figure-it-out.md (17941B)

1 # Rubik's cube: how to figure it out 2 3 So you decided to try and solve a 4 [Rubik's Cube](https://en.wikipedia.org/wiki/Rubik%27s_Cube). Maybe it 5 is for a bet. Maybe your parents grounded you for playing too much 6 Minecraft and now you have to find an off-line hobby. Maybe you have 7 been fascinated by this toy since you were a kid and now that you have 8 retired from work you finally have the time to try and solve it. 9 10 Of course it is and all you have to do it to learn how to solve 11 this puzzle is asking your favorite search engine (or AI? I guess search 12 engines are *so* 2021...). Sure you can do that, and you will find plenty 13 of tutorials that teach you how to build it up *layer by layer*, and with 14 the help of some pre-mmemorized sequences of moves you can easily learn 15 it in a few hours (or days). Also, there are probably apps that can take a 16 picture of a scrambled cube and tell you how to solve it move by move. 17 18 But that feels a bit like cheating, doesn't it? What if you want to 19 figure it out on your own, without relying on someone else's instructions? 20 21 This is where this page comes in. Here I will try to explain just enough 22 about the Rubik's cube so that you can try and tackle it alone. After 23 explaining some basics about how the cube works as a 3D puzzle, I will 24 introduce a couple of general-purpose techniques to help you move the 25 pieces around, without telling you directly what steps to take. 26 27 So, let's dive in! 28 29 ## How it works 30 31 The first thing you should do to understand how the cube works is taking 32 it all apart and inspect the pieces it is made of and the mechanism 33 holding them together. 34 35 Well actually don't do it, just look at the picture below: 36 37 ![A disassembled Rubik's cube](disassembled.jpg) 38 39 As you can see, there are three types of pieces: 40 41 * **The core**, consisting of 3 axes intersecting in the center. Some 42 screws and springs keep the **6 center pieces** attached to it, in such 43 a way that they can spin. This is what makes the faces turn. 44 * **Edge pieces**, with only two colored faces each. When the cube is 45 assembled, they lie between two center pieces. There are 12 of them. 46 * **Corner pieces**, with three colored faces each. When the cube is 47 assembled, each corner piece is adjacent to 3 edge pieces, and it 48 touches 3 center pieces "diagonally". There are 8 corner pieces. 49 50 So far so good. But what does this tell us about solving the cube 51 by turning its sides? 52 53 First of all, the fact that the centers are attached to the core implies 54 that their relative position does not change, ever. In other words, you 55 can think of the **centers as fixed** pieces, and **build the rest 56 around them**. 57 58 Another important thing that is made clear by looking at the disassembled 59 cube is that **you should think about pieces, not stickers** or colors. 60 When naïvely attempting to solve the cube without a clue, many would 61 think about "moving this color there". But what you have to always keep 62 in mind is that you can never move a single colored sticker: the other 63 colored stickers attached to the same piece, be it an edge or a corner, 64 will always move with it. 65 66 Now we are ready to move on to some fundamental techniques. If you have 67 actually disassebled your cube, put it back together *in a solved state*: 68 if you don't, there is a good chance (11 out 12) that you will reassamble 69 it in an **unsolvable state**, just like a 70 [15 puzzle](https://en.wikipedia.org/wiki/15_Puzzle) with the 71 last two number swapped. 72 73 ## Building blocks 74 75 Most methods to solve a Rubik's cube a divided into steps, where: 76 77 * In the first few steps, you put together blocks of pieces. This is 78 also known as **blockbuilding**. These steps are often "intuitive", 79 which means that they do not require memorizing sequences of moves 80 to apply blindly. 81 * In the last few steps, you need to move around a few remaining pieces 82 without destroying the progress made in the previous steps. Most of the 83 time steps like these rely on memorized sequences of moves that are known 84 to only affect the position of certain pieces; but there are alternative 85 approaches, as we will see later. 86 87 So, what do we mean exactly by "block of pieces"? To be precise we could 88 say that two or more adjacent pieces form a block when adjacent stickers 89 of different pieces have the same color. For example, the simplest kind 90 of block is a **pair**, that you can see in the picture below: 91 92 ![A corner-edge pair](pair.svg) 93 94 To be precise, the one above is a coner-edge pair. There are also center-edge 95 pairs, consisting of a center and an edge, but they are rarely referred to 96 as "pairs". In any case, they do fit our definition of "block". 97 98 A more complex example of a block is a **layer**, which is the result of 99 the first two steps of the classic "layer by layer" method. This one is 100 a correctly solved layer: 101 102 ![A layer](layer.svg) 103 104 On the other hand, this is **not a layer**: 105 106 ![A non-solved layer, but with a solve "face"](face.svg) 107 108 It is worth pausing here to reflect a bit. The last two pictures both 109 clearly show a solved white face, don't they? For a most people, they 110 could look equivalent. But remember what we said earlier: you should 111 think about *pieces*, not *stickers*. In the last picture, the pieces 112 have all white on top, but their side colors do not match (except for 113 the blue-white edge and the blue-orange-white corner). So they do not 114 form a block, which means that their **relative position** is incorrect. 115 By contrast, the second to last picture shows a block: all side colors 116 match, not only the white top. This means that the pieces are in correct 117 relative position with respect to each other. 118 119 With this in mind, you can starting making your own way through 120 the first few steps of your solving method: try to build multiple 121 blocks of pieces and put them together to make even bigger blocks. 122 123 If you want some more hints, Ryan Heise's website contains some nice 124 examples about building blocks in his 125 [fundamental techniques page](https://www.ryanheise.com/cube/fundamental_techniques.html). 126 127 ## Commutators 128 129 *In the rest of this page I am going to use the standard 130 [Rubik's cube notation](../notation) to write down sequences of moves. 131 You should familiarize with it at least a bit before continuing. Don't 132 worry, it's very easy.* 133 134 The more blocks you build, the harder it becomes to make progress. The 135 reason for this is that you want to avoid destroying the blocks you 136 have already built, so your options become more and more restricted 137 as you go. This is completely normal. 138 139 The most common speedsolving methods go around this issue by 140 prescribing the use of memorized sequences, somewhat improperly 141 called "algorithms". For example the layer by layer method relies 142 on blockbuilding to build the first layer, but on at least 5 143 "algorithms" to complete the last two layers. The more advanced 144 [CFOP](https://www.speedsolving.com/wiki/index.php/CFOP_method) method 145 uses blockbuilding for the first two layers, but it then requires 146 78 different "algorithms" to complete the last layer. 147 148 Here I am going to outline an alternative, more flexible approach, 149 based on 150 [commutators](https://www.speedsolving.com/wiki/index.php?title=Commutator). 151 They are generally considered an advanced technique, I believe that 152 they are perfectly fine to learn as a beginner. Ryan Heise's page that 153 I linked above has a 154 [section about them](https://www.ryanheise.com/cube/commutators.html), 155 too. 156 157 If you have not done it already, you should have a look at my page on 158 [the Rubik's cube notation](../notation) before continuing. 159 160 ### Corner commutators 161 162 Suppose that you manage, via blockbuilding, to reach the following state: 163 164 ![A commutator](comm1.svg) 165 166 First of all, this would be an amazing achievement! The whole cube is 167 solved except for three corners. The bottom-left corner (only one red 168 sticker visible) is white-green-red, and it should go to the place where 169 the white-red-blue corner is right now. The latter should in turn take 170 the place of the corner on the right whose visible stickers are orange 171 and yellow (the hidden sticker being green). In Mathematical terms, these 172 3 corner form a **permutation cycle of 3 pieces**, or 3-cycle for short. 173 174 Commutators are a general technique to solve 3-cycles of pieces. They 175 can be decomposed in 4 small steps: 176 177 1. **Interchange**: a single move that interchanges two of the three 178 pieces. 179 2. **Insertion**: a sequence of moves (usually 3) that inserts the third 180 piece into the place of one of the other two, without affecting the 181 "interchange" face of the cube in any other way. 182 3. **Inverse interchange**: the inverse of the move done in step 1. 183 4. **Inverse insertion**: the inverse of the sequence of moves done in step 2. 184 185 **Note:** step 1 and 2 can appear in the other order; if they do, steps 186 3 and 4 should also be swapped. 187 188 Let's look at an example. From the position in the picture above, you 189 can interchange the top two corners using the move U. More precisely, U 190 brings the orange-yellow-green corner into the position currently occupied 191 by the white-red-blue corner. The move U' also works, as it moves the 192 white-red-blue corner to the position of the orange-yellow-green one. 193 194 An interchange move is worth nothing without a compatible insertion 195 sequence. In this case, you can use R' D R as insertion: this sequence 196 of 3 moves moves the red-green-white corner to the place currently 197 occupied by the white-red-blue one and, very importantly, **it does not 198 affect any other piece in the U layer**. To put it in another way, 199 **the interchange and the insertion only "clash" on one corner**. 200 201 The last thing to decide before we put all of this together is which 202 one should go first: the interchange or the insertion? This is not hard 203 to figure out: I described both of them as "moving a certain piece into 204 a certain position"; only one of the two moves a piece in its correct 205 final position, and that is the sequence that must go first. In our case 206 it is the insertion, because the red-green-white corner's final position 207 is the one occupied by the white-red-blue one. 208 209 So our commutator looks like this: R' D R U R' D' R' U'. Let's split 210 this up to review it: 211 212 * **R' D R**: the insertion sequence, moving the red-green-white corner 213 to the position of the white-red-blue one. 214 * **U**: the interchange move, moving the orange-yellow-green corner 215 to the position now occupied by the red-green-white one. 216 * **R' D' R**: the inverse of the insertion sequence. To invert a sequence 217 of moves, you have to **read it backwards inverting every single move**. 218 Here we start with R', because it is the inverse of R, the last move of 219 the insertion sequence; then we have D', the inverse of the second move; 220 and finally R, the inverse of the first move of the insertion sequence. 221 * **U'**: the inverse of the interchange move. 222 223 To help understanding all of this, you can visualize this commutator 224 [alg.cubing.net](https://alg.cubing.net/?setup=%5BU,_R-DR%5D&alg=R-_D_R_%2F%2FInsertion%0AU_%2F%2FInterchange%0AR-_D-_R_%2F%2FInverse_insertion%0AU-_%2F%2FInverse_interchange). 225 226 **Note:** looking at the position of the pieces is not enough to 227 determine a correct commutator to permute them. Their **orientation** 228 is also important. For example, consider the following case: 229 230 ![Another commutator](comm2.svg) 231 232 The three corners are permuted in exactly the same way, so everything 233 we said above could be repeated word by word, move by move. However, 234 if you apply the commutator we constructed to this case, you'll get 235 something like this: 236 237 ![Two twisted corners](twist.svg) 238 239 What's wrong here? Well, obviously the cube is not solved. All the pieces 240 are in their correct position, but two corners are twisted in place! 241 242 To avoid situations like this when creating your commutators, you need to 243 keep track of **which sticker goes where**. I know, I know: I said at the 244 beginning that *pieces* are important, not *stickers*. This is still true, 245 but sometimes it is important to keep track of both. 246 247 Let's highlight the difference between the two 3-cycles. In the first one: 248 249 ![A commutator](comm1.svg) 250 251 1. The red-green-white corner must go to the place of the white-red-blue one, 252 *with the white sticker of the first going to the place of the white sticker 253 of the latter*. 254 2. The white-red-blue corner must go to the place of the 255 orange-yellow-green one, *with the white sticker of the former going to 256 the place of the orange sticker of the latter*. 257 3. The orange-yellow-green corner must go to the place of the red-green-white 258 one, *with the orange sticker of the former going to the place of the 259 white sticker of the latter*. 260 261 While in the second case: 262 263 ![Another commutator](comm2.svg) 264 265 1. The red-green-white corner must go to the place of the white-red-blue one, 266 *with the white sticker of the first going to the place of the* **blue** *sticker 267 of the latter*. 268 2. The white-red-blue corner must go to the place of the 269 orange-yellow-green one, *with the* **blue** *sticker of the former going to 270 the place of the* **green** *sticker of the latter*. 271 3. The orange-yellow-green corner must go to the place of the red-green-white 272 one, *with the* **green** *sticker of the former going to the place of the 273 white sticker of the latter*. 274 275 The main point here is that not only interchange and insertion moves 276 should swap the correct pieces around, but they must also move each 277 "reference sticker" to the position of the next "reference sticker". 278 For example, using the commutator R' D R U R' D' R U' for the second 279 case does not work, because the insertion sequence R' D R moves the 280 white sticker of the red-green-white corner to the position of the 281 red sticker of the white-red-blue one, while it should move it 282 to the position of the blue sticker! 283 284 I won't repeat the whole construction for the second commutator, 285 but you can visualize a solution 286 [here](https://alg.cubing.net/?setup=%5BR-,_UL-U-%5D&alg=U_L-_U-_%2F%2FInsertion%0AR-_%2F%2FInterchange%0AU_L_U-_%2F%2FInverse_insertion%0AR_%2F%2FInverse_interchange). 287 288 ### Edge commutators 289 290 So far I have only talked about *corner* commutators, but what if you 291 are also left with some unsolved edges? For example, consider this case: 292 293 ![A edge 3-cycle](edgecomm.svg) 294 295 The picture shows a 3-cycle of edges. You might think that the same 296 reasoning can be applied and that you can use commutators to solve 297 edge 3-cycles of edges. This is exactly the case, and this is why this 298 subsection is so short. 299 300 Let's see how to solve the case above. As interchange move, you can use 301 the **inner-layer move** E' (check out the [notation page](../notation) 302 if you are unfamiliar with these). The insertion sequence to be used 303 with it is L' U2 L. Putting everything together, you get 304 [E' L' U2 L E L' U2 L](https://alg.cubing.net/?setup=%5BL-U2L,E-%5D&alg=E-_%2F%2FInterchange%0AL-_U2_L_%2F%2FInsertion%0AE_%2F%2FInverse_interchange%0AL-_U2_L_%2F%2FInverse_insertion). 305 306 ### Commutators with set-up moves 307 308 At this point I have good news and bad news. 309 310 The good news is that commutators are so powerful that you could solve 311 the whole cube using just commutators and at most one single move (this 312 sentence might sound a bit strange, but it is Mathematically correct - 313 the best kind of correct). Although it would not be very efficient, you 314 could avoid blockbuilding altogether and move pieces around 315 only with commutators - this is how advanced 316 [blindsolving](https://www.speedsolving.com/wiki/index.php?title=Blindfolded_Solving) 317 methods work. 318 319 The bad news is that not every 3-cycle can be solved directly with a 320 commutator, at least not one of the form I described above. Sometimes 321 you need to use **set-up moves**, also known as 322 [conjugates](https://www.ryanheise.com/cube/conjugates.html). 323 324 Consider the following case: 325 326 ![A 3-cycle of corners requiring a set-up move](setup.svg) 327 328 No matter how much you try, you are not going to find valid interchange 329 and insertion moves as above. The fundamental problem is that you would 330 like to use U (or U', or U2) as an interchange move, but this move affects 331 all 3 of the corners. You might think of using R or F as interchange; they 332 do affect only two of the pieces, but they do not move the the stickers in 333 the correct position: any commutator based on R or F as interchange move 334 would lead not to a solved cube, but to some corners twisted in place. 335 336 So, how can we deal with this case? The solution is to use one or more 337 moves to set up a better case. These moves will be done at the 338 beginning and then undone at the end. 339 340 For example in this case you can start by doing L as a setup move. 341 This has the effect of moving the white-red-green corner out of the U 342 layer, so that you can then use U (or rather, U') as interchange move. 343 The insertion sequence that makes it all work here is R D2 R', and 344 putting it all together you get: 345 346 * Set-up: L 347 * Interchange: U' 348 * Insertion: R D2 R' 349 * Inverse interchange: U 350 * Inverse insertion: R D2 R' 351 * Inverse set-up: L' 352 353 **Note:** in this case the insertion coincides with its inverse. This 354 can happen and there is nothing particular about it. 355 356 As usual, you can visualize the final result on 357 [alg.cubing.net](https://alg.cubing.net/?setup=L2B2R-F-RB2R-FRL2&alg=L_%2F%2FSet%26%2345%3Bup%0AU-_%2F%2FInterchange%0AR_D2_R-_%2F%2FInsertion%0AU_%2F%2FInverse_interchange%0AR_D2_R-_%2F%2FInverse_insertion%0AL-_%2F%2FInverse_set%26%2345%3Bup) 358 359 ## Conclusion 360 361 With what you have learned so far, you can now try and solve the Rubik's 362 cube on your own, without further help. Granted, it won't be a walk 363 in the park: this short tutorial is not meant to explain everything. I 364 could have given you advice on which blocks to build first or on when to 365 stop building blocks and start using commutators, I could have shown you 366 many more examples, I could have told you how to address tricky cases 367 like permutation parity or pieces twisted in place. But I think it can 368 be more fun to try and figure all of this out by yourself - and if you 369 disagree, just look for a more complete tutorial online. 370 371 Happy cubing!