Added Rubik's cube tutorial

12 files changed, 705 insertions, 0 deletions

diff --git a/src/blog/2024-03-10-figure-out-cube/figure-out-cube.md b/src/blog/2024-03-10-figure-out-cube/figure-out-cube.md
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index 0000000..3ce3f38
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+# Rubik's cube: how to figure it out
+
+Today I worked on something that does not quite fit with the theme
+of this blog, and this is why I decided to have it as a separate page
+rather than as a blog post. Nevertheless, I thought it would be worth
+linking it here in case there are interested readers :)
+
+In preparation for a presentation I am going to give in a couple of
+weeks at a company meeting, I have written short tutorial about the
+Rubik's cube:
+
+[Rubik's cube: how to figure it out](../../speedcubing/figure-it-out).
+
+This is not a complete guide, but rather a short writeup aimed at giving
+you just enough tools to be able to solve it on your own. I think this is
+an interesting approach that you won't find elsewhere by simply Googling
+"how to solve a Rubik's cube".
+
+See you next time for a post on the usual topics!
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diff --git a/src/speedcubing/figure-it-out/face.svg b/src/speedcubing/figure-it-out/face.svg
new file mode 100644
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diff --git a/src/speedcubing/figure-it-out/figure-it-out.md b/src/speedcubing/figure-it-out/figure-it-out.md
new file mode 100644
index 0000000..4690c76
--- /dev/null
+++ b/src/speedcubing/figure-it-out/figure-it-out.md
@@ -0,0 +1,371 @@
+# Rubik's cube: how to figure it out
+
+So you decided to try and solve a
+[Rubik's Cube](https://en.wikipedia.org/wiki/Rubik%27s_Cube).  Maybe it
+is for a bet. Maybe your parents grounded you for playing too much
+Minecraft and now you have to find an off-line hobby.  Maybe you have
+been fascinated by this toy since you were a kid and now that you have
+retired from work you finally have the time to try and solve it.
+
+Of course it is and all you have to do it to learn how to solve
+this puzzle is asking your favorite search engine (or AI? I guess search
+engines are *so* 2021...). Sure you can do that, and you will find plenty
+of tutorials that teach you how to build it up *layer by layer*, and with
+the help of some pre-mmemorized sequences of moves you can easily learn
+it in a few hours (or days). Also, there are probably apps that can take a
+picture of a scrambled cube and tell you how to solve it move by move.
+
+But that feels a bit like cheating, doesn't it?  What if you want to
+figure it out on your own, without relying on someone else's instructions?
+
+This is where this page comes in. Here I will try to explain just enough
+about the Rubik's cube so that you can try and tackle it alone. After
+explaining some basics about how the cube works as a 3D puzzle, I will
+introduce a couple of general-purpose techniques to help you move the
+pieces around, without telling you directly what steps to take.
+
+So, let's dive in!
+
+## How it works
+
+The first thing you should do to understand how the cube works is taking
+it all apart and inspect the pieces it is made of and the mechanism
+holding them together.
+
+Well actually don't do it, just look at the picture below:
+
+![A disassembled Rubik's cube](disassembled.jpg)
+
+As you can see, there are three types of pieces:
+
+* **The core**, consisting of 3 axes intersecting in the center. Some
+screws and springs keep the **6 center pieces** attached to it, in such
+a way that they can spin. This is what makes the faces turn.
+* **Edge pieces**, with only two colored faces each. When the cube is
+assembled, they lie between two center pieces. There are 12 of them.
+* **Corner pieces**, with three colored faces each. When the cube is
+assembled, each corner piece is adjacent to 3 edge pieces, and it
+touches 3 center pieces "diagonally". There are 8 corner pieces.
+
+So far so good. But what does this tell us about solving the cube
+by turning its sides?
+
+First of all, the fact that the centers are attached to the core implies
+that their relative position does not change, ever. In other words, you
+can think of the **centers as fixed** pieces, and **build the rest
+around them**.
+
+Another important thing that is made clear by looking at the disassembled
+cube is that **you should think about pieces, not stickers** or colors.
+When naïvely attempting to solve the cube without a clue, many would
+think about "moving this color there". But what you have to always keep
+in mind is that you can never move a single colored sticker: the other
+colored stickers attached to the same piece, be it an edge or a corner,
+will always move with it.
+
+Now we are ready to move on to some fundamental techniques. If you have
+actually disassebled your cube, put it back together *in a solved state*:
+if you don't, there is a good chance (11 out 12) that you will reassamble
+it in an **unsolvable state**, just like a
+[15 puzzle](https://en.wikipedia.org/wiki/15_Puzzle) with the
+last two number swapped.
+
+## Building blocks
+
+Most methods to solve a Rubik's cube a divided into steps, where:
+
+* In the first few steps, you put together blocks of pieces. This is 
+also known as **blockbuilding**. These steps are often "intuitive",
+which means that they do not require memorizing sequences of moves
+to apply blindly.
+* In the last few steps, you need to move around a few remaining pieces
+without destroying the progress made in the previous steps. Most of the
+time steps like these rely on memorized sequences of moves that are known
+to only affect the position of certain pieces; but there are alternative
+approaches, as we will see later.
+
+So, what do we mean exactly by "block of pieces"? To be precise we could
+say that two or more adjacent pieces form a block when adjacent stickers
+of different pieces have the same color. For example, the simplest kind
+of block is a **pair**, that you can see in the picture below:
+
+![A corner-edge pair](pair.svg)
+
+To be precise, the one above is a coner-edge pair. There are also center-edge
+pairs, consisting of a center and an edge, but they are rarely referred to
+as "pairs". In any case, they do fit our definition of "block".
+
+A more complex example of a block is a **layer**, which is the result of
+the first two steps of the classic "layer by layer" method. This one is
+a correctly solved layer:
+
+![A layer](layer.svg)
+
+On the other hand, this is **not a layer**:
+
+![A non-solved layer, but with a solve "face"](face.svg)
+
+It is worth pausing here to reflect a bit. The last two pictures both
+clearly show a solved white face, don't they? For a most people, they
+could look equivalent. But remember what we said earlier: you should
+think about *pieces*, not *stickers*. In the last picture, the pieces
+have all white on top, but their side colors do not match (except for
+the blue-white edge and the blue-orange-white corner). So they do not
+form a block, which means that their **relative position** is incorrect.
+By contrast, the second to last picture shows a block: all side colors
+match, not only the white top. This means that the pieces are in correct
+relative position with respect to each other.
+
+With this in mind, you can starting making your own way through
+the first few steps of your solving method: try to build multiple
+blocks of pieces and put them together to make even bigger blocks.
+
+If you want some more hints, Ryan Heise's website contains some nice
+examples about building blocks in his
+[fundamental techniques page](https://www.ryanheise.com/cube/fundamental_techniques.html).
+
+## Commutators
+
+*In the rest of this page I am going to use the standard
+[Rubik's cube notation](../notation) to write down sequences of moves.
+You should familiarize with it at least a bit before continuing. Don't
+worry, it's very easy.*
+
+The more blocks you build, the harder it becomes to make progress. The
+reason for this is that you want to avoid destroying the blocks you
+have already built, so your options become more and more restricted
+as you go. This is completely normal.
+
+The most common speedsolving methods go around this issue by
+prescribing the use of memorized sequences, somewhat improperly
+called "algorithms".  For example the layer by layer method relies
+on blockbuilding to build the first layer, but on at least 5
+"algorithms" to complete the last two layers.  The more advanced
+[CFOP](https://www.speedsolving.com/wiki/index.php/CFOP_method) method
+uses blockbuilding for the first two layers, but it then requires
+78 different "algorithms" to complete the last layer.
+
+Here I am going to outline an alternative, more flexible approach,
+based on
+[commutators](https://www.speedsolving.com/wiki/index.php?title=Commutator).
+They are generally considered an advanced technique, I believe that
+they are perfectly fine to learn as a beginner. Ryan Heise's page that
+I linked above has a
+[section about them](https://www.ryanheise.com/cube/commutators.html),
+too.
+
+If you have not done it already, you should have a look at my page on
+[the Rubik's cube notation](../notation) before continuing.
+
+### Corner commutators
+
+Suppose that you manage, via blockbuilding, to reach the following state:
+
+![A commutator](comm1.svg)
+
+First of all, this would be an amazing achievement! The whole cube is
+solved except for three corners. The bottom-left corner (only one red
+sticker visible) is white-green-red, and it should go to the place where
+the white-red-blue corner is right now. The latter should in turn take
+the place of the corner on the right whose visible stickers are orange
+and yellow (the hidden sticker being green). In Mathematical terms, these
+3 corner form a **permutation cycle of 3 pieces**, or 3-cycle for short.
+
+Commutators are a general technique to solve 3-cycles of pieces. They
+can be decomposed in 4 small steps:
+
+1. **Interchange**: a single move that interchanges two of the three
+pieces.
+2. **Insertion**: a sequence of moves (usually 3) that inserts the third
+piece into the place of one of the other two, without affecting the
+"interchange" face of the cube in any other way.
+3. **Inverse interchange**: the inverse of the move done in step 1.
+4. **Inverse insertion**: the inverse of the sequence of moves done in step 2.
+
+**Note:** step 1 and 2 can appear in the other order; if they do, steps
+3 and 4 should also be swapped.
+
+Let's look at an example. From the position in the picture above, you
+can interchange the top two corners using the move U. More precisely, U
+brings the orange-yellow-green corner into the position currently occupied
+by the white-red-blue corner. The move U' also works, as it moves the
+white-red-blue corner to the position of the orange-yellow-green one.
+
+An interchange move is worth nothing without a compatible insertion
+sequence. In this case, you can use R' D R as insertion: this sequence
+of 3 moves moves the red-green-white corner to the place currently
+occupied by the white-red-blue one and, very importantly, **it does not
+affect any other piece in the U layer**. To put it in another way,
+**the interchange and the insertion only "clash" on one corner**.
+
+The last thing to decide before we put all of this together is which
+one should go first: the interchange or the insertion? This is not hard
+to figure out: I described both of them as "moving a certain piece into
+a certain position"; only one of the two moves a piece in its correct
+final position, and that is the sequence that must go first. In our case
+it is the insertion, because the red-green-white corner's final position
+is the one occupied by the white-red-blue one.
+
+So our commutator looks like this: R' D R U R' D' R' U'. Let's split
+this up to review it:
+
+* **R' D R**: the insertion sequence, moving the red-green-white corner
+to the position of the white-red-blue one.
+* **U**: the interchange move, moving the orange-yellow-green corner
+to the position now occupied by the red-green-white one.
+* **R' D' R**: the inverse of the insertion sequence. To invert a sequence
+of moves, you have to **read it backwards inverting every single move**.
+Here we start with R', because it is the inverse of R, the last move of
+the insertion sequence; then we have D', the inverse of the second move; 
+and finally R, the inverse of the first move of the insertion sequence.
+* **U'**: the inverse of the interchange move.
+
+To help understanding all of this, you can visualize this commutator
+[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).
+
+**Note:** looking at the position of the pieces is not enough to
+determine a correct commutator to permute them. Their **orientation**
+is also important. For example, consider the following case:
+
+![Another commutator](comm2.svg)
+
+The three corners are permuted in exactly the same way, so everything
+we said above could be repeated word by word, move by move. However,
+if you apply the commutator we constructed to this case, you'll get
+something like this:
+
+![Two twisted corners](twist.svg)
+
+What's wrong here? Well, obviously the cube is not solved. All the pieces
+are in their correct position, but two corners are twisted in place!
+
+To avoid situations like this when creating your commutators, you need to
+keep track of **which sticker goes where**. I know, I know: I said at the
+beginning that *pieces* are important, not *stickers*. This is still true,
+but sometimes it is important to keep track of both.
+
+Let's highlight the difference between the two 3-cycles. In the first one:
+
+![A commutator](comm1.svg)
+
+1. The red-green-white corner must go to the place of the white-red-blue one,
+*with the white sticker of the first going to the place of the white sticker
+of the latter*.
+2. The white-red-blue corner must go to the place of the
+orange-yellow-green one, *with the white sticker of the former going to
+the place of the orange sticker of the latter*.
+3. The orange-yellow-green corner must go to the place of the red-green-white
+one, *with the orange sticker of the former going to the place of the
+white sticker of the latter*.
+
+While in the second case:
+
+![Another commutator](comm2.svg)
+
+1. The red-green-white corner must go to the place of the white-red-blue one,
+*with the white sticker of the first going to the place of the* **blue** *sticker
+of the latter*.
+2. The white-red-blue corner must go to the place of the
+orange-yellow-green one, *with the* **blue** *sticker of the former going to
+the place of the* **green** *sticker of the latter*.
+3. The orange-yellow-green corner must go to the place of the red-green-white
+one, *with the* **green** *sticker of the former going to the place of the
+white sticker of the latter*.
+
+The main point here is that not only interchange and insertion moves
+should swap the correct pieces around, but they must also move each
+"reference sticker" to the position of the next "reference sticker".
+For example, using the commutator R' D R U R' D' R U' for the second
+case does not work, because the insertion sequence R' D R moves the
+white sticker of the red-green-white corner to the position of the
+red sticker of the white-red-blue one, while it should move it
+to the position of the blue sticker!
+
+I won't repeat the whole construction for the second commutator,
+but you can visualize a solution
+[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).
+
+### Edge commutators
+
+So far I have only talked about *corner* commutators, but what if you
+are also left with some unsolved edges? For example, consider this case:
+
+![A edge 3-cycle](edgecomm.svg)
+
+The picture shows a 3-cycle of edges.  You might think that the same
+reasoning can be applied and that you can use commutators to solve
+edge 3-cycles of edges. This is exactly the case, and this is why this
+subsection is so short.
+
+Let's see how to solve the case above. As interchange move, you can use
+the **inner-layer move** E' (check out the [notation page](../notation)
+if you are unfamiliar with these). The insertion sequence to be used
+with it is L' U2 L. Putting everything together, you get
+[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).
+
+### Commutators with set-up moves
+
+At this point I have good news and bad news.
+
+The good news is that commutators are so powerful that you could solve
+the whole cube using just commutators and at most one single move (this
+sentence might sound a bit strange, but it is Mathematically correct -
+the best kind of correct). Although it would not be very efficient, you
+could avoid blockbuilding altogether and move pieces around
+only with commutators - this is how advanced
+[blindsolving](https://www.speedsolving.com/wiki/index.php?title=Blindfolded_Solving)
+methods work.
+
+The bad news is that not every 3-cycle can be solved directly with a
+commutator, at least not one of the form I described above. Sometimes
+you need to use **set-up moves**, also known as 
+[conjugates](https://www.ryanheise.com/cube/conjugates.html).
+
+Consider the following case:
+
+![A 3-cycle of corners requiring a set-up move](setup.svg)
+
+No matter how much you try, you are not going to find valid interchange
+and insertion moves as above. The fundamental problem is that you would
+like to use U (or U', or U2) as an interchange move, but this move affects
+all 3 of the corners. You might think of using R or F as interchange; they
+do affect only two of the pieces, but they do not move the the stickers in
+the correct position: any commutator based on R or F as interchange move
+would lead not to a solved cube, but to some corners twisted in place.
+
+So, how can we deal with this case? The solution is to use one or more
+moves to set up a better case. These moves will be done at the
+beginning and then undone at the end.
+
+For example in this case you can start by doing L as a setup move.
+This has the effect of moving the white-red-green corner out of the U
+layer, so that you can then use U (or rather, U') as interchange move.
+The insertion sequence that makes it all work here is R D2 R', and
+putting it all together you get:
+
+* Set-up: L
+* Interchange: U'
+* Insertion: R D2 R'
+* Inverse interchange: U
+* Inverse insertion: R D2 R'
+* Inverse set-up: L'
+
+**Note:** in this case the insertion coincides with its inverse. This
+can happen and there is nothing particular about it.
+
+As usual, you can visualize the final result on
+[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)
+
+## Conclusion
+
+With what you have learned so far, you can now try and solve the Rubik's
+cube on your own, without further help. Granted, it won't be a walk
+in the park: this short tutorial is not meant to explain everything. I
+could have given you advice on which blocks to build first or on when to
+stop building blocks and start using commutators, I could have shown you
+many more examples, I could have told you how to address tricky cases
+like permutation parity or pieces twisted in place. But I think it can
+be more fun to try and figure all of this out by yourself - and if you
+disagree, just look for a more complete tutorial online.
+
+Happy cubing!
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diff --git a/src/speedcubing/figure-it-out/setup.svg b/src/speedcubing/figure-it-out/setup.svg
new file mode 100644
index 0000000..fa2360a
--- /dev/null
+++ b/src/speedcubing/figure-it-out/setup.svg
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diff --git a/src/speedcubing/figure-it-out/twist.svg b/src/speedcubing/figure-it-out/twist.svg
new file mode 100644
index 0000000..79e0a02
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diff --git a/src/speedcubing/speedcubing.md b/src/speedcubing/speedcubing.md
index ee2fe4b..93d1945 100644
--- a/src/speedcubing/speedcubing.md
+++ b/src/speedcubing/speedcubing.md
@@ -27,6 +27,9 @@ Below you can find some links
 
 * [Rubik's cube notation](notation), good to know before reading any
 text-based tutorial.
+* [Figure it out](figure-it-out): a "tutorial" on how to figure out
+the Rubik's cube by yourself, without spoilers. It will not teach you
+how to solve it step by step, but it will point you in the right direction.
 
 ## Software