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 Post subject: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Thu Sep 16, 2010 9:21 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Many members have expressed that the Crazy 4x4x4 Type II is one of the most entertaining mass-produced puzzles ever.
I share this opinion and can recommend that puzzle, if you have not yet solved it.
If you are familiar with the puzzle, or if you want to solve it on your own, you can skip this post completely. If you want to go on, be warned it has got a bit lengthy (but many pictures!)

The text and pictures below are the result of discussions via PM’s between TP member robertpauljr and me.
In the course of this discussion, I’ve learnt that it is not completely trivial to create the right view on the different pieces of a Crazy II (from now on I’ll use this very often as a synonym to “Crazy 4x4x4 Type II”).

This will NOT be a tutorial, but I want to create a basic understanding how to view the pieces of the Crazy 4x4x4 Type II.

If you are looking for a tutorial you can go to the Youtube videos below
Crazy 4x4x4 II tutorial part 1
Crazy 4x4x4 II tutorial part 2
Crazy 4x4x4 II tutorial part 3

I have not watched all of that, but it seems very detailed.

robertpauljr had solved the puzzle several times, when we started our discussion. Interestingly, he had thought about the little triangles inside the circle as being equivalent to edges on a normal 4x4x4. I could convince him, that a different view is fitting better.

Basically, I want to create the understanding what the square centres are and what the little triangles inside the circle are comparing it with a normal 4x4x4 (following I'll use "Normal" as a sysnonym) or a 4x4x4 Supercube (following I'll use "Supercube" as a synonym).
How the corners and the outer edges compare to a Normal is trivial: They are completely equivalent!

What I mean by a 4x4x4 Supercube is this:

Each and every centre piece has its specific location.

I have disassembled my Crazy II and have taken some pictures.

This picture should make it clear, why three of the square centres are building a "corner of the inner 2x2x2".
(This is one physical piece.)

Probably, you have disassembled a Rubik's Revenge. Inside it you'll find a ball.
The piece above is just an octant of that ball with three "visible centres". Actually you could shrink them to flat stickers and would have a rounded corner of that "inner 2x2x2".
On a Crazy II you do not have centres like on a Normal.
On a Normal you cannot see the inner 2x2x2.

But you can view the pair of inner edges that travels together as a strange kind of a center. You can NOT see it as a usual center (because one tile of the inner 2x2x2 replaces the square centre pieces of the Normal ), but you can see it from the side in the two faces it is exposed to.

Here are all moving pieces (the whte plastic is some hidden internal piece):

And here are two adjacent inner edges (not a logical pair in this case!):

A "logical pair" (two inner edges (= little triangles inside the circle) travelling always together) is not physically but logically connected.

Now I want to convince you that we have to view an inner edgepair as a kind of "virtual centre piece".

The logical pair white/red to which the arrows are pointing is virtually connected to the location under the green centre (and we know this not a centre at all but 1/3 part of a corner of the inner 2x2x2)

I've found this quote in the thread about "Most entertaining puzzles"
bmenrigh wrote:
For me the most entertaining from a twisty-puzzle perspective is the Crazy 4x4x4 II. Those virtual face centers (I like to think of them as "holographic" pieces) are a real brain twister.
...

Isn't that nicely worded?
If you can develop that holographic view - looking through the square centre and recognizing the "virtual centre" underneath - you can solve the puzzle like a Supercube!

We can say "inner edge pair" = "virtual centre" = normal centre on a Supercube.

With the following sequence (you are probably familiar with it) I will show that the analogy of "inner edge pairs are equivalent to virtual (partially hidden) centres" is correct.

On a 4x4x4 Supercube I can make a 3-cycle of 3 centres e.g. with
r' d' r U' r' d r U (a simple commutator, BTW I've found this algorithm back in 1981 or 82 when I've got my first Rubik's Revenge. I've solved it by myself and it was almost equally hard than solving the Rubik's Cube without help in the first place)

This is the result (U =white F =green):

Three centres have moved bdR -> Urf -> Ulf -> bdR.

You'll understand why I'm using a Supercube showing this: On a Normal you would see a swap of two centres only.

The following picture shows both cubes after the sequence above:

You'll see by this photo and the following from different angles that the following is true for the Crazy II:
The virtual, partially hidden centre sitting under the location bdR (In the following I'll use a name like "bdR" describing the inner edge pair. In this case, the inner edge pair yellow/blue. BTW because there is always another edge pair with the same colour pair, I'll apply the following rule: When I look at the first colour, the second will be to my right) has travelled to Urf. Urf (edge pair green/red) is now at Ulf location and Ulf (orange/green) has arrived at bdR.

Now, with this understanding, you can solve it e.g. in the following order (like a 4x4x4 Supercube)
1. solve the inner 2x2x2 (skip that on a Supercube, because it is not visible)
2. solve the inner edge pairs = virtual centres = centres on a Supercube
3. pair the outer edges
4. solve it as a 3x3x3 Supercube

When you have solved the circles, this means that you have solved the inner 2x2x2 AND all the virtual 4x4x4 centres (= edge pairs).
Now, you’ll pair the outer edges and solve the Crazy II as a 3x3x3 Supercube.
Translate this to the situation on a "normal" 3x3x3 Supercube:
As long as you use face turns only, the inner 1x1x1 (which is the inner 2x2x2 on the Crazy II) remains untouched. You start with all centres correctly oriented, but whenever you turn a face you rotate the 3x3x3 centre (all virtual centres related to that face as an entity). And because you start with the correct orientation (e.g. on a 3x3x3 picture cube the piece of the picture on the centre is correct) you have to maintain that correct orientation while you are making progress towards the solution.

If you have ever solved a 3x3x3 supercube, you'll find your way easily .
I do it like this
1. Cross (e.g. white face)
2. Corners
3. Second layer (or connecting 2 and 3 as F2L)
4. orient the edges for the yellow cross (You'll NEVER have the situation with an uneven number of yellow edges. This is due to the fact that the inner 2x2x2 has been solved early in the game.)
5. position yellow edges
6. yellow corners
Here you can possibly have the situation, where you have to swap two outer edge pairs.
See my Note at the end of this post!

BTW, you can position the inner edges at the very end using that commutator for the "virtual centres" of the Crazy II. If you do it earlier, you'll find much shorter sequences. This is identical to solving the Normal 4x4x4. You can do the centres at the very end, but because there are so many, it is less time consuming to fix them at the beginning.

Alternatively you could not care about the correct orientation of the centres at the beginning and rotate them correctly at the very end.
This is a bit a matter of taste. I find it better to keep the solved circles correctly during the very short phase "Solve a 3x3x3 Supercube".

Ask yourself, how you are doing an 3x3x3 Supercube (e,g, picture cube) and stay with your preferred method.
As I have pointed out earlier, you can even solve the virtual centres of the Crazy II in the last step.

On many shape shifting 3x3x3 variants (which are Supercubes indeed) I prefer to orient the centre pieces very early. This is just a personal preference.

Note: Close to the final solution, you can end with a situation where you have to swap two outer edges. I have posted something about this and have dug it up:
konsassen wrote:
ubuntucuber wrote:
u L' U' L U F U' F' u' d' F U F' U' L' U L d

that is the parity algorithm

only is the pieces that need to be switched are on the front-left and front-right

This parity algorithm is quite similar to the parity algorithm of Michael Gottlieb that I had mentioned
in my post October 14.
I repeat it here mirrored to show better the similarity to ubuntucuber's sequence:
u L' F U' L F' u' d' F L' U F' L d (14 moves).
I find the sequence quite elegant because 1.) it is easy to see what's going on,
2.) needs almost no memorization (Because you understand what's happening) and 3.) because it is short.

For those who are interested I'll explain the algorithm in detail:

Why do I say "easy to see what's going on"?
Let's put some brackets into the sequence:
u (L' F U' L F') u' d' (F L' U F' L) d
The part X1=(L' F U' L F') swaps the pair of edges at FL (named in the following Flu and Fld) and the part (F L' U F' L)
is just X1' (inverse sequence of X1).
(It's like changing the orientation of an FL edge on a 3x3x3.)

The first u creates a mixed pair of FL and FR edges (Fru goes to the location Flu).
Then X1 swaps this mixed pair of edges (original cubie Flu is now at its final location Fld, original cubie Fld
at location Flu).
The rest of the cube looks a bit scrambled but the u and d layers have changed at location Flu and Fld only.
u' brings the original Fld (currently at Flu) to its final destination at location Fru.
d' moves cubie Frd to location Fld.
X1' swaps again the pair of edges at FL (Original Frd is now at final destination Flu) AND sets back the scrambled rest of the cube to its original state.
d inverses the d' (Original Flu goes to final destination Frd) and we are done.

In ubuntucuber's algorithm the part (L' U' L U F U' F') is X1 and (F U F' U' L' U L) is X1'.
The rest of the logic is identical.

Works fine on 4x4x4 and 5x5x5 supercubes as on the Crazy 4x4x4.

EDIT: robertpauljr is writing a blog where he reports about his experiences with this puzzle.

EDIT2: If you have problems solving a 4x4x4 Supercube, you may want to look at Michael Gottlieb's blog:
http://michael-gottlieb.blogspot.com/20 ... cubes.html
Please, be informed that he is using not WCA notation but SiGN notation!!! e.g. u is (Uu) in WCA.

EDIT3: I have not said explicitly that I'm using WCA notation. If you are not familiar with it, please, have a look here http://www.worldcubeassociation.org/reg ... /#notation
Small letters are slice moves.

_________________
My collection at: http://sites.google.com/site/twistykon/home

Last edited by Konrad on Sun Mar 27, 2011 11:02 am, edited 2 times in total.

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 Post subject: Re: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Thu Sep 16, 2010 3:32 pm

Joined: Mon Aug 27, 2007 3:50 pm
Location: Copenhagen, Denmark
Thank you Konrad! I will finally be able to solve this puzzle. With some more in detail study of your post of course

_________________
Tony Fisher wrote:
I believe it would work best with black plastic.

My puzzles in the Museum
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 Post subject: Re: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Thu Sep 16, 2010 4:17 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Hi Konsassen, thanks for writing this this guide on the puzzle. This is the guide I always wanted to make but never got around to starting it. Your photos and description are great.

I should point out that Carl (wwwmwww) did some great work thinking about how the Crazy 4x4x4 series works here and here.

Also, I should point out that there are a lot more circle cubes with really crazy pieces on Gelatinbrain's website here. Puzzle 3.1.9 is the Crazy 4x4x4 I and 3.1.10 is the Crazy 4x4x4 II. By changing what slice causes the circle to turn you can get all sorts of neat virtual/holographic pieces including 2x2x2 corners, 3x3x3 edges, 4x4x4 centers, the + centers of a 5x5x5, etc. I found 3.1.19 to be the hardest because it contains so many different types of virtual pieces. Just figuring out what they are is a huge challenge!

If you like the Crazy 4x4x4 series, there are so many more good puzzles just like it a few click away .

EDIT: fixed the link to point to the post rather than the image -- thanks!

_________________
Prior to using my real name I posted under the account named bmenrigh.

Last edited by Brandon Enright on Thu Sep 16, 2010 4:54 pm, edited 1 time in total.

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 Post subject: Re: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Thu Sep 16, 2010 4:50 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
I should point out that Carl (wwwmwww) did some great work thinking about how the Crazy 4x4x4 series works here and here.

Nice... I was just about to link to those pics as well. Well at least the second one. Your first link is to the gif icon that appears in the top conner of all posts. LOL!!!

Before I had come up with these images:

http://wwwmwww.com/Puzzle/Crazy1.png
http://wwwmwww.com/Puzzle/Crazy2.png
http://wwwmwww.com/Puzzle/Crazy3.png
http://wwwmwww.com/Puzzle/Crazy4.png

I had barked up a few overly complex ideas too. Check out my equivalent 4x4x4 idea here:

http://twistypuzzles.com/forum/viewtopic.php?p=181159#p181159

And before that I was even toying with the need to have a 6x6x6 to explain all the pieces:

http://twistypuzzles.com/forum/viewtopic.php?p=181157#p181157

Carl

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 Post subject: Re: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Thu Sep 16, 2010 5:26 pm

Joined: Tue Feb 16, 2010 12:15 pm
Location: Sandnes, Norway
When I saw that this thread was created, I immediately decided to solve the Crazy 4x4x4 just to be able to join in on the discussion and to be able to comment on your outline. Turns out I solved the wrong Crazy 4x4x4 So I solved the first version, and it was hard until I realized that it's basically a 2x2x2 solve, 4x4x4 solve and then a 3x3x3 solve (if you use reduction to solve the 4x4x4, of course). But this one seems even more complicated. And complicated puzzles tend to be fun to solve, therefore this puzzle should by this reasoning be fun to solve

I will solve the correct one and then properly join in on the discussion Judging by the length of your post, konsassen, it seems I will likely not have to much to add. Though I did not read your post, that way when I attempt my solve I'll get to make my own discoveries.

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 Post subject: Re: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Thu Sep 16, 2010 5:57 pm

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
Carl's pictures really gave me the "AHA!" moment in understanding circle puzzles. Your post is a bit word for my tastes, but still good in explaining. I like the picture of the Crazy 4x4 next to the super-stickered 4x4.

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 Post subject: Re: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Fri Sep 17, 2010 5:50 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Thanks for the comments!
And thanks to bmenrigh for the links and giving to Carl all the credit he deserves.
And thanks to Carl for all his pictures and the links to them!
I had already seen his pictures and explanations before I got my Crazy II as early as it was on the market.
I had given the link to the Crazy 4x4x4 thread to robertpauljr and he was a bit overwhelmed by the long thread.
Therefore, I have not thought of linking it myself. My text is long enough, already!
GuiltyBystander wrote:
Carl's pictures really gave me the "AHA!" moment in understanding circle puzzles. Your post is a bit word for my tastes, but still good in explaining. I like the picture of the Crazy 4x4 next to the super-stickered 4x4.
I agree, that it is a long text. (When I got no reply for what I thought a long time, I said to myself "OK, you have explained the obvious with too many words". )
You have to see the history of it: I've connected existing text portions and pictures out of many PM's I had exchanged with robertpauljr. This reflects the fact, that for some people a single picture explains it all and sometimes it is not so easy getting the right view on things.
I want to emphasize, that I consider it a great achievement of robertpauljr that he solved this puzzle completely on his own. He had never looked at the Crazy 4x4x4 thread or any other material on the Internet!
We started, when he had made several solves already and he asked me, how I solve it myself. I tried explaining it and he could not follow the paradigm "inner edge pairs are virtual centres", immediately.
My post had not been addressed to the experts - and I've warned you at the beginning - , but I'll be glad if I could help any beginners or if I could motivate some people to get this puzzle.

EDIT: I want to add that the Crazy 4x4x4's have a special sentimental meaning to me, because I have made my very first post here at the TP forum, almost a year ago.
EDIT2: I'll certainly get a Crazy 4x4x4 Type III as soon as possible, even concluding from Carl's pictures that is not really anew challenge.

_________________
My collection at: http://sites.google.com/site/twistykon/home

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 Post subject: Re: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Fri Sep 17, 2010 10:07 am

Joined: Sun Aug 12, 2007 8:28 pm
Location: Northern Central California
If you want a more complete story of my adventure with the Crazy II by all means check it out here. But here is a brief recap.

I fiddled with it for a few days trying to find a commutator that would 3-cycle pieces. I found right away that the outer corners and outer edges could be manipulated without scrambling pieces inside the circle. I call squares inside the circle centers, and the little pieces inside the circle inner edges. But I couldn't find a 3-cycle for inner edges or centers. Everything I tried moved 6 if not 9 pieces around.

Finally after further experimentation it dawned on me that the centers were actually corners, and the little edges were much like outer edges in that they travel in pairs. So just like an outer edge has 2 sides to it, the inner edges also have 2 sides. It is just that they are not connected.

With this in mind I realized that I had indeed found several 3-cycles already. I just hadn't seen it until I realized the centers were corners and the inner edges traveled together.

I was ready to scramble and solve the puzzle. And I did. Then I wanted to share with someone, so PMed Konrad. That is where my self-discovery and analysis turned into the guided discovery and instruction that Konrad posted above.

Before working through the concepts of the inner edges corresponding to normal centers, I just viewed them as edges and solved them as such. Basically I tried solving it like I solve the 4x4x4. Because of my experimenting I had the idea that outer edges could be viewed as normal centers. Crazy, but it worked. I do not use a reduction method. I do not pair outer edges.

1. Solve the non-white, non-yellow outer edges.
2. Solve all the squares inside the circles. That is, the inner corners.
3. Solve the white and yellow inner circle edges. I use a keyhole technique for doing this so each piece can be put home with 3 to 5 twists.
4. Solve one row of unsolved inner circle edges. f2 R' f2 R2 f2 R f2 is used on each piece that isn't already home after a simple setup move to get it adjacent to its home spot.
5. Solve the other row of unsolved inner circle edges. One or two 8-move 3-cycles accomplish this.
6. Solve the white and yellow outer edges.
7. Solve the outer corners. 3 or 4 applications of a commutator and we're done.

After learning the proper correspondence of pieces on a Crazy II to a Normal, I applied this method thusly, first solving the inner 2x2x2, then the 4x4x4:

1. 2x2x2 (centers)
2. All the inner edges that don't have any white or yellow.
3. Corners. I had to put one of the non-white non-yellow sides down to keep the already solved pieces out of the way. It worked.
4. White and yellow outer edges. I use a keyhole technique. Each piece can be put home with 3 to 5 twists.
5. One row of middle layer outer edges. F2 r' F2 r2 F2 r F2 is used on each piece that isn't already home after a simple setup move to get it adjacent to its home spot.
6. The other row of middle layer outer edges. One or two 8-move 3-cycles accomplish this.
7. 3-cycle the remaining inner edges home.

I have since tried pairing up outer edges and solving the cube as a 3x3x3 picture cube, but have not successfully completed a solution this way. The methods above both work with no parity issues and since I am well practiced at solving the 3x3x3 by the Corners First Keyhole Method I see no reason to try to learn a new way to solve the Crazy II.

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 Post subject: Re: Hint: How to solve a Crazy 4x4x4 Type II?Posted: Sun Sep 26, 2010 5:31 pm

Joined: Sun Aug 12, 2007 8:28 pm
Location: Northern Central California
I'm curious. Have any of you counted how many moves it takes you to solve the Crazy 4x4x4 II once you are past the 2x2x2? I just solved it twice, and the 4x4x4 part of the solve took about 270 twists.

After the 2x2x2 I started counting. I solved the middle layer outer edges first. Then all the inner edges inside the circles. Then the white and yellow outer edges. I did not make edge pairs before solving any of the edges. The first time I solved the corners last. The second time I solved them near the beginning.

Edit: Move counts. When I say inner edges, I mean the teeny pieces inside the circles. When I say non-white, I mean neither of the two that travel together is white. I consider the white and yellow layers as the top and bottom so the middle layer edges are the outer edges on the two layers between the top and bottom layers.

1. 2x2x2: 31 twists
2. non-white, non-yellow inner edges: 33 twists
3. Corners: 15 twists
4. White and yellow edges: 54 twists
5. Middle layer edges: 27 twists
6. White and yellow inner edges: 64 twists

I do not speed cube, nor am I into fewest move challenges, but I wanted to stick with the most efficient of the ways I had come up with to solve this thing. I think this method will work the best for me, especially once I get step two down. I am curious to know how many twists others put into solving this puzzle by their various methods, if anyone is up to the challenge of counting. And for the record, I count Rr as one, and I count r as 1 even though in reality to do r I do Rr R'.

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