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      1 <script src="https://cdn.cubing.net/v0/js/cubing/twisty" type="module"></script>
      2 
      3 # Rubik's cube: how to figure it out
      4 
      5 So you decided to try and solve a
      6 [Rubik's Cube](https://en.wikipedia.org/wiki/Rubik%27s_Cube).  Maybe it
      7 is for a bet. Maybe your parents grounded you for playing too much
      8 Minecraft and now you have to find an off-line hobby.  Maybe you have
      9 been fascinated by this toy since you were a kid and now that you have
     10 retired from work you finally have the time to try and solve it.
     11 
     12 Of course it is and all you have to do it to learn how to solve
     13 this puzzle is asking your favorite search engine (or AI? I guess search
     14 engines are *so* 2021...). Sure you can do that, and you will find plenty
     15 of tutorials that teach you how to build it up *layer by layer*, and with
     16 the help of some pre-mmemorized sequences of moves you can easily learn
     17 it in a few hours (or days). Also, there are probably apps that can take a
     18 picture of a scrambled cube and tell you how to solve it move by move.
     19 
     20 But that feels a bit like cheating, doesn't it?  What if you want to
     21 figure it out on your own, without relying on someone else's instructions?
     22 
     23 This is where this page comes in. Here I will try to explain just enough
     24 about the Rubik's cube so that you can try and tackle it alone. After
     25 explaining some basics about how the cube works as a 3D puzzle, I will
     26 introduce a couple of general-purpose techniques to help you move the
     27 pieces around, without telling you directly what steps to take.
     28 
     29 So, let's dive in!
     30 
     31 ## How it works
     32 
     33 The first thing you should do to understand how the cube works is taking
     34 it all apart and inspect the pieces it is made of and the mechanism
     35 holding them together.
     36 
     37 Well actually don't do it, just look at the picture below:
     38 
     39 ![A disassembled Rubik's cube](disassembled.jpg)
     40 
     41 As you can see, there are three types of pieces:
     42 
     43 * **The core**, consisting of 3 axes intersecting in the center. Some
     44 screws and springs keep the **6 center pieces** attached to it, in such
     45 a way that they can spin. This is what makes the faces turn.
     46 * **Edge pieces**, with only two colored faces each. When the cube is
     47 assembled, they lie between two center pieces. There are 12 of them.
     48 * **Corner pieces**, with three colored faces each. When the cube is
     49 assembled, each corner piece is adjacent to 3 edge pieces, and it
     50 touches 3 center pieces "diagonally". There are 8 corner pieces.
     51 
     52 So far so good. But what does this tell us about solving the cube
     53 by turning its sides?
     54 
     55 First of all, the fact that the centers are attached to the core implies
     56 that their relative position does not change, ever. In other words, you
     57 can think of the **centers as fixed** pieces, and **build the rest
     58 around them**.
     59 
     60 Another important thing that is made clear by looking at the disassembled
     61 cube is that **you should think about pieces, not stickers** or colors.
     62 When naïvely attempting to solve the cube without a clue, many would
     63 think about "moving this color there". But what you have to always keep
     64 in mind is that you can never move a single colored sticker: the other
     65 colored stickers attached to the same piece, be it an edge or a corner,
     66 will always move with it.
     67 
     68 Now we are ready to move on to some fundamental techniques. If you have
     69 actually disassebled your cube, put it back together *in a solved state*:
     70 if you don't, there is a good chance (11 out 12) that you will reassamble
     71 it in an **unsolvable state**, just like a
     72 [15 puzzle](https://en.wikipedia.org/wiki/15_Puzzle) with the
     73 last two number swapped.
     74 
     75 ## Building blocks
     76 
     77 Most methods to solve a Rubik's cube a divided into steps, where:
     78 
     79 * In the first few steps, you put together blocks of pieces. This is 
     80 also known as **blockbuilding**. These steps are often "intuitive",
     81 which means that they do not require memorizing sequences of moves
     82 to apply blindly.
     83 * In the last few steps, you need to move around a few remaining pieces
     84 without destroying the progress made in the previous steps. Most of the
     85 time steps like these rely on memorized sequences of moves that are known
     86 to only affect the position of certain pieces; but there are alternative
     87 approaches, as we will see later.
     88 
     89 So, what do we mean exactly by "block of pieces"? To be precise we could
     90 say that two or more adjacent pieces form a block when adjacent stickers
     91 of different pieces have the same color. For example, the simplest kind
     92 of block is a **pair**, that you can see in the picture below:
     93 
     94 ![A corner-edge pair](pair.png)
     95 
     96 To be precise, the one above is a coner-edge pair. There are also center-edge
     97 pairs, consisting of a center and an edge, but they are rarely referred to
     98 as "pairs". In any case, they do fit our definition of "block".
     99 
    100 A more complex example of a block is a **layer**, which is the result of
    101 the first two steps of the classic "layer by layer" method. This one is
    102 a correctly solved layer:
    103 
    104 ![A layer](layer.png)
    105 
    106 On the other hand, this is **not a layer**:
    107 
    108 ![A non-solved layer, but with a solve "face"](face.png)
    109 
    110 It is worth pausing here to reflect a bit. The last two pictures both
    111 clearly show a solved white face, don't they? For a most people, they
    112 could look equivalent. But remember what we said earlier: you should
    113 think about *pieces*, not *stickers*. In the last picture, the pieces
    114 have all white on top, but their side colors do not match (except for
    115 the blue-white edge and the blue-orange-white corner). So they do not
    116 form a block, which means that their **relative position** is incorrect.
    117 By contrast, the second to last picture shows a block: all side colors
    118 match, not only the white top. This means that the pieces are in correct
    119 relative position with respect to each other.
    120 
    121 With this in mind, you can starting making your own way through
    122 the first few steps of your solving method: try to build multiple
    123 blocks of pieces and put them together to make even bigger blocks.
    124 
    125 If you want some more hints, Ryan Heise's website contains some nice
    126 examples about building blocks in his
    127 [fundamental techniques page](https://www.ryanheise.com/cube/fundamental_techniques.html).
    128 
    129 *Note: I have updated this page to use [twizzle](https://alpha.twizzle.net)
    130 applets instead of simple pictures. These applets can be played to show the
    131 solution. To experiment around with this solution, click on
    132 the TW at the bottom right to open them in
    133 [twizzle](https://alpha.twizzle.net/). In this way you can also
    134 check which moves correspond to the written notation, if you
    135 are not familiar with it.*
    136 
    137 ## Commutators
    138 
    139 *In the rest of this page I am going to use the standard
    140 [Rubik's cube notation](../notation) to write down sequences of moves.
    141 You should familiarize with it at least a bit before continuing. Don't
    142 worry, it's very easy.*
    143 
    144 The more blocks you build, the harder it becomes to make progress. The
    145 reason for this is that you want to avoid destroying the blocks you
    146 have already built, so your options become more and more restricted
    147 as you go. This is completely normal.
    148 
    149 The most common speedsolving methods go around this issue by
    150 prescribing the use of memorized sequences, somewhat improperly
    151 called "algorithms".  For example the layer by layer method relies
    152 on blockbuilding to build the first layer, but on at least 5
    153 "algorithms" to complete the last two layers.  The more advanced
    154 [CFOP](https://www.speedsolving.com/wiki/index.php/CFOP_method) method
    155 uses blockbuilding for the first two layers, but it then requires
    156 78 different "algorithms" to complete the last layer.
    157 
    158 Here I am going to outline an alternative, more flexible approach,
    159 based on
    160 [commutators](https://www.speedsolving.com/wiki/index.php?title=Commutator).
    161 They are generally considered an advanced technique, I believe that
    162 they are perfectly fine to learn as a beginner. Ryan Heise's page that
    163 I linked above has a
    164 [section about them](https://www.ryanheise.com/cube/commutators.html),
    165 too.
    166 
    167 If you have not done it already, you should have a look at my page on
    168 [the Rubik's cube notation](../notation) before continuing.
    169 
    170 ### Corner commutators
    171 
    172 Suppose that you manage, via blockbuilding, to reach the following state:
    173 
    174 <center>
    175 <twisty-player experimental-setup-alg="[U, R' D R]"
    176 alg="R' D R   // Insertion
    177      U   // Interchange
    178      R' D' R   // Inverse insertion
    179      U'   // Inverse interchange"></twisty-player>
    180 </center>
    181 
    182 First of all, this would be an amazing achievement! The whole cube is
    183 solved except for three corners. The bottom-left corner (only one red
    184 sticker visible) is white-green-red, and it should go to the place where
    185 the white-red-blue corner is right now. The latter should in turn take
    186 the place of the corner on the right whose visible stickers are orange
    187 and yellow (the hidden sticker being green). In Mathematical terms, these
    188 3 corner form a **permutation cycle of 3 pieces**, or 3-cycle for short.
    189 
    190 Commutators are a general technique to solve 3-cycles of pieces. They
    191 can be decomposed in 4 small steps:
    192 
    193 1. **Interchange**: a single move that interchanges two of the three
    194 pieces.
    195 2. **Insertion**: a sequence of moves (usually 3) that inserts the third
    196 piece into the place of one of the other two, without affecting the
    197 "interchange" face of the cube in any other way.
    198 3. **Inverse interchange**: the inverse of the move done in step 1.
    199 4. **Inverse insertion**: the inverse of the sequence of moves done in step 2.
    200 
    201 **Note:** step 1 and 2 can appear in the other order; if they do, steps
    202 3 and 4 should also be swapped.
    203 
    204 Let's look at an example. From the position in the picture above, you
    205 can interchange the top two corners using the move U. More precisely, U
    206 brings the orange-yellow-green corner into the position currently occupied
    207 by the white-red-blue corner. The move U' also works, as it moves the
    208 white-red-blue corner to the position of the orange-yellow-green one.
    209 
    210 An interchange move is worth nothing without a compatible insertion
    211 sequence. In this case, you can use R' D R as insertion: this sequence
    212 of 3 moves moves the red-green-white corner to the place currently
    213 occupied by the white-red-blue one and, very importantly, **it does not
    214 affect any other piece in the U layer**. To put it in another way,
    215 **the interchange and the insertion only "clash" on one corner**.
    216 
    217 The last thing to decide before we put all of this together is which
    218 one should go first: the interchange or the insertion? This is not hard
    219 to figure out: I described both of them as "moving a certain piece into
    220 a certain position"; only one of the two moves a piece in its correct
    221 final position, and that is the sequence that must go first. In our case
    222 it is the insertion, because the red-green-white corner's final position
    223 is the one occupied by the white-red-blue one.
    224 
    225 So our commutator looks like this: R' D R U R' D' R' U'. Let's split
    226 this up to review it:
    227 
    228 * **R' D R**: the insertion sequence, moving the red-green-white corner
    229 to the position of the white-red-blue one.
    230 * **U**: the interchange move, moving the orange-yellow-green corner
    231 to the position now occupied by the red-green-white one.
    232 * **R' D' R**: the inverse of the insertion sequence. To invert a sequence
    233 of moves, you have to **read it backwards inverting every single move**.
    234 Here we start with R', because it is the inverse of R, the last move of
    235 the insertion sequence; then we have D', the inverse of the second move; 
    236 and finally R, the inverse of the first move of the insertion sequence.
    237 * **U'**: the inverse of the interchange move.
    238 
    239 **Note:** looking at the position of the pieces is not enough to
    240 determine a correct commutator to permute them. Their **orientation**
    241 is also important. For example, consider the following case:
    242 
    243 <center>
    244 <twisty-player experimental-setup-alg="[R', U L' U']"
    245 alg="R' D R   // Insertion
    246      U   // Interchange
    247      R' D' R   // Inverse insertion
    248      U'   // Inverse interchange"></twisty-player>
    249 </twisty-player>
    250 </center>
    251 
    252 The three corners are permuted in exactly the same way, so everything
    253 we said above could be repeated word by word, move by move. However,
    254 if you apply the commutator we constructed to this case, you'll get
    255 something like this (you can also see this by playing the applet above):
    256 
    257 <center>
    258 <twisty-player experimental-setup-alg="[U, R' D' R D R' D' R]">
    259 </twisty-player>
    260 </center>
    261 
    262 What's wrong here? Well, obviously the cube is not solved. All the pieces
    263 are in their correct position, but two corners are twisted in place!
    264 
    265 To avoid situations like this when creating your commutators, you need to
    266 keep track of **which sticker goes where**. I know, I know: I said at the
    267 beginning that *pieces* are important, not *stickers*. This is still true,
    268 but sometimes it is important to keep track of both.
    269 
    270 Let's highlight the difference between the two 3-cycles. In the first one:
    271 
    272 <center>
    273 <twisty-player experimental-setup-alg="[U, R' D R]"
    274 alg="R' D R   // Insertion
    275      U   // Interchange
    276      R' D' R   // Inverse insertion
    277      U'   // Inverse interchange"></twisty-player>
    278 </center>
    279 
    280 1. The red-green-white corner must go to the place of the white-red-blue one,
    281 *with the white sticker of the first going to the place of the white sticker
    282 of the latter*.
    283 2. The white-red-blue corner must go to the place of the
    284 orange-yellow-green one, *with the white sticker of the former going to
    285 the place of the orange sticker of the latter*.
    286 3. The orange-yellow-green corner must go to the place of the red-green-white
    287 one, *with the orange sticker of the former going to the place of the
    288 white sticker of the latter*.
    289 
    290 While in the second case:
    291 
    292 <center>
    293 <twisty-player experimental-setup-alg="[R', U L' U']"
    294 alg="U L' U'   // Insertion
    295      R'   // Interchange
    296      U L U'   // Inverse insertion
    297      R   // Inverse interchange"></twisty-player>
    298 </twisty-player>
    299 </center>
    300 
    301 1. The red-green-white corner must go to the place of the white-red-blue one,
    302 *with the white sticker of the first going to the place of the* **blue** *sticker
    303 of the latter*.
    304 2. The white-red-blue corner must go to the place of the
    305 orange-yellow-green one, *with the* **blue** *sticker of the former going to
    306 the place of the* **green** *sticker of the latter*.
    307 3. The orange-yellow-green corner must go to the place of the red-green-white
    308 one, *with the* **green** *sticker of the former going to the place of the
    309 white sticker of the latter*.
    310 
    311 The main point here is that not only interchange and insertion moves
    312 should swap the correct pieces around, but they must also move each
    313 "reference sticker" to the position of the next "reference sticker".
    314 For example, using the commutator R' D R U R' D' R U' for the second
    315 case does not work, because the insertion sequence R' D R moves the
    316 white sticker of the red-green-white corner to the position of the
    317 red sticker of the white-red-blue one, while it should move it
    318 to the position of the blue sticker!
    319 
    320 ### Edge commutators
    321 
    322 So far I have only talked about *corner* commutators, but what if you
    323 are also left with some unsolved edges? For example, consider this case:
    324 
    325 <center>
    326 <twisty-player experimental-setup-alg="[L' U2 L, E']"
    327 alg="E'   // Interchange
    328      L' U2 L   // Insertion
    329      E   // Inverse interchange
    330      L' U2 L   // Inverse insertion"></twisty-player>
    331 </twisty-player>
    332 </center>
    333 
    334 The picture shows a 3-cycle of edges.  You might think that the same
    335 reasoning can be applied and that you can use commutators to solve
    336 edge 3-cycles of edges. This is exactly the case, and this is why this
    337 subsection is so short.
    338 
    339 Let's see how to solve the case above. As interchange move, you can use
    340 the **inner-layer move** E' (check out the [notation page](../notation)
    341 if you are unfamiliar with these). The insertion sequence to be used
    342 with it is L' U2 L. Putting everything together, you get
    343 E' L' U2 L E L' U2 L.
    344 
    345 ### Commutators with set-up moves
    346 
    347 At this point I have good news and bad news.
    348 
    349 The good news is that commutators are so powerful that you could solve
    350 the whole cube using just commutators and at most one single move (this
    351 sentence might sound a bit strange, but it is Mathematically correct -
    352 the best kind of correct). Although it would not be very efficient, you
    353 could avoid blockbuilding altogether and move pieces around
    354 only with commutators - this is how advanced
    355 [blindsolving](https://www.speedsolving.com/wiki/index.php?title=Blindfolded_Solving)
    356 methods work.
    357 
    358 The bad news is that not every 3-cycle can be solved directly with a
    359 commutator, at least not one of the form I described above. Sometimes
    360 you need to use **set-up moves**, also known as 
    361 [conjugates](https://www.ryanheise.com/cube/conjugates.html).
    362 
    363 Consider the following case:
    364 
    365 <center>
    366 <twisty-player experimental-setup-alg="[L: [R D2 R', U']]"
    367 alg="L   // Set-up
    368      U'   // Interchange
    369      R D2 R'   // Insertion
    370      U   // Inverse interchange
    371      R D2 R'   // Inverse insertion
    372      L'   // Inverse set-up"></twisty-player>
    373 </center>
    374 
    375 No matter how much you try, you are not going to find valid interchange
    376 and insertion moves as above. The fundamental problem is that you would
    377 like to use U (or U', or U2) as an interchange move, but this move affects
    378 all 3 of the corners. You might think of using R or F as interchange; they
    379 do affect only two of the pieces, but they do not move the the stickers in
    380 the correct position: any commutator based on R or F as interchange move
    381 would lead not to a solved cube, but to some corners twisted in place.
    382 
    383 So, how can we deal with this case? The solution is to use one or more
    384 moves to set up a better case. These moves will be done at the
    385 beginning and then undone at the end.
    386 
    387 For example in this case you can start by doing L as a setup move.
    388 This has the effect of moving the white-red-green corner out of the U
    389 layer, so that you can then use U (or rather, U') as interchange move.
    390 The insertion sequence that makes it all work here is R D2 R', and
    391 putting it all together you get:
    392 
    393 * Set-up: L
    394 * Interchange: U'
    395 * Insertion: R D2 R'
    396 * Inverse interchange: U
    397 * Inverse insertion: R D2 R'
    398 * Inverse set-up: L'
    399 
    400 **Note:** in this case the insertion coincides with its inverse. This
    401 can happen and there is nothing particular about it.
    402 
    403 ## Conclusion
    404 
    405 With what you have learned so far, you can now try and solve the Rubik's
    406 cube on your own, without further help. Granted, it won't be a walk
    407 in the park: this short tutorial is not meant to explain everything. I
    408 could have given you advice on which blocks to build first or on when to
    409 stop building blocks and start using commutators, I could have shown you
    410 many more examples, I could have told you how to address tricky cases
    411 like permutation parity or pieces twisted in place. But I think it can
    412 be more fun to try and figure all of this out by yourself - and if you
    413 disagree, just look for a more complete tutorial online.
    414 
    415 Happy cubing!