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

http://www.youtube.com/watch?v=6vjOkMc98nkCrazy 4x4x4 II tutorial part 2

http://www.youtube.com/watch?v=EwZJNogX ... re=relatedCrazy 4x4x4 II tutorial part 3

http://www.youtube.com/watch?v=lPPux5OLEI8&NR=1I 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.htmlPlease, 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 ... /#notationSmall letters are slice moves.