Solving the Edges
Step 1: Centers | Step 2: Edges | Step 3: Fix parity
Intro | 2 pair chain solving | Other pairing methods

2 pair "chain" solving

This is the method that I would use about 90% of the time. In reality I use a combination of this idea and the 6 pair solving idea. I will provide examples of when in the appendix at the bottom of this page. Below I will continue our example centers solve using this edge method.

If you have just come from the centers page your cube is already correct, skip to the edges stuff below.

If you have the same color scheme as I do then scramble with yellow on top and green on front. To get to this scramble from a solved cube do:

b F (Ll) u2     B U' F2 R B     F' l2 D2 l2 B     L B' d2 b r     l2 f u' f ' R     D' B u2 B' R'     u F2 b2 U2 (Ff)'     F' R' D r d     F2 U2 (Rr) F B     (Rr) L2 (Bb)2 F (Uu)'     F2 (Uu) (Bb)' D' R'     (Bb) D' (Bb) L (Bb)'     L (Bb) R2 (Bb) L2     U (Bb) L (Bb) L2     (Bb)' y

Now your cube should look like this:
Front/Top View
Back/Bottom View

If your cube does not look like the diagram, please solve and rescramble with that alg above. I know it is long, but I double checked and it does get to the position shown here.


How to solve 2 pairs and chain each one together

What we are going to do for this method is solve two pairs at a time, but also make sure that the end of the 2nd pair for one round leads right into the beginning of the first pair for the 2nd round. So we'll sort of chain our way through all the edges until we're done. The benefits of this method are that there are no bad or problem cases really, however the downside is that we only solve two pairs at a time. So it all sort of balances out I guess.

Ok so we've just finished solving the centers. Now in the last few moves for the centers you should be looking for two edge pieces that belong together when the cube is solved. While finishing the centers I probably would have noticed the blue white edge pieces first. These pieces are in dFR and bDR. The first thing we need to do is to put them into the middle layers. It is in the middle layers (u and d) where we will pair up edges. The only real way to describe this is to just jump right into an example, so let's solve blue/white.

When I notice a pair I have a very regimented, unwavering set of recognition steps. I'll detail those here.

Top View
Bottom View

First I just find the two blue white pieces. Now I need to do the move D2 L' to get those two pieces into the middle layer such that I can line them up. However, as I am doing the D2 move (which I do as (D')2 so that the pieces pass in front of my eyes) I see the piece connected to the blue white piece that will end up in the d layer. So let's do the first D' ,

Top View
Bottom View

Now remember we are literally in the middle of the (D')2 turn, and at no point are we going to slow down. Basically I am highlighting the edges that I look for at the point that I would recognize it. So you see the edges at the same time that I would.

Now at this point I glance at the edges on the visible parts of the U, F and R faces and try to find the other white green edge. *NOTE* remember we are actually still doing the setup moves to solve white/blue, but we are almost ready to solve green/white too!

This is what I call an easy case, since the other white/green edge is easily visible at fUR. I am going to draw the next diagram with the (D')2 turn completed, since realisticly but the time we found the other green/white piece we would have already completed that turn.

Top/Left View
Top/Right View
Left/Bottom View

Now remember, the moves we're doing do not line up with the pieces we're following mentally. Physically we are just about to do the L' move to ready blue/white to be solved. Mentally though we're going to plan on how we're going to solve green/white! At this step you are now, mentally, 6 moves ahead of what your hands are doing!

So let's plan 6 moves ahead. Now remember our very next move is going to be L' which will ready the blue/white pieces in the u and d layers such that a (Dd)' move will align them in the FL position of the cube. Now here's the key, notice that the green/white piece that is next to the blue/white one (the green/white piece that in the above diagram is in bDL) that will be in dFL after our L' move will be kicked back to the dBL position. After I have paired up blue white with the (Dd)' move at FL, replacing that solved pair with the edge pair at UR with the move L' U2 L will mean that undoing our d layer turn with (Dd) will solve green/white in the FL slot. If you're having trouble following this here is a diagram of what we're going to do. Remember, we haven't actually done any of these moves yet, no net even the L' move. This is all stuff that should be in your head ready to do.

Ok so mentally we are picturing this (read the steps in order),

1) After the L' move we want to do (Dd)'. This will pair up blue/white...2) ...and kick that green/white piece to dBL.

3) Aftering pairing up blue/white we want to replace that solved edge group with the green/white at fUR such that it ends up in the u layer (the circled spot in the diagram). Notice that by doing so the green white piece at dBL (the one with the "X" on it) will pair up with it after undoing our d layer turn with (Dd).

Ok so that is what we're going to do, but remember as of right now our cube still looks like this,

Top/Left View
Top/Right View
Left/Bottom View

So based on the diagrams and examples above, the move we want to do to pair up both blue/white and green/white is L' (Dd)' L' U2 L (Dd). *NOTE* Don't actually do this move yet! So now you know what we are trying to do, but I haven't actually been fully honest with you yet. My recognition is actually even a bit more complicated. I didn't want to do too much at once, so now let me step you're through exactly how I would solve those two pairs. At this point please actually follow along and do the moves as I do.

1) The first thing I notice is blue/white. It stands out to me, I don't know why.2) I twist the cube around like a madman until I spot the other blue white. However I don't just look at the blue/white piece, I always notice the piece that is next to the one I'm looking for. This is the secret to making the chain method work. At this point I also notice that (Dd)2 L' will place the D layer blue/white such that I can pair it up with the one already in the u and d layers.

3) So I just finished doing the (Dd)2 move, which I would still have done as [(Dd)']2 in case I missed the green/white edge the first time. This gives me a chance to double check should I need it. Ok so by this time I would have spotted the other green/white and the blue/yellow piece next to it. Remember, when I am looking for a specific piece somewhere on the cube, once I see it I always remember which piece is next to it. At this point I know that L' (Dd)' L' U2 L (Dd) will solve both the blue/white and green/white edge pairs, and leave that blue/yellow piece I spotted at dFR. Notice this is exactly where we started with the blue/white piece! The only difference is that the colors of our base piece have changed, and we have two more solved edges on the cube!

So now our cube looks like this,

Top/Left View
Top/Right View

However, again I haven't been completely honest with you. Again my recognition is a bit more complicated than this. If you have already done the moves on your cube to solve both blue/white and green/white, then undo right now with (Dd)' L' U2 L (Dd). In other words don't undo all the way, just to where we haven't paired up blue/white yet.

Now in order to continue the chain solving we have to find the other blue-yellow piece before we finish doing the move (Dd)' L' U2 L (Dd). This is very important. Now you are going to be twisting the cube around like crazy, so it's better to get the (Dd)' turn out of the way before you start twisting, as double layer turns done while twisting are very hard to do. So do (Dd)'.

Now twist around frantically looking for the other blue/yellow piece while at the same time doing the moves to solve the blue/white and green/white edge.

Now I actually didn't notice the other piece until just after I had done the (Dd) at the very end, as I happened to be looking at the bottom left side of the cube at the time. Sometimes you can spot the piece immediately and track them through all the moves you do, and other times like this example you may finish the moves before you spot the piece. Just try to make sure you have no delays in transition from one chain to the next.

OK so here is our new cube, with all the new pieces we've recognized drawn in.

Top/Left View
Top/Right View
Left/Bottom View

So these are the pieces I've noticed on this solve. Notice that I am also still drawing all the solved edge groups too. This is because that's another really important thing you have to focus on as well. Trust me, it is hard enough to try to chain the edges together that you will not be able to count the number of solved edges without it potentially slowing you down. Notice how in DB we have the green/yellow edge pair solved. This was solved right when we started the edges, so even if we were counting we would undercount by at least one edge unless we tried to take into account the colors of edge groups we've solved and catch any ones we haven't solved before.

Trust me, it is not economical to count edges during a speed solve, in my opinion it would only slow you down. Just try to keep a "feel" of how many solved edges there are. Also, keep in mind you do two edges at a time each time, so you should have to chain around 5 times on average before you have to worry if there are any other unsolved edges left. So I'm drawing the solved edges since you really need to keep an eye on how many are solved in order to get an idea if you are done or not.

Ok so notice how I've not only drawn the other blue/yellow piece, but also the one that was next to it. Remember, when looking for a specific piece on the cube, once you find it always remember the piece that is next to it, as it becomes the next step in your chain. Ok so now looking at where the two blue/yellow pieces are I would rotate the cube with the rotation y, or if you don't know xyz notation spin the whole cube as if you were doing the move U. Now do D' R. Here's the trick though, while you are doing the cube rotation and the two moves to setup blue/yellow, look frantically around for the other white/orange piece (the one that was next to the blue/yellow piece in the D layer). Now to be honest I didn't notice it while doing these setup moves and the cube rotation, which happens sometimes. However, it's ok because there is a cool trick to use here.

Now remember our cube is setup to solve blue/yellow already, but we have no idea where the other white/orange piece is!

Front/Left View
Front/Right View

Ok now here's what we're going to do. We need to look for the other white/orange piece. Now what is the most devasting place it could be in? Remember the most visible locations to look are the U and D layers. So an edge is hard to find if it is in either BL or BR, in fact being in these spots also throws off our chain solving. So it is devastating to have the edge we need in either BR or BL. So to solve this problem simply check there first. Rotate your whole cube towards you as if you were doing the move L. Is the piece we need in either BL or BR? The answer is no! There are only two scenarios here. Either the piece we need is not in BL or BR (that's good), or the piece we need is in BR or BL (that's bad). If the piece is in either BR or BL then do a quarter turn on the B face such that the piece ends up in the U layer. If the piece is in BL then do B' and if it is in BR then do B. If the piece is not in either BR or BL then do the required (Dd) move to pair up the pieces you have setup to pair.

Ok so blue/yellow will pair up in FR if we do (Dd), and the other white/orange piece is not in either BR or BL, so we can safely do (Dd)right here. After you have done the (Dd) move, then you can safely start looking for the other white/orange piece. The reason for this is, had we done the (Dd) turn right at the start, and the white orange piece was in either BR or BL, then when we finally did finish scanning the U and D faces, we would find it in the middle layer and the centers would be misaligned. This means there's nothing we can do but to either replace the solved blue/yellow pair with some random pair, solving only 1 pair, or undo our (Dd) move, get white/orange out of the middle, then redo the (Dd) move. So now you see why it is so terrible to have the piece you need in either BR or BL.

Now our example was actually a lot easier than this. After rotating the cube towards you to see if white/orange was in BR or BL I'm sure you saw that it was right in front of your face at what had become UF! Now 11/12 times that won't happen, so when it does that's good! Just keep in mind that having the piece you need in either BR or BL is so terrible, that you always, always, always check there first if you haven't found a piece by the time you are ready to solve the first pair in your chain.

Ok so we found the other white orange piece, it is in lUB. Now what else should we know by now? Correct, that the piece next to white/orange is red/yellow.

Ok so the chain goes like this, (Dd) to pair up blue/yellow and kick our middle layer white/orange to dBR, R U R' to replace our solved blue/yellow edge pair such that undoing our first d turn will also solve white orange and kick our red/yellow piece to dFL. So now, before doing any moves, we know where the red/yellow piece will end up. So scan the cube frantically looking for the other red/yellow piece as you do the moves (Dd) R U R' (Dd)'. Now I noticed the other red/yellow piece immediately after having finished the (Dd)' turn. I was scanning the bottom face as I did that move (I always scan the U face first since I can already see it, then the D face afterwards). Now what else do we need to know at this point too? Correct, the piece next to the D layer red/yellow piece is green/orange.

So our cube looks like this now,

Front/Left View
Front/Right View

Ok so now notice that the setup move to get ready to solve red/yellow is R. Do that move now. Ok now at this point we don't know where the other green/orange piece is yet. So the first thing you do is to glance where? Exactly, at the BR and BL slots. Is the other green/orange piece in either spot? No, it isn't. So now do the move (Dd), and at the same time look around frantically for the other piece. I always look at the U face first, since you are already looking at it, then spin to the bottom. Now if you do that, scan the U and D faces you won't find our piece. Ok now this is important. You know exactly where it is now, and we haven't even seen the piece! Remember, you checked the BR and BL slots already, so it isn't there. Now you have scanned the entire U and D layers and it wasn't there either. You know the red/yellow piece that starts the chain is in dFL, so our piece has got to be in uFL! This is a very, very important trick for you to learn, using the process of elimination like this. It is exactly for that reason that I always scan BR and BL, then the U layer, then the D layer. If after having done that I know 100% for a fact that my piece is connected to the first piece in my chain! Pretty cool huh?

Ok think about this. How do we handle it when the piece that we need to complete the chain is in a place such that we cannot use it the way we have been doing our chain before? We're going to have to break our chain right? Wrong! This case is actually awesome, because we will now solve 3 pairs in the same sequence rather than 2!

Ok so here is how this works. I know now by the process of elimination that the piece I need is in uFL. Well if you look real hard I could either pair red/yellow up at FR with (Dd) or green/orange at FL with (Dd)'. Is there a way to take advantage of this? Yes there is! Find a random unpaired edge in the U layer, and do it fast! Now if the U layer contains nothing but solved edges at this point, then find a random unpaired edge in the D layer. Notice on our cube that all the edges in the U layer are solved. So the edge I would pick is in the D layer at DB. Now I am picking this edge for several reasons. First off notice that the red/white piece is in the DB edge at rDB. And also notice that the other red/white edge is very visible right now at bDR. So we are going to solve not only red/yellow and green/orange, but red/white as well!

Ok, so our cube looks like this right now (including all the pieces that we have recognized, and all our solved pairs).

Front/Left View
Front/Right View

From here do the move (Dd) to pair up red/yellow at FR. Now remember, we did the double layer turn, so the edge we're using to replace our solved pair went from DB to DL. So place the DL edge into FR such that the red/white piece ends up in the d layer. So do F D F'.

Now our green/orange piece in the d layer is in dBR and we want it to be in dFL to pair with the other green orange, so go ahead and do that. Do the move (Dd)2.

Now replace the solved pair in FL with the other red/white edge such that the red/white one ends up in the u layer. So do the move L D2 L'. Now do (Dd) to line up the centers again, and viola we've also paired up red/white at FL! 3 pairs at once for only 3 more moves than our regular way! This is actually very good, since it is impossible to pair an edge and leave the centers aligned properly in 3 moves. We utilized this special case to pair an extra pair in extremely few moves!

Now that we did 3 pairs the way we did we are thrown completely out of the chain and have to get back into a chain solve. So look for two pieces that belong together, and the ones I saw are green/red because they are so close together at dFR and fDR.

Top View
Bottom View

Notice that the edge next to the green/red already in the middle layer I don't care about, but the one next to the edge in the D layer I do care about, it is yellow orange.

Again we have no idea where that edge could be right now since we were thrown out of a chain solve. So where is the very first place we look? Exactly, BR and BL. So check there and viola our edge is in BR!

We can actually setup green/red to solve, and get our other orange/yellow edge out of the middle layer with the move D B. However if this were an actual speed solve I would do this move as (U u d') R. So do the move (U u d') R on your cube right now.

Now remember by now we should have noticed that the piece next to the other green/red is blue orange. Well remember, we've done a lot of 2 pair solves by now, and lots of edge pairs are solved, so check at the edge above our first green/red (the one now at dFL). It is also blue/orange! So this tells us that these three edges will solve togehter, a good indicator that we might almost be done.

So your cube should look like this now,

Front/Left View
Front/Right View

To solve those three edges together pair up green/red and kick the middle layer orange/yellow back with (Dd). Then replace the solved pair with the U layer orange/yellow such that the orange/yellow piece ends up in the u layer with the move F' U F. Now undo the d layer move with (Dd)' and those three are solved.

Actually if you noticed, we had two edge pairs solved right from the start too, so now we're done!

Front/Top View
Back/Bottom View


Other endings for the edges step

Other than ending on a special case like in the appendices below, here are two other endings that I see as well.

4 edge pair ending - two double swaps

At this point there are only four edges left to solve on the cube, and they are in the configuration below,


This is an uncommon way of doing 4 edges at a time. What we will do is pair up the pairs at FL and BR with a (Dd)2 move. Notice also that the two edges in the U layer can pair up the same way, so replace them in the u and d by doing the move L' U L B U2 B'.

Now your cube should look like this,


And lastly a (Dd)2 turn will line up the other two edges.

Two edge ending

This ending is very common. If you noticed, for our chain solving we actually use 3 edge groups, even though we only actually solve 2. So what do you do if you end with 10 edges solved and 2 unsolved?

First off get them adjacent to eachother like this, or also on opposite diagonals at FR and BL works just as well, just replace all the (Dd) moves in the following alg with (Dd)2 if you start with them at opposite diagonals of the middle layer.



Now do the move (Dd) to move the dFL piece over to dFR, and also kick the dFR piece back to dBR.



Now if we flip the edge in FR, undoing our d layer turn will solve those two edges. To flip simply do the move R F' U R' F. For flipping the edge some people prefer R U2 R' F' U' F but I would definitely suggest the first, shorter alg.



From here do a (Dd)' to pair up the two edges again and you're done!

Appendix A

Special case scenarios when I would use the 4 or the 6 pair method

At the top of this page I mentioned that I only use the 2 pair method about 90%-95% of the time, so here is what I am doing the other 5%-10%. Predominantly I would use the 4 pair method, which is in between the 2 and 6 pair methods, but there are some very rare special cases where I would use the 6 pair method.

So here is an example of when I would use the 4 pair method. I would say I use this about 5%-9% of the time, with the less than 1% left over left for the 6 pair method.

Here is the scramble to set up what I am going to do. If you want this to look like a real solve, you can do this. Scramble your cube then solve the centers. Now solve 4 edge groups and place them in the UL, DF, FR, and BR locations. See picture below,

Start with the solved edges here,


Now apply this scrambling algorithm and you will be setup for this example: (Dd) R U2 R' B' D B (Dd)'

Remember this is only a special case for me, so my approach to this case is still to just do a 2 pair chain solve. So let's start with a 2 pair chain solve where I am ready to pair up dFL with uFR by doing a (Dd) move. I will include diagrams, but remember that our colors are very likely to be different since there are 495 different ways to solve 4 edges ;-)

So here is all I see so far thinking in my regular 2 pair mode,

Front/Left View
Front/Right View

Let's say also that this happened at the right situation from the previous chain that I have no idea where the other green/orange piece is. If that happens where's the very first place I look? You gotta remember this, it's very important. Exactly, I look in BR and BL. Now let's look in BR and BL and viola green/orange is in BR and not only that it is in uBR! That's even better! Notice now that a (Dd) move will make a pair in FR and BR! Now I could do my regular thing and kick that green/orange to the U layer by doing B then pairing up green/yellow and chaining this, but in actuality the 4 pair method is perfect to use in this situation!

Now again your colors are very likely to be different from mine, so we need another diagram. After noticing the other green/orange remember that I always look at the piece next to the one I am looking for, there is never an exception to this.
Front/Right View
Right/Back View

Now on my cube the other color next to the green/orange piece is the blue/orange. Ok now we're going to take a concept that I've detailed in the example solve and expand it to this addendum example. There is one position that the other blue/orange can be in that would absolutely destroy the special case taking advantage of a position for a good end and turn this into a move wasting approach. The usual places I check are BR and BL since they are the spots in the middle layer not being used. Well now we have pieces in FL, FR, and BR that are very important to our chain. So the horrible spot would be if the other blue/orange was in BL. So check BL right now for the other blue/orange piece. It's not there, so immediately pair up the two edge pairs at FR and BR with the move (Dd).

Now the rest continues like a chain solve. We need to find the other blue orange piece and put it into BR such that the blue orange is in the u layer and we'll solve blue orange when we undo our d layer turn. That edge is right now in DL. Ok now what is the other thing you should have noticed already as well? Correct, the edge next to the blue/orange piece is blue/white. Ok now while we are placing the blue/orange and blue/white edge group into BR such that the blue/orange piece ends up in the u layer we will be looking around for the other blue/white piece. To place that edge into BR correctly you need to do B' D' B, so do that now, while looking for the other blue/white.

The other blue/white edge is in UL. Again what else should you have noticed by now? Exactly, the edge next to blue/white is red/yellow (or whatever it is on your cube). Now do the move R U2 R' and finish with (Dd)' to pair up four more edges. Your four correct edges should now be in the spots shown below.


To extend this to six pairs you would do this. After realizing that the first two edge pairs at FR and BR will pair up with a (Dd) move, you would check BL to make sure that the piece that pairs with the dBR piece isn't there. If the piece that pairs with it is in uBL then you will pair three edge pairs with the (Dd) move. At this point you will have checked the edge in dBL. If it pairs with the piece in uFL then you will do four pairs with the (Dd) turn (extremely low probability). So now chain your way through all the edges on the way back, the same way I've described for the 4 pair example and you will do 6 pairs at once. If you are able to do 4 pairs on either the first d layer turn or on the second then you can do 7 pairs at once. If you can do 4 on both d layer turns (extremely ridiculously low probability) then you can do 8 pairs at once.


Appendix B

Knowing when to break the chain

It isn't always beneficial to continue the chain when 2 pair solving. In fact sometimes it is better to break out of the chain you are in and start a completely new one. Consider this example,

Front/Left View
Front/Right View

We have just solved two pairs and they ended up in BL and UF. The last piece in our chain, the one that leads into the next one is blue/red. However, you have not yet found the other blue/red piece. Now, while scanning the U layer you noticed that the two green/yellow pieces are ready to be paired up with an (Ff) move. So why continue looking for blue/red when you can already pair up two pieces right now?

Here's the key to making this work though. First you have to rotate your cube such that your new chain has the pieces to be paired up in the u and d layers. So do x' if you know rotation notation, or if not rotate the whole cube towards you one quarter turn. Now you have to create a new chain, so the move (Dd) after our rotation will pair up green/yellow at FR, so look at which piece is in dFR. This is all detailed on the diagrams below,

Front/Left View
Front/Right View

Now we're starting a new chain, so we have no idea where the other blue/orange piece is, it could be anywhere right now. In that case where is the very first place we look always? Exactly, in BR and BL. So glance in BR and BL first, if you see the piece you need you have to either kick it to the U layer, or use it in a 4 or 6 pair chain special case. If the piece you need is not in BR or BL then go ahead and do (Dd) to pair up green yellow and continue searching for the piece. Now just continue your chain solving from here, you are well into your new chain already.
Appendix C

Avoiding opposite diagonal pairing of edges

This is something I've come to realize from solving the 4x4x4 for a while, and that is that pairing over an opposite diagonal is harder than pairing on adjacent spots and should be avoided as much as possible.

Here is what I mean,

Adjacent pairing
Opposite pairing

Try as hard as you can to avoid setting up edges such that they pair across opposite diagonals. However, if you notice that two edges are ready to pair but are on opposite diagonals, don't do any setup moves to place them adjacent just solve them as is. The time it is ok to setup two edge such that they are on opposite diagonals is if they start off both in the u and d layers adjacent to eachother in the same layer. Such as if the pieces were in uFR and uFL, or dFR and dFL. In that case it's ok to just do a double turn to get one into the other middle layer.

The only other thing you need to remember is that solving opposite diagonal edges means your "problem" locations change. Remember, our problem spots for adjacent pairing are BR and BL, but for opposite solving they are FR and BL, or FL and BR. Below is a diagram,

Front/Left View
Front/Right View
Problem spots for adjacent pairing are in BR and BL

Front/Left View
Front/Right View
Problem spots for opposite pairing will either be FR and BL or FL and BR


Appendix D

What to do if you miss 2 edge pairs

Ok, so here's the scenario. You're done with the edges step and you start solving the 3x3 step with the F2L method. You're over halway through with your F2L when you realize you only solved 10 of the edge groups, 2 are still unsolved! That's bad right?! ... Well ... not really actually.

If you do run into this situation there is a 19/33 = 57.58% chance that one of the misaligned edges will be one of your cross edges, so you would spot it before even starting the F2L. However, what do you do on the 14/33 = 42.42% chance that your cross edges are fine? You may not notice the messed up edges until well into your F2L solve.

Well here is what you do on the 14/33 chance that you get well into your F2L solve without noticing the misaligned edges.

Front/Left View
Front/Right View

The way to solve this is easy, put each unsolved edge into an F2L slot, such that the pieces that belong together end up in the same layers, like so,

Front/Left View
Front/Right View

Now just do a regular 2 edge ending, but to flip the edge you have to use an F2L-preserving alg. So I would do (Dd) R U' R' F' U2 F (Dd)'. And now you can continue your F2L solve.

Definitely try to avoid this situation, but don't freak out if this happens. Remember, you would have had to solve those edges earlier anyway, so the only way you are wasting moves is that the setup alg to get the pieces next to each other will be around 6 moves instead of 1-3. Also your flip move is 1 extra move long now since you have to preserve the F2L. So you've wasted only about 4-5 moves this solve, which translates to about 1-2 seconds. So don't freak out, just take care of it.

Step 1: Centers | Step 2: Edges | Step 3: Fix parity
Intro | 2 pair chain solving | Other pairing methods
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