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Arnaud's 5x5x5 Edge-Pairing method: Examples

AvGalen

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I thought it would be a good idea to demonstrate how I solve the edges on the 5x5x5. I think this method is at least as fast as the bigcubes.com method. I average around 60 seconds with the "advanced" variation of my method.

The basic idea is to use the 2-at-a-time method that is used by a lot of people on the 4x4x4. If you don't know it, I suggest you learn it first from bigcubes.com. For the 5x5x5 you don't match a wing with another wing, but you match a wing with it's center-edge. If a wing is matched with it's center-edge I call this a semi-pair. If you try this method on a 5x5x5 you will see that it works very good sometimes, but very bad other times, especially if you try to use 4-at-a-time or 6-at-a-time. This is because a center-edge has 2 wings. To avoid those problems you start with a semi-pair and keep it there all the time. If a semi-pair is not available directly after doing centers you have to creat one, which is really easy. The way I perform the edge-pairing is that 0. I put the semi-pair on the upper half of the Left-Front tredge-spot. This means my "1st working wing" is in the lower part of the Left-Front tredge-spot.
1. I then put the first edge-center at the Front-Right-spot
2. I then do a d-move to form a semi-pair at the Front-Right-spot. The "2nd working wing" has been moved to the Right-Back-spot
3. I then replace the first edge-center at the Front-Right-spot with the 2nd edge-center.
4. I then do a d-move to form another semi-pair at the Front-Right-spot. The next "1st working wing" is now at Left-Front again.
There are 3 special cases "Parity before the end", "flipped wing", "full 3-cycle" and they can be handled in different ways. Analyze the examples to find out how I do that.

I have developed a scramble that keeps the centers solved, the edges scrambled pretty well and that is easy and fast to execute. To follow my examples you should have a 5x5x5 cube with a "regular" Blue-Orange-Yellow colorscheme. The scramble should be executed with white on top and green on front. DFRBLU are outer face moves, dfrblu are double layer moves. I don't use slice moves at all.

To make it easier to clarify any questions that people might have I will update this first post with general information. I will use the next five posts for the following examples: Beginner, Intermediate, Advanced, Expert, Other additions such as an example with parity and a worse beginning.

OK, lets get started:

Scramble (57 moves in total, but a wrote them in an easy to follow manner)
r U r' U r U2 r'
F' U
l' U2 l U l' U l
L2 R2 F2 B2

r U r' U r U2 r'
F' U
l' U2 l U l' U l
L R' F2 D' B2

r U r' U r U2 r'
F' U
l' U2 l U l' U l
 
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AvGalen

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Beginner

Beginner (117 moves used): Pre-setup to Up-Back, put at Front-Right using (F' U2 F) or (R U R')

Red-White: D B2, (F' U2 F)
Green-Red: B, d (R U R') d'

White-Red: U2, (F' U2 F)
White-Green: B, d (F' U2 F) d'

Orange-Yellow: D2 B2, (F' U2 F)
Blue-Yellow: U2, d (R U R') d'

Green-Yellow: U', (R U R')
Yellow-Blue: U2, d (F' U2 F) d'

Green-White: U, (R U R')
Easy piece (will not match yet) Yellow-Green: d (R U R') d' (special case "Parity before the end")

Yellow-Green: U', (R U R')
Red-Yellow: D B2, d (F' U2 F) d'

Blue-Orange: D2 B2, (R U R')
Yellow-Red: U2, d (F' U2 F) d'

Orange-Green: U, (F' U2 F)
Green-Orange: d (F' U2 F) (R U R') d' (special case "flipped wing")
Red-Blue: x' y (special case "full 3-cyle")

Yellow-Orange: D B2, (R U R')
Easy piece (will not match yet) Red-Green:U d (F' U2 F) d' (special case "Parity before the end")

White-Orange: B' (F' U2 F)
Red-Green: d (F' U2 F) d'
Orange-Blue: y2 z' (special case "full 3-cycle")

Blue-White: U2 (F' U2 F)
White-Blue: d (F' U2 F) (R U R') d' (special cases "flipped wing" and "full 3-cycle")
Orange-White: (special case "full 3-cycle")
 

AvGalen

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Intermediate

Intermediate (86 moves used): Put at Front-Right in few moves

Red-White: F D' F'
Green-Red: B', d R U R' d'

White-Red: U F' U' F
White-Green: B, d F' U2 F d'

Orange-Yellow: B R2
Blue-Yellow: d R U' R' d'

Green-Yellow: R
Yellow-Blue: d F' U F d'

Green-White: F D' F'
Red-Blue: d' F U' F' d (special case "Parity before the end")
Red-Yellow: B, d R U R' d'

Blue-Orange: D R
Yellow-Red: d F' U F d'

Orange-Green: B R2
Green-Orange: d (F' U2 F) (R U R') d' (special case "flipped wing")
Yellow Green x2 (special case "full 3-cycle")

White-Orange: D R
Red-Green: d' L' U L d (special case "Parity before the end")
Orange-Blue: d R' D2 R d'
Yellow Orange: y' x (special case "full 3-cycle")

Blue-White: F D F'
White-Blue: d (F' U2 F) (R U R') d' (special case "flipped wing")
Orange-White: (special case "full 3-cycle")
 

AvGalen

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Advanced

Advanced (75 moves used): Put at Front-Right in fewest moves (4u means the top 4 layers or D y)

Red-White: 4b R' (= F z' R')
Green-Red: d R' D2 R d'

White-Red: B R2
White-Green: d F D F' d'

Orange-Yellow: B' R2
Blue-Yellow: d F' U' F d'

Green-Yellow: U' R'
Yellow-Blue: d F D2 F' d'

Green-White: 4b' R (= F' z R)
Red-Blue: d' F' D F d2 (special case "Parity before the end")
White-Orange: F' U' F d'
Red-Green: z2 (special case "full 3-cycle")

Blue-White: D R
White-Blue: d (F' U2 F) (R U R') d' (special case "flipped wing")
Orange-White: x' (special case "full 3-cycle")

Red-Yellow: B' R2
Blue-Orange: d F' U' F d'

Yellow-Red: R'
Orange-Green: B d R U R' d'

Green-Orange: R 4b' R (= R F' z R)
Yellow-Green: d' L' U2 L (special case "Parity before the end")
Yellow-Orange: d2 R' D2 R d'
Orange-Blue: (special case "full 3-cycle")
 

AvGalen

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Expert

Expert (73 used): Put at Front-Right in fewest moves (4u means the top 4 layers or Dy), use 4-at-a-time on the final edges

Red-White: 4b R' (= F z' R')
Green-Red: d R' D2 R d'

White-Red: B R2
White-Green: d F D F' d'

Orange-Yellow: B' R2
Blue-Yellow: d F' U' F d'

Green-Yellow: U' R'
Yellow-Blue: d F D2 F' d'

Green-White: 4b' R (= F' z R)
Red-Blue: d' F' D F d2 (special case "Parity before the end")
White-Orange: F' U' F d'
Red-Green: z2 (special case "full 3-cycle")

Blue-White: D R
White-Blue: d (F' U2 F) (R U R') d' (special case "flipped wing")
Orange-White: x' (special case "full 3-cycle")

Red-Yellow: B' R2
Blue-Orange: d F' U' F d'

Yellow-Red + Orange-Green: R' d y
Green-Orange + Yellow-Green: F' U2 F R U R'
Yellow-Orange: y' d2 L' U L
Orange-Blue: d2 F D2 F' d'

I would actually recommend against using this method on this example. As you can see it is pretty difficult to see what is happening and you only save a couple of moves. I just included this to show that it is possible.
 

AvGalen

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Other additions

Reserved room to be used later for parity and other additions.

For now it is important to know that (with an unlikely exception, the single edge flip) you will have 50% chance of no parity and 50% chance of the "edge flip/swap" that is explained on this page. If you get it during the solve (not at the end) you shouldn't use this parity fix! For beginners there is an easier algorith to change that parity into a 3-cycle: (d B2)*5. Again, only use the parity fix at the end of the edge-pairing step!
 

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thanks for this Arnaud. I didn't realize the only difference in your method (from beginner to advanced) is just the insertion moves. Now I just have to work on execution and practice these moves. Hopefully I can get it down to be able to sub-2, but with a 30+ second 3x3, it seems almost impossible.
 

AvGalen

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Finally, someone writes a reply :)

The difference is not just the insertion moves!

For beginner you "pre-setup" at Up-Back, you don't do that for intermediate and further. This saves a lot of moves, but it also means there are a lot more insertion moves. With a bit of practice these insertion moves will be easy and short (intermediate) or even easier and shorter (advanced) if you use some cube rotations/4-layer-turnes.

Another difference are the way you handle (special case "Parity before the end"). For beginner you just solve 1 single piece piece instead of 2. For intermediate and further you solve 3 pieces instead of 2 (with some added moves)

And the way you solve the final parity is different, but that doesn't happen with this scramble.

I think the limit for this method will be around 40 seconds (about 2 moves per second). I can do it in 60, Erik did it in 52 at the Chech Open and has improved a lot since then.

Just remember, don't look at the pieces you are solving, always look for the next center-edge. If you do it correctly you don't stop at all during the 24 semi-pairs and you have only looked at the Front-Left-Down Edge-Wing once!
 

Mike Hughey

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Sorry I didn't reply sooner, Arnaud! I hadn't had the chance to actually try out your examples to see how they went.

Thank you very much for this - it comforted me by satisfying me that I'm doing it right - my solution would have looked very much like your Advanced one, except for the fact that I would have gone wrong on the very first move by setting up with F D' F' instead. I have everything else optimized by turning the cube, but somehow I missed this one. (I would even get it right if it was on the other side - funny how you develop blind spots for things like this!) So that might help me a little.

One thing I wonder about: If you have a case of pairs of what BigCubes.com calls the "double edge flip", do you have any simpler way of solving them other than to do the algorithm they have there? For that matter, I guess just a "single edge flip" could happen as well. If I ever encounter that (which happens often enough I've had it a number of times - once even with 2 pairs in a single solve!), I just perform that algorithm when I get to the end. But I was wondering if there's a better way that you know of.
 

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"double edge flip" and "single edge flip" are extremely rare. Basically they will only occur if they were already there after doing centers. They will never be formed during the edge-pairing as long as you keep using a semi-pair as your starting position.

If they do happen to occur, fix them directly at the beginning of the edge-pairing phase with the following moves:

Single edge flip:
* Put the wrong edge at the Front-Right (mostly without a move by using cube rotations)
* Make sure the Front-Left doesn't have a semi-pair (that is why you should do this as soon as possible. Most of the time there isn't a semi-pair at Front-Left or an L or L' move will do)
* Just do what you would do in (special case "flipped wing") with two added double layer u turns so d (F' U2 F) (R U R') d' becomes d u' (F' U2 F) (R U R') u d'
* You have used 10 moves for 2 semi-pairs

Double edge flip:
* Put the wrong edge at the Front-Right (mostly without a move by using cube rotations)
* Make sure the Front-Left has the other wrong edge (2 moves at most)
* Just do what you would do in (special case "flipped wing") with two added double layer u turns so d (F' U2 F) (R U R') d' becomes d u' (F' U2 F) (R U R') u d'
* You have used 10-12 moves for 4 semi-pairs

Yes, that's right it's the same!
 

Mike Hughey

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Okay, Arnaud! Thanks! I see that for the "double edge flip", this really isn't any different from doing the BigCubes.com solution, but for the "single edge flip", this is a real savings, so that's helpful. The only problem I see with it is that I would think it would be easy to not notice that one of these is around when starting the edges, and since it's so rare, it doesn't seem worthwhile to actually look for it. But I guess the idea should be that, as soon as you spot one, you can stop what you're doing, fix that, and then go back to what you were doing.

Yes, I know those are extremely rare, but believe it or not I actually had a solve once where I was using your method and I had 4 of them (2 "double edge flips") in the same solve. I couldn't believe my bad luck. But I see now that it's not really all that big of a deal.
 

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If you knew the double edge flip, that would be really good luck! You could put them all in the E ring, do u d' then flip two edges diagonal of each other, solving four full edges (or eight semi pairs)!
 

Mike Hughey

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I see what you mean now. I guess it really was pretty easy even the way I knew how to do it, and now it's even easier to think about.

Arnaud, I was wondering if you ever considered the following possible optimization to your method. The idea is, instead of just using "d" to pair up the first piece in a pair, also allow "d2", IFF it turns out that would require less setup moves. I would give examples, but right now I don't have time to put one together - I can try later (if you don't immediately shoot the idea down because of something I'm missing). I tried to find one in your example, but unfortunately, in your example, it doesn't really help. You can see an example of it for the very first pair (one setup move instead of 2 from your advanced example), but unfortunately in that case, the second piece is in the way and so it doesn't really help (or hurt). But often times I think it helps, and I've been playing with it again since you posted this, and I think it might really be possible to make this work and pay off. Have you ever tried it?
 

Jack

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I do that sometimes, mainly if the the edge is already in that spot, or if it is in the FR but flipped (so that R2 will put it in BL correctly). The only problem is when you have the middle edge you will need after doing d2 in the FR or BL spots, and you have to do three moves to take it out before doing d2.
 

Mike Hughey

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I do that sometimes, mainly if the the edge is already in that spot, or if it is in the FR but flipped (so that R2 will put it in BL correctly). The only problem is when you have the middle edge you will need after doing d2 in the FR or BL spots, and you have to do three moves to take it out before doing d2.
Right. That's what always kept me from doing it before. But you can check for those 2 edges right away when it happens, and if it's not there, you can go ahead with it. And if it IS there, you can either take 3 moves to get it out, which I think is just as good as moving it to the normal place, or you can just move it to the normal place. Question: am I right that if the edge is already in the right spot for d2, it takes 3 moves to get it in the right place for d? Or is there a way to get it there that's quicker than 3 moves?

If you need just one move to get it to the correct place for d2, and it's not in the FR but flipped position, you can look ahead for the other piece you'll need, and get it out of the way before making the move to setup for d2. Once I discovered that this week, I started deciding it was worth more attention. I'm starting to really like it now. I'll probably use it for this week's competition (which means my times will probably get worse again - oh well).
 

AvGalen

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I have put quite a lot of time in analyzing the allowance of d2 (and d') matching. In most cases (8/10) it will be ok, but in the other cases very bad things happen to look-ahead, move count and overal flow. Because the gains are marginal and the risk is high I recommend against using d2 and d'.

You can see some examples in the final fases of the expert-stage. You will immediately see how messy it becomes and how few moves it saves.
 

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I will be giving it a shot soon. How are you holding the cube when you scramble?

Thanks
 

AvGalen

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About the scramble: White on top, Green on Front like all "official" scrambles. Have fun.

Question: am I right that if the edge is already in the right spot for d2, it takes 3 moves to get it in the right place for d?
Answer: Yes, that would require 3 moves or more. I would do y R U F' y'.

About d2: An example of how it could backfire: d R U R' d' U' R B' U'. If you would do b' to lign up for d2 you have a big problem. If you would use 1 extra move (U R') you avoid the problem. Besides, U R' is probably faster then doing B' and d is definately faster than d2

The best example of how this could backfire is this: F2 d' F2 d' F2 R2 d' R2 u F2 u' F2 d2 F2 R U'. What would you do after the B' move?

There are also case where it would actually be faster and save moves (simple example: d2 R' U2 R B U B' d2) but the wins are minimal and the loss can be very great.
 
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Mike Hughey

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About d2: An example of how it could backfire: d R U R' d' U' R B' U'. If you would do b' to lign up for d2 you have a big problem. If you would use 1 extra move (U R') you avoid the problem. Besides, U R' is probably faster then doing B' and d is definately faster than d2

The best example of how this could backfire is this: F2 d' F2 d' F2 R2 d' R2 u F2 u' F2 d2 F2 R U'. What would you do after the B' move?

There are also case where it would actually be faster and save moves (simple example: d2 R' U2 R B U B' d2) but the wins are minimal and the loss can be very great.
(I would just like to say in advance - sorry for the long post. I hope it's not too inappropriate for me to post this.)

Yes, I know of these cases. And I can see how they would make you reject the idea, especially considering how smooth your lookahead is.

However, my lookahead isn't nearly as smooth. I can usually get 6 or 8 pieces put together in any given solve with the same kind of smoothness that you manage, but invariably just as I think I'm going great, I hit a piece I just can't "see". I'll look at it 3 or 4 times, and still not see it. By the time I've found it, I've wasted 10 seconds, and then I waste 10 seconds more getting going again. Which is the primary reason why I'm a minute and a half slower than you.

Anyway, at my speed, I'm finding that just taking an extra half-second to "double-look-ahead" allows me to use d2 (or actually, in my case, r2, since I use the M-slice - I'm translating everywhere here to the E-slice) very effectively, and fixes all the issues that can come up.

For instance, in your first example d R U R' d' U' R B' U', if you're "double-looking-ahead", you can see the problem coming and do D' first, then B', and you're in great shape! d2 R D2 R' d2. Same number of moves as with d - not an improvement, but at least not so bad. I guess it's bad that there are so many double moves, but with an Eastsheen, it's just about as fast as a single move. The bad thing is having to do the double lookahead, but really you only need to worry about what 2 spots look like: FR and BL. If they aren't problems, you won't have any problems.

As for your second example, I would probably have already noticed I was doing my last pair, so I wouldn't even try for d2. But if not, I'd probably just undo the B' move, and do it the regular way. (I wonder if there's a good algorithm for this position? There probably is, but I'm not sure it's worth the trouble to learn it. But if this method works out well, maybe it is.)

If you weren't on the last pair, say with your case twice with a z' thrown in: F2 d' F2 d' F2 R2 d' R2 u F2 u' F2 d2 F2 R U' z' F2 d' F2 d' F2 R2 d' R2 u F2 u' F2 d2 F2 R U', now you can just do it like you would with d, only with d2:
B' d2
F U F' (special case "parity before the end") (and now you can see what piece you need to swap with at the BR location)
B U' B' d2
and you got all 4 in just 9 moves.

There are some cases where d2 clearly hurts you, but I'm wondering if it wouldn't be possible to be selective and just use d2 in the best cases.

For instance, with your original example in this thread, we could do something like this:
4b R' (recognize that BL is "poisoned", so just do the regular d method here)
d R' D2 R d'

B R2
d F D F' d'

B' R2
d F' U' F d'

U' R'
d F D2 F' d'

R (not "poisoned" this time at FR or BL, so give it a try)
d2 F U2 F' (special case "parity before the end")
U R' U' R d2 (this worked out the same as with d - 10 moves total, so no advantage, but at least no disadvantage)
y (special case "full 3-cycle")

R2 B
d R U R' d'

U' R (not "poisoned", so try it)
d2 D' B' D B d2 (well that didn't work out well because the D slice wasn't well positioned, but the step that's about to come is compensation)

R2 (I love this case, because it's so easy to see! You can see it coming a mile away - such an easy setup move)
d' F' D2 F (special case "parity before the end")
d2 F D F' d'
z2 (special case "full 3-cycle")

R
d (F' U2 F) (R U R') d' (special case "flipped wing")

So this one works out no better than usual. But it's honestly a pretty unlucky case for using d2.

Anyway, sorry for the long post, but I'm having real luck with allowing d2 as well as d. I've had some of my best times ever with it now. I think that's mainly because it's forcing me to double-look-ahead, and I think that's proving to be good practice for my lookahead. So I'm planning on using it for a while, even if only as a training tool.

I realize your method's biggest advantage is its simplicity - you have very few decisions to make, so you can concentrate on looking for the right piece. And introducing d2 complicates things significantly. So probably when you're as fast as you are, it mostly doesn't help. But I still think it might be worth allowing it in the good cases, if you can narrow those down to a manageable set of rules. For instance, if you have a "flipped wing" you can do with d2 - that one might be worth it, since it's so easy to see and there are no complications.
 
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