L F'R 18,18,16 [DV] MG.1982BsU F'LsD F'U'BsR 12,12,9 [DH]B R B'R!U'F'L U R D L U'B'L U L! 18,16 [DSS'] p86 Twisted RingsLsF!R!D'FsU'F!sD LsDsB!F'U!R!Us 28,21,15 [DSS']B'R'U'LsF R U!F R BsU'F'L' 16,15,13 [DHH'] snake R. Schoof= C.u.C'^, C = B'R'U'LsF R UU B!L D FsL'DsRsF'D!R' 16,14,11 [DHH'] p121 Improve RCC 3.9.31U'R F'R'F L F'DsBsL U'R'D R D'F 18,18,16 [DOO'] NánásyU'R'DsBsL D'R'U'R DsR'D R U R'U F 20,20,17 [DOO'] MG.1982U L D R!U R U'L DsBsR'B L'B'L!B'D'F' 22,20,18 [DVH] MG.1997L F L D'B D L!F!D'F'R U'R'F!D 18,15 [DVSS'] (cube in cube)= C.rf.C^, C = L F L D'B D L!F'U!F!R!U'L!D B R'B R'B R'D'L!U' 20,15 [DVSS'] p7a (K)F U!F'R'F!R!F'R U L D F'D'R!L'U'F U!F' 24,19 [DVSS'] MG.1982F R B'L B R DsL'F'L UsB'R B R'U'R'U R'F' 22,22,20 [DVSS'] MG.1982U F D'F!D F'R'F!R U'aB'U B!U'B L B!L'D 24,20 [DVSS']U F'U!B U B'U!F U'LsB D'B!U B'U'B!D L' 24,20,19 [DVHH']L!R'F D!L'F'DsBsD R F!D'L R! 20,16,14 [DVOO'] p1 6 X of order 3= C.urb.C', C = L!R'F D!L'F'DsR'F!sR!U!sR'DsBsU'R!sU!F!sD'Ls 30,20,13 [DVOO'] chessboardFsR!sU B!sU!R!sU L'U!sR!B!sL'Bs 32,20,13 [DVOO']p1aRsF'DsB'D!sF DsBaDsF D!sB'DsB'Ls 28,24,16 [DVOO'] MG.1982? [DSS'H]B'L'D L'U'BsLsU'R DsR'D R U R'D L 20,20,17 [DHOO'] MG.



LsDsBsLs = DsBsLsDs = BsLsDsBs 8,8,4 [D]R'DsBsLsD = U'BsLsDsB = F'LsDsBsL 8,8,5 [D]L DsBsLsU'= D BsLsDsF'= B LsDsBsR' 8,8,5 [D]

Specials

F RsU!F!sU'aFsL'U R!sU!F!sD 26,18,13 chessboard of order 6 (K)p2 6 X of order 6

SEMI-REGULAR PATTERNS

Semi-regular ornaments

R'F'L BsD'F'D FsL'F!R 14,13,11 [S] (easy)R U!L'UsB U'B'DsL U'R' 14,13,11 [S]U F'U'F U RaU F U'F'R'aU' 14 [S]R F!B'D'S'fD!SfD'B F!R' 16,13,11 [S]R U F'U!sB U'B'U!sF R' 16,12,10 [S] p196D R B'L!B R'B'L!B D' 12,10 [O] (easy)D'L B!L'F'L B!L'F D 12,10 [O]R'U F!D'F'D F!U'F R 12,10 [SO] (easy)R'F!L'D'L F'L'D L F'R 12,11 [SO]R'F'L F'D'F'D F L'F!R 12,11 [SO]R'F'L BsD'F'D B'L'B L F L'B'F!R 18,17,16 [SO'] MG.1982D L U'F'L'B'U B L F R!B'R'B R'aD' 18,17 [SO'] MG.1997U R'F!R F D'F!D F'U' 12,10 [HO] (easy)U F R'F R F D'F'D F!U' 12,11 [HO]U'L!D B'D'L D B D'L U 12,11 [HO]FsU'RsU'R!sD RsU Bs 16,14,9 [SH] MG.1982B!R U!B!L'B'L B'U R'U B! 16,12 [VS] JF.1982B!R U'R B'R'B!U'B'U R'U B! 16,13 [VS]B!F'U F U'R U'B U'B'U R'U B! 16,14 [VS]B!F'U F U'R U'R'U R'F R F'B! 16,14 [VS]B'L F'L!FsR'B R F'LsB'R B!L F L' 20,18,16 [VO] MG.1982F D R D'L'B'U'B D B'U B R'D'F'R'F L F'R 20 [VO]LsU FsD F!sU'FsU'F L'FsU F'U'BsR 24,22,16 [SHH'] MG.1982



R U F'U!B U'FsD F'U!F D'R' 16,14,13 [SOO'] Nánásy.1982R'F'L F'D'F'D L'B L F L'B'F!R 16,15 [SOO'] MG.1982R'F!L'D'L F!B D B'D'FsL B L!D L F'R 22,19,18 [SS'O] MG.R'D B D B R'U!R B'D'R!B'D'R 16,14 [SOH] MG.1982R'D B R'U'B L U!L'B'U R'B'D'R 16,15 [SOH]U'FsUsR F!R'DsR'B R!F'R U 18,16,13 [SOH]R'F'L FaD'F D FsL'U F!U'R 16,15,14 [SO'H] Nánásy.1982R'F'L BsD'F'D B'L'F U F U'F R 16,16,15 [SO'H] Nánásy.1982B D L'D!R D'FsUsR U'L!U L'D' 18,16,14 [VSH] MG.1982R'D'R!B R'F'R B'R!F R'D R F R F' 18,16 [VSO] MG.1982? [VSO']BsR'D B!D'R F!sU L'B!L U'Bs 20,16,13 [OHH']B!D L'F'D!F L D'L'D F L!F'D'L B! 20,16 [OHH']B!D L'F'D!L'F U L!U'F'L F L D'B! 20,16 [OHH'] MG.1983B D R'B'D L F'D'L B!L'D F L D'B R D'B' 20,19 [OHH']D'L F R U!L U!R!U!L'U R!U R'F'L'D 22,17 [VHO] MG.1982

Semi-regular ornaments - supplements

R F!B'D'BsRaDsF'R!L U' 16,14,12 [DS] MG.1982B'D L!D'R'D L!U'BsLsD L 16,14,12 [DO] MG.1982D'L B!L'F'L B!R'DsBsL B 16,14,12 [DO] MG.1982R'U FaLsDsF'R'B R!F'R U 16,15,13 [DSO] MG.1982R'F'L F'D'F'D F R'DsBsRaU 16,16,14 [DSO] MG.1982R'F'L BsD'F'D B'L'B L F R'DsBsR U 20,20,17 [DSO'] MG.1982U R'F!L DsBsL B'R!B R'F' 16,14,12 [DHO] MG.1982U R'F!R B LsDsF'R!B R'F' 16,14,12 [DHO] MG.1982FsU'RsU'R!sD!U'BsLsF Ls 20,17,11 [DSH] MG.1982BsLsD U!RsU'R!sD RsU Bs 20,17,11 [DSH] MG.1982B!R U!B!L'B'L F'LsDsB U'F L! 20,16,14 [DVS] MG.1982B!R U!B!R'DsBsR'D L'F U'F L! 20,16,14 [DVS] MG.1982B'L F'R'aDsBsU'L U R'DsL'U L!D R D' 22,21,18 [DVO] MG.1982F!L'F R UaF'aRaD'L D'L!R'U!F!L 22,18 p20 Mark's Pattern 1 (K)LsU FsD F!sU'FsU'F R'DsB R'F'LsU 24,22,16 [DSHH'] MG.1982R'F'L F'D'F'D L'B L F R'DsBsR U 18,18,16 [DSOO'] MG.1982? [DSS'O]



R'BsL F!R'DsR'B R!F'R U 16,14,12 [DSOH] Nánásy.1982U'B L B L U'F!U L'B'U!L'F'LsDsB 20,18,16 [DSOH] MG.1982R'F'L B!LsDsF'R B RsD'F R!F'U 20,18,15 [DSO'H] Nánásy.1982D L B'L B L U'F L'F'L'U L'D' 14 [DVSH] MG.1982D L U'L!U L'FsL F'L!B L'D' 16,14,13 [DVSH] MG.1982R'D'R!B R'F'R B'R!F R'D L DsBsL U R 22,20,18 [DVSO] MG.1982R'D B D L'FsU B U'BsLsU!R B'D'R!B'D'R 24,22,19 [DVSO] Nánásy.1982? [DVSO']B'L!F L'D'B'U RsB'D F D'B L B L'B D 20,19,18 [DOHH'] MG.1982? [DVHO]

Ornaments composed from U2, U3 or U4

D F'UsL'DsF UsL!F'L DsF'LsD BsR'D B'R' 26,25,19 [DU3U4]U L F'L DsF'LsD BsR'D LsU'L'U'F'aU'BsL F!U 28,27,22 [DU2U3]=[LU3]

Semiregular semi-ornaments

Even positions

RsD!F R!DsB!R BsD!L!U F!sU' 24,17,13 [WOS(1324)S'(14)(23)] (K)p138 Mark's Pattern 5

D'F'D F DsB L'D L UsB'D'B R'F'D'F'D 20,20,18 [WORKS(14)] MG.1982B R D!L'D R!D'L D!R'B R!B! 18,13 [WRR']U'F!U F!R'B'U'F!sD B L'F!U'R!D 22,16,15 [WRR'S(1423)S'(1423)]

p158 Mark's Pattern 14U'B!RaU!R'B'U L'B UsR'B U'R B'DsB!R!U 26,22,20 [WS(14)S'(23)] WalkerB'U'FaR U'R!U R'F'aU B 14,13 [OKK'S(14)]B'D'RaB L'F!L B'R'aD B 14,13 [OKK'S(14)]L'DsF D'R!D F'UsL 12,11,9 [ORKS(14)]D R!U'aF'aR!FaR!U 14,11 [RK]D R!sU!B D!R!sU!F D 18,11,9 [RK]

p181 Mark's Pattern 18R'(D B D'B')^3 R 14 [RR']FaR!sF'aU'F!UaF!U D! 18,13,12 [RR'S(14)(23)]D'F!D'aR!sB!R!sD!U 18,11,9 [RR'S(14)(23)]D FaR!F'aD!R'aB!RaD 16,13 [RR'S(14)(23)S'(14)(23)]



4 T's type 1 Nánásy.1982D'B!RaF!UaB!U'aB!R'aD 18,14 [RR'KK'S(14)S'(23)](F!U F!sU B!)!Us 22,14,10 [RR'KK'S(1342)S'(24)]

p163 4 ARM FullU'L'B'U'R!sD F L'R!D!F!U!sF!U' 24,16,14 [RR'KK'OS(12)]

p147 ML's 6 ARM Pattern (K)U RaF!R'aU' 8,7 [RKS(14)(23)]D FaR!F'aD' 8,7 [RKS(14)(23)]U R'aB!RaU' 8,7 [RKS(14)S'(23)]D F'aL!FaD' 8,7 [RKS(14)S'(23)]U'R F!sL!F!sR'U 14,9,7 [RKS(14)S'(14)]D L!D'aBaL!B'aL!U = D'RaB!R'aB!UaB!U' 14,11 [RKS(23)S'(23)]U'R!UaR!D' 8,6,6 [KK'S(14)(23)]D'F!DaR!sB!R!sU' 16,10,8 [KK'S(14)S'(23)]U'FaR!F'aU!R'aF!RaU' 16,13 [KK'S(14)(23)S'(14)(23)]

4 T's type 2 Nánásy.1982RsDsR UsF!LsD = SrSfD SbD!SlD 12,11,7 [S(13)S'(24)](U L!B!R!D)!L!R!B!F! 24,14,12 [S(13)(24)S'(12)(34)]

p165 2 Swap, 4 H fullD'R!F'aD!FaR!F!D 14,10 [S(14)(23)]D F!R!F!D!R!F!R!D 16,9 [S(14)(23)]D'R!FaU!F'aR!F!D 14,10 [S(14)S'(23)]Sf!Sr!D Sf!D!Sr!D 20,11,7 [S(13)(24)S'(13)(24)]Sr!Sf!D Sf!D!Sr!D 20,11,7 [S(13)(24)S'(13)(24)]U'R!UaR!sD'aL!U 14,10,9 [S(14)(23)S'(14)(23)](D R!U )^4 16,12 p162 2 X, 4 H full(F!sD F!sU)! 20,12,8 [S(134)S'(134)]

p170 2 edge swap + 2 HR'B R!L'DsFsU RsB'D'L'D Rs 18,14,11 [DS(14)]RaF!sR'aDs 10,8,6 [DWO](D R!FaU)^3 U!sR!U!sL! 30,21,19 [DWORK]

p178 2L, 2 Fours, 2 armD'L!FaD!F'aL!D!sF!U!D' 20,13,12 [DWOS(14)(23)]U!F!D!U!F!Da = U F!U!D!F!D!U 12,7 [DWOS(12)S'(12)]U F!D!sF!D!U 12,7,6 [DWOS(14)S'(14)]



F!U!RsD!sLsU!B!Us 18,12,8 [DWOS(12)(34)]S'(12)(34)]U!F!R!sD!R!sD!F!Da 20,11,9 [DWOS(12)(34)]S'(12)(34)]B'U'R F'U RsB'U R'B U'RsFaUs 18,18,15 [DO]R D'B L'D F!sU'R B'U L D!sR!Us 22,17,14 [DO]U R'B'U'F!sD B L'D F!R!F!U'F!D' 22,16,15 [DORR'] (K)

p93 Mark's Pattern 2R F U FsR'F'R F'R DsB'L'R!D'F' 18,17,15 [DOS'(143)] MG.1982D R!UsB!R!B RsD'R!D LsB'U' 20,16,13 [DOS(14)S'(14)] MG.1981D R!UsB!U'R!B RsD'R!D LsB' 20,16,13 [DOS(14)S'(14)]

p81 Snake type 1D R!UsB'LsB'R'B RsD'R D B'U' 18,17,14 [DOS(14)S'(14)] MG.1981D R!UsB RsB R B'LsU R'U'B U' 18,17,14 [DOS(14)S'(14)] MG.1981F!sLsB'DsL!F!L'FsU B!D' 20,15,11 [DS(13)(24)S'(13)] (K)

p146 Mark's Pattern 9

Odd positions

R D'L!B L B'L U!R'F R U!D R' 17,14 [OK]R!F!D F D'F L!B'U B R!s 17,12,11 [KS(14)]L!F L'D'L F'L'D F'D'F'D F L' 15,14 [KS(14)]L F'D'aF D F'U F!D'F D F'L' 15,14 [KS(14)]F!D F!D L!D F!D'F!D'L!D!F!D'F!D' 25,16 [KS(14)]R!U'F'U F'L!B D'B'L!F!R! 17,12 [KS(13)]F'D'F L!B'U B L!F!D F! 15,11 [KS(12)]RaDaR'D'R U'R!D R'D'Rs 15,14,13 [KS(23)]D!B!R!D'R'D R'B!L U'L'D! 17,12 [RS(13)]U!F!L!D'L'D L'F!R U'R'U! 17,12 [RS(13)]U R'F R'B'R'B R'F'U'R U R'U! 15,14 [RS(13)]U R'B U'B'U'B U R U R'B'R U! 15,14 [RS(13)]U!L!U B!U'B!D L!D'B!D B!U! 21,17 [RS(1324)]R D'F!D F'R F UsF DsF'R!D F D'R D F'D' 23,21,19 [ORKK'S(14)(23)] MG.1983D'RaB!L'U'aL!B L B'L U!R'F R DsR'D 23,20,19 [ORR'K'S(23)S'(14)] MG.1983F!D!sF R'D R!D'R F'U R!U!R!D!F! 25,16,15 [ORR'K'S(23)S'(14)] (K)

p119 Mark's Pattern 3R U'F R'B'D!R'U'BsLsD B L U'R'U'F U'Ls 23,22,19 [DK]



Cyclic patterns

R!sU F!sU!R!sD'F!sD! 22,12,8 crossesR!sD!F!sD R!sU!F!sU' 22,12,8 crossesU R!sU'R!sU!R!sD'R!sU' 22,13,9 crosses + 2 X'sR'B'RsF'R'SdR F LsB R 14,14,11 4 U's MG.1981F R'B'D!L'UsB D!R F L' 14,12,11 p112 4 U order 4R'B'RsF'R'U R F LsB R 13,13,11 4 T's MG.1981R!sF!sD'F!sR!s 17,9,5 4 Y'sD R!sDsR!sU'a 13,8,5 4 Y's + 2 X'sFsL F!L!UsB U!RsB!D L!U' 20,15,12 p114 3 ARM Order 4U'B'R'U F'UsL U'F R'U R!sD'L!D 20,17,15 4 X's & 2 K's (K)

p122 Mark's Pattern 4D'R F R'F!B L D'F U L'B R!sF!R 20,16,15 4 O's & 2 K's

p144 Mark's Pattern 7

Three-color regular patterns

R!sUsRsBsUs 12,10,5 blossomesB R'D!R D'B!D R'D!F!R U'B!U R'F!R F (F) 24,18 6 Stripe type 1U F B R DsRsD'R'aB R!D!R!B!R 21,17,15 p120 6 Stripe (1)L U R'B!R B'U!B R'B!D!R F'U!F R'D!R U'R 26,20 6 Stripe type 2RaD R'aD F'aU'FaRaU'R'aD'F'aD Fa 22 diagonals (4 Peak flip)

U'F!L!B!D!F!R'UaF'LsBsR U' 21,16,14p69 Tetra Peak exch (K)

D'R'BULsB'UB'U'BL'BLB'UB'U'BRsDB'DsF'UsLB 30,30,26 giant meson 1L D R!D'L!U B'D'B D'R'D R DsL D'B!D 22,19,18 giant meson 2R!L'DBR'D'BLBL'D'B'D!LDL'U'FD'F'RFL'FLF!R'D!U some about g. meson 1DFRsU'B'D'R'aD'LsBDFD!F!DF'R'B'L'DLBF!RD'R!D some about g. meson 2

RaDaR'aF'aDaFa 4 Peak flip + 4 twists(Pinwheels)

U'L'B L'B'L'D!U'FsRsF'U F D F'U B 20,19,17 [D'HOO'] (see [HOO'])LsD L'DsF D'B'RsD!U' 14,13,10 [D'H]F R!sU!sF!B'R!U F!sR!sU!D'L! 28,16,12 p82 8 Crn Twist 1U R U'R D!B!U'L'D R FsUaL!U!sB! 24,18,16 [DV]+chessboard s.p. (K)



p125 ML's Multicolor 1B!D'F!U'L!D!F!U'F R B U!F!R F!D FsU'F' 28,20,19 Triple Threes (K)U B!D'L!F!R!U'B U'L D F!L U'B'F'R!U'F! 26,19,19 Orthogonal bars (K)UsRsFsU L UsB'DaB'D!sF U!R U!sL'D!U 30,24,18 (similar to [D'HH']

Three-color regular ornaments

B'L'D L'D'L'F R FsUsF'D'F RsU'F B!D 22,21,18 [D'OHH'] (see [OHH'])B!R!D!R!B'D!B'L!F U!L'D L D!R'F L!F!D B' 30,20,20 Treep's Skein (K)U!B'D!B'U!B!U!B'D!B R!L B R D'R F!D U!R B 30,21,21 Z-snake type 1 (K)

More-colored regular patterns

F!D'F!L!U'F!R!F!D F'U!LsFsU Fs 25,18,15 Perry's Pinwheel (K)D!B!R!F!R!D!U L!F!D RaF'L F'aD'B!L!U' 30,20,20 Six L's - 6 colors (K)U!L!U L!D'L!U B!U'F'aRsF'D!L!U'B'D B 27,20,19 Two diamonds (K)

More-colored nonregular patterns

B R'D!R D'B!D R'D!F!R U'B!U R'F!R B' 24,18 4 StripeD RaD!F LsBsR'D'aB U!RsB! 20,17 p92 4 Stripe (Full)R'B'L!BD'R!D B'L!U!B U'R!U B'U!B R 24,18 4 Stripe + 2 X's MG.1982R!F!U'L!U!F!U'F!L!F!R'F R!B L F'R!B'R D!R! 33,21,21 Four T's (K)R'U'B'R!B R B U'FsU F'L'F'L!F U L 20,18,17 2 Peak twist MG.1983U R'F'R!F D R U'R D'R!L'U'RaFaU'F'a 22,20 3 Peak twist MG.1982

Flips and twists

(U R'F R )^5 20 p171 6 Edge Flip(F R'D'R)^3 (L'B D B')^3 24 6 flip(RaUaFa)! 12 8 edge flipR U'R!U'R U'aR U'R'D!R'U'R U'a 18,16 p183 6 TwistU F!L F!sU!sR'L!B!U!D'F!sR!s 28,16,12 p155 8 Crn Twist 2 (K)(L R!F!B')^4 24,16 p156 8 Crn Twist 3RaD!B'L!F!R!DsR'D!F'aD'F!D'R!U'F!D' 28,20,19 p3 12 flip (K)D F!U'B!R!B!R!L B'D'F D!F B!U F'RaU!F' 28,20 p6 12 flip, 8 twist (K)FsR UsB!R!B'LsF!sU'F!D'F!sD'R!D' 28,20,15 p139 6 X + 12 flip (K)



B'L D L F'U'aL B'D'F'RsF'UaL!UaB! 22,20,19 p139a (K)F R'DsR'B D'F!R!L F R'B'L D!B!U'R!sD'R!s 30,22,19 p141 Superfliptwst + 6X (K)

F R L!U'R!L'U'D!R!F D B D F!U'R'D'F!D!L! 28,20 p141a (K)

Patterns possible by disassembly

U L U L'U'F L'U'F'U'F'U L U L!U'F'L' + FLU+ 19,18 2 Peak exch MG.1983R'D!B!L B L'B D'R D' + BDR- 12,10 1 Peak twist MG.1982R'D R'B U'B U B!R!D' + BDR- 12,10 MG.1982R'D R'B R B!D B D'R D' + BDR- 12,11 JF.1983R'D R'B R B'R D'R F'R'F + BDR- 12 MG.1982F D'F'D R'D B'D B D'R D' + BDR- 12 MG.1982

Ornaments

U'R D BsR'U'L Ds + (UR,LF+) 10,10,8 3 U's + 3 O's MG.1982L'UsL'U R FsD'R'U B + (UF,BL+) 12,12,10 3 U's + 3 O's MG.1982D'R'aD RsF'aUsR FaL + (UR,BD+) 14,14,12 3 U's + 3 O's MG.1982F U RsB'D'F'RsF'LsD F + (RD,RF) 14,14,11 6 U's type 1 MG.1982U'F U RsB'D'F'RsF'LsD F R + (UR,FR) 16,16,13 6 U's t2 MG.1982U'F U RsB'D'F'RsF'LsD F R + (UR+) 16,16,13 3 U's + ring MG.1982

Regular semi-ornaments

L D R!D'L!B'D'B UsR'D R DsL D'B!D +(LDF,LFU) 22,19,17 cube within a cube

Semiregular semi-ornaments

R F U F!U'R'F' + (FD,RD+) 8,7 [VRKS(14)]U'B!L'F D'F'L B!R U R + FU+ 13,11 [VRS(14)]F!R'D R D'R F!D R'D'R' + (FD,RF) 13,11 [RS(14)(23)]F!R'D R D'R F!D R'D'R' + (FD,FR+) 13,11 [VRS(14)(23)]R F U F!U'R'B'D'F'D BsR'F R + FD+ 16,15,14 [VRKS(14)(23)]D!F UsL'D!L!FsU!D FsL' + (BL,DL) 18,14,11 [D]R'F L DsBsD LsB'U'R'U Ls + (DF,UL) 16,16,12 [D]

Cyclic patterns



RF'DR'U'B'UB'R'B'RBR'D'R'FR'Us + (LUB,LBD) 19 O's MG.1982

B'D R D'RsF R'F'R'F'UsL B R + (BU,LU) 16,16,14 O's MG.1982F L B UsR'U'B'U FsR'B R F'L' + (BU,LU) 16,16,14 O's MG.1982R'B D F'RsU F'D'F'D'LsB U F + (LU,LB) 16,16,14 O's MG.1982

Simple patterns

It is quite a qood excercise to try to get back the Cube from these positions. The number at each sequencemeans a number of 'long moves' like Fs,R!,Da. Three characters in the column 'position' describe what youcan see on faces F,R,U. Only position with O,H,I,X,+,_ on all faces, where O,I,+ occur in pairs, (inparentheses in the column 'position') are possible.

slice moves positionR!D'sR!sD'sL! 6 (H,H,H)FaR!F!sD!sL!F'a 8 (H',H,H)R!sF!sD!s 6 (X,X,X)RsFsRsFsRsFs 6 (X,X,X)RsF!sD!sRs 6 (X,X,X)R!sF!R!sB! 6 (_,_,H)D!RsF!sR'sU! 6 (_,_,H)DaF!sD'sF!sD! 7 (_,_,X)R!sF!R!F!sR!F! 8 (_,_,X)R!sD'sF!sD's 6 (O,O,_)RsD'sF!sD'sR's 6 (O,O,_)D!FsD!sFsD! 6 (O,O,H)D!R!sF!sD! 6 (O,O,X)R!F!DsF!sDsB!R! 8 (H,H,_)RaD!sR'aF'aD!sFa 8 (H,H,_)(F!B!R!L!U!)! 10 (H,H,_) p160aD!sRsF!sR's 6 (H,_,H)F'sR'sFsR'sF'sRs 6 (H,_,H)FsD!F!D!sF!D!Fs 8 (_,H,H)R!B!R!sB!L! 6 (I,I,_)R!s 2 (I,_,I)R!F!R!sF!R! 6 (_,I,I)U!RaD!sRaD! 6 (_,I,I)



R!F!sR! 4 (I,I,H)R!FsD!sF'sR! 6 (I,H,I)DsFaD!sFaDs 6 (I,H,I)RaD!sRa 4 (H,I,I)R!B!DsF!sD'sB!R! 8 (H',I,I)F!D!R!sF!sU!B! 8 (H',I,I)R!sF!s 4 (I,I,X)RaD!sRsD!sR! 7 (I,X,I)R!FaR!sFaR! 6 (X,I,I)R!sDsR!sDs 6 (H,H,X)RsFsR!sFsRs 6 (H,H,X)R!F!sD!sL! 6 (H,X,H)R!F!sR!F!D!sB! 8 (X,H,H)DsRaU!R'aD'sF'aU!Fa 8 (+,+,_)FaUaR!sUaFaU!s 8 (+,+,_) p16aR!F!R!DsF!R!F!Us 8 (+,+,_) p16cR!D!sF!D!sF!R! 8 (+,+,H)D!sR!D'sR!sD'sL! 8 (+,+,H)R!sD'sR!sDs 6 (+,+,X)RsF'sR'sFsRsF's 6 (+,+,X)R!F!R!sF!L!D!s 8 (X,X,_)D!sR!F!sR! 6 (X,X,H)RaD!R'aFaD!F'a 6 (U,U,_)RaD!R'aF'aU!Fa 6 (U,U~,_)D'aB!DaL!D!R! 6 (U,U',_)R'aD!RaF!D'aR!DaF! 8 (Z,Z,_)R'aD!RaF!DaR!DaB! 8 (Z,Z',_)R'aF!D'aL!DaF!RaD! 8 (Z',Z',_)D'R'aD RaU!R'aD'RaD' 9 (Z',Z',_)F!sDaF!sD'a 6 (_,H,X)R!sF'aD!sF'a 6 (_,H,X)L!F!R!sF!D!sL! 8 (H,X,_)DaF!sD'a 4 (_,O,I)D!R!sU! 4 (O,_,I)U!R'sD!sRsD! 6 (_,O,+)



F!R!sF!R!s 6 (_,I,+)F!R!DsR!sD'sR!B! 8 (I,+,_)DsF!sDs 4 (H,O,I)D!R!sF!sD!F!s 8 (O,H,I)D!R!sU!R!F!sL! 8 (O,I,H)F!RaD!sRaF! 6 (H,I,O)RsF!D!sB!R's 6 (H,O,+)DaF!D!sR!sB!Da 8 (O,+,H)R'sD!sRs 8 (H,I,+)R!F!sD!sL!D!s 8 (+,I,H)RsU!F!D!sF!U!R's 8 (+,I,H)R!D'sR!sDsR! 6 (I,+,H)R!F'sR!sFsR!D!s 8 (+,H,I)R!sF!D!sB! 6 (X,O,I)F!U!F!sU!R!sB! 8 (X,I,O)R!F!sR!F!D!sF! 8 (X,O,+)FaR!sFaR!s 6 (X,I,+)DsFaD!sFaD's 6 (X,+,I)F!L F!U!sB!R B! 14,8 4 T'sF!L B!U!sF!R B! 14,8 4 T's

One little square of opposite color

RsU!RaFaU!F'aR! 14,11,10 [E4]L!D S!rD!S!rD L! 16,9,7 [E4]D!F!U!L!F!L!U!F!L!D! 20,10 [E4]RaD'aB!D'aR!D!Ls 14,11,10 [E6]U R!F!sL!D'BsL!BsD! 18,12,9 [E6]F!U!L!D!F!L!F!D!F!U!F!L! 24,12 [E6]B!U!R!B!R!B!L!B!U!B!R!s 24,12 [E6]B!U!B!L!B!R!B!R!U!B!R!s 24,12 [E6]R D'L B L!D!R F R'D!L B'D R' 17,14 [C2E2]D'L!F!U F!U'F!L!U'L!D F!D'L!D 23,15 [C2E2] Domino alg.U R'F'R!F R U!R'F'R!F R U 16,13 [C3]

Two corner little squares of opposite color



RaD!B!D!B!D!LsB!L! 18,11,10 [C3+3]

FaU!FsR!U!R!U!R!F! 18,11,10 [C3+3]F!D!R!D!F!D!F!R!F!D! 20,10 [C3+3]D'aFsD!BsUsFsD!Bs 16,14,9 [C2+4]R!F!D!R!F!D!R!F!D! 16,14,9 [C2+4]BsU!FsR!U!R!U!R! 16,10,8 [C6]U!R!U!F!U!F!R!F!U!F! 20,10 [C6]

Odd simple patterns

F'D!R!F R F'R U!L'B L D!sF D! 21,15,14R B'D RaD'L'D R'D!L D'L'D B R'D! 19,17F!R!U!F!R!U'R!D!B!U'L!U!L!B!U 27,15 p123 Cube in a cube 2

Algorithms signed (p..) where .. replaces a number were taken from Michael Reid's PATTERNS.TXT -August 24, 1996.

Algorithms with (K) as the author were probably produced by a computer program based on Kociemba's 2-step algorithm. The authors of those algorithms are not known to us. Please, if you know who obtained thosealgorithms for the first time, let us know. Thank you.

Algorithms signed (MG.) are Mirek Goljan's results, JF. denotes Jessica Fridrich.

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