cage/cage variants sub minute average potential?

[rachmaninovian]

let me propose perhaps a better form of your revised 4SCS:

for step 1, allow adjacents too. according to my brute force speed approach, adjacent on average has better times than opposites, which are harder to execute and recognize.

actually to be honest, I would really like to direct solve everything, because that way you have a clearer view of what to solve next.
In fact, after my exams (like 2 months later) I would try to find combos...like the "double niklas" (case 3,adjacent, or the opposite counterpart).

regardless of whether to preserved solved pieces or not, you would only need to find cases for step 2 and 3, because step 1 and 4 would use the same algs. based on my experience, allowing step 1 to include adjacent cases would be much faster in general. some centres cases would be better to be done adjacent than opposite.

to be efficient and to perhaps to decrease the number of algs needed, I suggest only finding cases for centres with already 2 centre pieces done.

right now I guess I might just as well come up with a similar approach, but with more freedom: 2 look 1st centre, 1 look 2nd centre, finish last 2 centres (be it adjacent/opposite). argh never mind.

 

[cubacca1972]

let me propose perhaps a better form of your revised 4SCS:

for step 1, allow adjacents too. according to my brute force speed approach, adjacent on average has better times than opposites, which are harder to execute and recognize.

actually to be honest, I would really like to direct solve everything, because that way you have a clearer view of what to solve next.
In fact, after my exams (like 2 months later) I would try to find combos...like the "double niklas" (case 3,adjacent, or the opposite counterpart).

regardless of whether to preserved solved pieces or not, you would only need to find cases for step 2 and 3, because step 1 and 4 would use the same algs. based on my experience, allowing step 1 to include adjacent cases would be much faster in general. some centres cases would be better to be done adjacent than opposite.

to be efficient and to perhaps to decrease the number of algs needed, I suggest only finding cases for centres with already 2 centre pieces done.

right now I guess I might just as well come up with a similar approach, but with more freedom: 2 look 1st centre, 1 look 2nd centre, finish last 2 centres (be it adjacent/opposite). argh never mind.

Just for the sake of clarity, the current 4SCS method (once completed) will consist of the following steps:

Step 1 (if necessary): For the first center, direct solve any of its center cubies which are in the opposite center (e.g., if solving blue center first, solve any blue center cubies that are in the green face).

Step 2: Direct solve the first center (any remaining center cubies on the adjacent faces).

Step 3: Solve the opposite center

Step 4: Solve the remaining centers

Here is my rationale for doing steps 1 and 2 in this particular way. First, if the opposite center does not contain any of the first center cubies, you simply skip to Step 2 and direct solve the first center.

The reason for drawing from the opposite center for Step 1 is to avoid the need to learn a set of algorithms which draw from an adjacent center and the opposite center and a set of algorithms which draw from both adjacent centers and preserves the opposite center. Once you clear the opposite face of the first center cubies, you can reliably go to the 15 algs which will direct solve the opposite face.

To keep the number of required algorithms for the system to a reasonable number, and to keep the system as linear as possible (no time wasting forks in the road: should I solve this way, or should I solve that way?), Steps 1 and 2 need to solve the first center. Step 3 needs to solve the next center, and Step 4 needs to finish the solve. I can forsee 3 variations on this idea, and in my opinion, only 2 would be reasonable.

Variation 1

Steps 1 and 2: Solve first center.
Step 3: Solve the opposite center.
Step 4: Finish the solve.

Variation 2

Steps 1 and 2: Solve first center.
Step 3: Solve an adjacent center.
Step 4: Finish the solve.

Variation 3

Steps 1 and 2: Solve first center.
Step 3: Decide which of the remaining 3 centers you want to solve next, draw from the correct algorithm set and solve it.
Step 4: Finish the solve.

Basically, variation 3 has way too many ways to screw up. You need to know more algorithms. You need to waste time deciding which center to do next (do I commit to this center and just go for it, or should I look at the other one, or the other one? Which adjacent center is the best? and so on). Along this line of thinking, Variation 2 gets a bit tricky as well, as you need to decide which adjacent center to go for in step 3.


For Steps 1 and 2, I want to avoid any sort of decision tree, hence the rigid order and cubie location. I guess I am trying to prune as many branching paths as possible with this method in order to achieve the goal of a fast (hopefully), straightforward system which guarantees a specific low number of steps.

 

[rachmaninovian]

step 3 would require the adjacent faces algs =P

 

[TMOY]

I am not too sure how old TMOY's method is, but the first version of separating centers then sollving center pairs that I have seen was on Jaap's puzzle page, from which I first learned the corners first/cage method.

I did my very first 4^3 solves in 1982. I think I already did separation then center pairs at that time, but I can't remember it with certainty.
My current method is about 1 year old. Note that I've never claimed it was an entirely new and original method.

 

[cubacca1972]

I am not too sure how old TMOY's method is, but the first version of separating centers then sollving center pairs that I have seen was on Jaap's puzzle page, from which I first learned the corners first/cage method.

I did my very first 4^3 solves in 1982. I think I already did separation then center pairs at that time, but I can't remember it with certainty.
My current method is about 1 year old. Note that I've never claimed it was an entirely new and original method.

Thanks for the clarification. I don't know if the history of bigger cube solutions gets as much documentation or attention as the 3x3x3.

step 3 would require the adjacent faces algs =P

Most definately. Variation 2 of 4SCS:

Steps 1 and 2: Solve first center.
Step 3: Solve an adjacent center.
Step 4: Finish the solve.

would also require the opposite face algs.

Thankfully, either way you go (Variation 1 or 2), there would be some shared algorithms, so you wouldn't have to wipe the slate clean if you learned one variation, then decided to switch. I do still believe that if you tried to go with Variation 3, there would be too many potential screw ups introduced. If I learned one method (Variation 1, of course), and decided to change things, I would probably find/add the algorithms to make it 3SCS.

A couple of different lines of thought I have had regarding some of the algorithms that have been found:

There seems to be a fair number of variations and reflections available for some of the algorithms, which would reduce some of the adjustment steps, similar to the Niklas variations.

The other line of thought has to do with the order of face adjustments. At the start of Step 2, there would be up to 3 face adjustments needed, depending on the case. I think the face that contains the first center to be finished should always be adjusted last. This is to avoid the situation where you set up the faces, solve the center, undo all the setups, move on to the next center, then end up redoing one or two face setups to solve the next center. Instead, once the first center is solved, just undo that face adjustment, move on to the next center, and adjust faces if necessary. This would probably save on the total number of face adjustments necessary during the center solves.

 

[rachmaninovian]

i meant that even in variation 1, say you solve the blue centre, and you want to solve the green centre next, there is always chances that all the greens available are on say, the white face. =P

TMOY: ah wells, you did your first 4x4 solves 10 years before i was born...xD

right now i'm concentrating on reducing my cage times and not waste time on centre control; sometimes i waste too much time thinking on how to control the centres....wells centre control has given a 50s cage a sub 1 solve: only 3 single cubie swaps was done.
I've become more optimistic of the method I'm using for the middle edges: lots of freedom, and easy recognition. and 3 kinds of centre control can be achieved. the first kind is intuitive, like pairing up edges with centres already attached to one piece. the second kind...is of course, while solving the last 2 dedges, affect the centres differently. I'm still not too good at this. the last one is a very recent one, but a rather good one: using double layer turns like (Uu)2 in algs instead of U2, etc, to preserve centres. execution just a tad slower, but centres are much easier, with more single swaps compared to block swaps.

I'd be coming up with more stuff with centres soon. I've already got some ideas in mind...

 

[cubacca1972]

i meant that even in variation 1, say you solve the blue centre, and you want to solve the green centre next, there is always chances that all the greens available are on say, the white face. =P

Yes, that is what I thought you meant. The same would apply to doing variation 2, and running into the opposite face set:

If you solve the blue centre, and want to solve the yellow centre next, all the available yellow cubies might be in the white face. With either variation, you would need to know both the adjacent face series and opposite face series.

right now i'm concentrating on reducing my cage times and not waste time on centre control; sometimes i waste too much time thinking on how to control the centres....wells centre control has given a 50s cage a sub 1 solve: only 3 single cubie swaps was done.
I've become more optimistic of the method I'm using for the middle edges: lots of freedom, and easy recognition. and 3 kinds of centre control can be achieved. the first kind is intuitive, like pairing up edges with centres already attached to one piece. the second kind...is of course, while solving the last 2 dedges, affect the centres differently. I'm still not too good at this. the last one is a very recent one, but a rather good one: using double layer turns like (Uu)2 in algs instead of U2, etc, to preserve centres. execution just a tad slower, but centres are much easier, with more single swaps compared to block swaps.

I'd be coming up with more stuff with centres soon. I've already got some ideas in mind...

Regarding your cage completion-

When you say dedge, do you mean a middle edge pair? If so, I am guessing that you solve one edge pair, then solve an adjacent edge pair, then solve the last 2 edge pairs at the same time. Is this correct? I can't check your site myself, because of the freeze ups I get when trying to access your site.

Would you be able to cut and paste some of the algs you use to finish your edge pairs? In particular, I am curious about how you solve the case where you need to swap UFr and UBr.

 

[rachmaninovian]

dedges = double edges = middle edges pairs. =)

and yes, I solve the bottom 2 dedges (4 pieces) in 2 small steps: solve 3 pieces, insert last piece. sometimes I do the first dedge, followed by the other. I like the first approach more, but the second approach is better sometimes..

to switch UFr and UBr?
I do l' U2 l' U2 (l' r) U2 l' U2 l U2 r' U2 l2 [alg by Lucas Garron] most of the time
or (x) B2 D2 B' r' B D2 B' r B'

I used to use the 2nd one a lot, but I realized that the first one is faster.
I've started to use the 2nd one again, so that I can control centres a bit.
fyi, the first alg is used for parity in BLD or other direct solving methods.

check out http://rachmaninovian.webs.com/step4b.htm for the complete set of algs for the 15 (or 16) cases, and the various algs I know for each case.

Speaking of which I particularly love the "PLL parity" because it is the easiest case for me to control centres...
about the other centre control I discovered (or perhaps people have discovered before me...), say, try "OLL parity"
r' U2 l F2 l' F2 r2 U2 r U2 r' U2 F2 r2 F2, which is the pure one, or
r' Uu2 l Ff2 l' Ff2 r2 Uu2 r Uu2 r' Uu2 Ff2 r2 Ff2, and you will see that this alg treats the edge and the 2 adjacent centre cubies beside it as a single big edge.
If you are lazy...try r l D2 l' U2 l D2 l' U2 r' to see what it does, undo it, and try r l Dd2 l' Uu2 l Dd2 l' Uu2 r'. the wonder of centre control

I've started using this simple technique even when solving the bottom 2 dedges..
to illustrate how good this is, I actually broke my pb average (58.62, 10 of 12, and no lucky cases at all) using this technique (obviously not on all solves...but I think at least 5-6 solves.), because it created quite some 3 alg finishes, or extremely easy 4 alg finishes (like 3 niklas' and 1 block swap). Of course I am not able to do it all the time...but wells, good enough. The idea is to preserve as many centres as you can while forming the cage, and using this technique, be it for the bottom 2 dedges or the last 2 dedges. The fastest solve of the average, 52.90, was achieved using this technique..I got a free first centre, but I preserved it intentionally...so nonlucky =) if I hadn't used it, I would have at least 1 more alg to do, and I could say byebye to a sub 55 solve.

 

[rachmaninovian]

found some algs...for centres

(r U2) *6 - 4 cycle blocks

(r U l' U') *4 - 4 cycle single cubies

 

[cubacca1972]

Thanks for posting those algorithms.

It looks like I do a variation of B2 D2 B' r' B D2 B' r B':

U2 F2 U r U' F2 U r' U r.

I was thinking about methods for finishing the dedges (lredges, rledges, medges). As I have posted previously and added to my site, I have a 2 step solution which first sorts ledges and redges into their respective layers in the first step, then permutes them in the second step. One of the drawbacks of this idea is the difficulty in recognizing when an edge cubie is in the wrong layer.

So instead of sorting ledges and redges into their appropriate layers, why not sort Uedges and Dedges into their appropriate layers? The recognition would be way easier (edge cubies with blue facelets belong on the U layer, green on the D layer). Then solve the Uedges, then solve the Dedges. I am guessing that there would be too many permutations possible to hope for a set of "permute the Uedges and Dedges" algorithms (kinda like the Ortega permute the corners). Maybe after sorted into Uedges and Dedges: build dedges then permute them, or sort again to F and B faces, then permute.

I know in advance that this might simply add too many steps, but I wonder if this would give you more center control opportunities.

 

[rachmaninovian]

To be honest I picked my current midges method because of mizzle's post about his roux by four on an earlier thread. The previous method I was using is 1. finish l edges (in like 2 or more short algs) 2. permute r edges and I found the recognition to be crappy. The reason being you need to look for too many pieces around a larger area of space. The good thing is that it only requires 3 algs..

wells very importantly to get fast solves centre control is needed to either preserve centres, or affect them. I do not think that other systems would allow as much centre control, besides the double layer turns (eg Uu2) technique.
TMOY has a method that orients the pieces, then permute. I guess it is a rather efficient method as well, but I don't know about the centre control it can achieve.
Even if centre control does not reduce the number of algs, it gives easier centre cases which are faster to execute than the harder ones...

btw, to your Uedges and Dedges thingy, there are 16 cases for both faces, including the solve state. thus 16^2 algs are needed? I don't think it is very feasible...because really, the key to solving midges fast is recognition. I believe I can sub 10s average for midges already. something like 8s?

 

[cubacca1972]

Yeah, the Uedge Dedge thingy was just a random brainwave. The sort part would be really easy, but the actual permutation part would probably be too long. Definately 2 steps instead of one giant permute for Uedges and Dedges. Oh well.

Based on the times you are getting for the medges, there doesn't appear to be much room for improvement, other than discovering shorter algs (i.e., via computer search).

 

[rachmaninovian]

wells I generated algs for tricky cases to solve the bottom 2 dedges...and I haven't learn them yet. I kinda made sure that most of my algs are optimal anyway...by searching on ron's cube solver...
I'm slowly replacing some of the algs I hate with the <r U2 B2> algs because they are the fastest for me.
Of course, execution is rather crucial...
I guess the next step would be learning some of the tricky cases...like some of TMOY's permutation algs. For example, sometimes I only have to cycle 3 oriented edges after say, 1 move. These would be outside the system, but again, for those cases it would be better if I do it 1 look than 2 look as I now do them..

 

[cubacca1972]

wells I generated algs for tricky cases to solve the bottom 2 dedges...and I haven't learn them yet. I kinda made sure that most of my algs are optimal anyway...by searching on ron's cube solver...
I'm slowly replacing some of the algs I hate with the <r U2 B2> algs because they are the fastest for me.
Of course, execution is rather crucial...
I guess the next step would be learning some of the tricky cases...like some of TMOY's permutation algs. For example, sometimes I only have to cycle 3 oriented edges after say, 1 move. These would be outside the system, but again, for those cases it would be better if I do it 1 look than 2 look as I now do them..

The first 2 dedges in one look...How many cases is that?

 

[rachmaninovian]

theres no point in having 1 look for the first 2 dedges for the middle edges..

after all you can pair the first one in like 3 moves..

wells there are 9 cases for the 2nd one...including reflections/mirrors.
and they are all trivial algs. well...most of them are. I cheated for some by generating algs because my solutions are too long...
the max number of moves needed is 7, without setup moves.

 

[rachmaninovian]

r U2 r U2 r U2 r2 U2 l2 U2 l' U2 l2 U2 r U2 r'(17)
l U2 l' U2 r2 U2 r U2 r2 U2 l2 U2 l' U2 l' U2 l'(17)
r' U2 r U2 l2 U2 l' U2 l2 U2 r2 U2 r U2 r U2 r(17)
l' U2 l' U2 l' U2 l2 U2 r2 U2 r U2 r2 U2 l' U2 l(17)

3 cycle entire centres; I don't this are is an optimal solution but because it is limited to <r, l, U2> it can be done very fast...
found this with clement's 4x4 solver..
I generated this because I had to do l' U' r U l U2 l' U r' U2 r U l U' r' U' 3 TIMES. if i knew this 3 cycle centres I could have used it, and finished with a modified niklas. probably cost me a sub 50 solve..I got 55.01 for this solve =( that 16 move alg takes 4s to execute...without lag time. bleah.

 

[TMOY]

I know this one:
U2 l U2 l' r' U2 r U2 D2 l' D2 l r D2 r' D2
16 moves only Probably not faster than yours though.

 

[rachmaninovian]

yeaps I ran the solver and for <r, l, U2, D2> the optimal solution is 16...

 

[cubacca1972]

r U2 r U2 r U2 r2 U2 l2 U2 l' U2 l2 U2 r U2 r'(17)
l U2 l' U2 r2 U2 r U2 r2 U2 l2 U2 l' U2 l' U2 l'(17)
r' U2 r U2 l2 U2 l' U2 l2 U2 r2 U2 r U2 r U2 r(17)
l' U2 l' U2 l' U2 l2 U2 r2 U2 r U2 r2 U2 l' U2 l(17)

3 cycle entire centres; I don't this are is an optimal solution but because it is limited to <r, l, U2> it can be done very fast...
found this with clement's 4x4 solver..
I generated this because I had to do l' U' r U l U2 l' U r' U2 r U l U' r' U' 3 TIMES. if i knew this 3 cycle centres I could have used it, and finished with a modified niklas. probably cost me a sub 50 solve..I got 55.01 for this solve =( that 16 move alg takes 4s to execute...without lag time. bleah.

Which 4x4 solver is that?

 

[rachmaninovian]

http://www.speedsolving.com/forum/showthread.php?t=11409

use on mac or linux; doesnt work on windows.