# Message #803

From: matthewsheerin <damienturtle@hotmail.co.uk>

Subject: Re: Length 4 Pentagonal Duoprism

Date: Wed, 02 Dec 2009 17:27:08 -0000

Hi Chris,

Nice write-up there, and well done on suitably conquering the {5}x{4} duoprisms :). I did skip the paragraph on the parity though, I fully intend to get around to solving these myself and I’d rather figure it out for myself (not to mention trying to topple the twist counts ;) )

I am fully of the belief that if you enjoy the 4D puzzles a lot and can solve them, then 5D is certainly worth a go for the extra challenge. It is just the same principle as 4D, just taken a bit further and as such, more confusing to start with. However, you may be pleased to know that I think that 3D visualisation skills (and 4D ‘visualisation’ as well as much as it exists) are still perfectly valid, as long as you put in enough effort to get your head around how the different axes behave. Just play around with it for a while and you get the hang of it, that’s what I did. Also, all the techniques from 4D work too.

I’ll be looking out for where your name crops up next for solving something around here.

Matthew

— In 4D_Cubing@yahoogroups.com, Chris Locke <project.eutopia@…> wrote:

>

> Hello everybody! This is Chris again, and I thought some of you might like

> to hear an update on how my length 4 solve of the {5}x{4} duoprism went.

>

> I thought that with the length 5 down, I would take a bit of a break, but

> the gap in the record table felt so tempting, and I had be thinking about

> how I already have done most of the work needed to solve it by doing the

> length 3 and 5 puzzles. Turns out that my algorithms for 3 cycling face and

> edge pieces were perfectly usable in the length 4 case to build up all faces

> and edges. Only thing that caught me a little off during this phase was the

> reappearance of, what I guess could be called degenerate pieces. In the

> length 2 and 4 puzzle, there are some pieces in the edge and face blocks

> that don’t have the same number of colors as the other pieces. As in, there

> are some 1c pieces that are equivalent to corresponding length 5 2c pieces,

> where the color that is part of the square torus is gone due to the shape of

> the cuts. Likewise, there are 2c edge pieces sandwiched between two 3c edge

> pieces. Turns out this isn’t too much of a problem, and it actually kind of

> makes it easier as you don’t have to worry about an extra color matching up,

> and the algorithms are exactly the same as the length 5 case.

>

> The real problem with the length 4 is the possibility of parity issues. A

> parity issue, for those unaware, occurs in even length puzzles precisely

> because you don’t have ‘central’ pieces to each block, so when you build

> your centers, faces, and edges, you can’t be sure that you are putting them

> together in the correct permutation to be solvable or not when you reduce it

> to its equivalent length 3 puzzle. Oh yeah, if you want to solve the puzzle

> for yourself, you might want to opt out of the following discussion on the

> parity issues present. In that case, you can skip the next paragraph.

>

> For faces in this puzzle, it turns out that with a 3 cycle algorithm for

> each of the kinds of face pieces, you can actually solve any potential

> parity issue with just those macros alone. The faces between pentagonal

> facets (2c(5,5)) seem to have no issues. The 2c(4,4) faces are pretty easy

> to fix if you end up with a case where you need to flip one or swap two

> blocks. This is because while those two cases have odd parity when you

> consider the reduced length 3 puzzle, in terms of the individual pieces they

> have even parity so can be fixed by the even 3 cycles (remember that 3 cycle

> is an ‘even’ permutation). If you consider the 2c(5,4) faces, there are

> basically two kinds of pieces. Pieces on a given diagonal within the face

> can be cycled by algorithms, but you can never place a face piece outside of

> its diagonal. You can discover this by playing around with the various

> kinds of possible twists. So this case is similar to the 2c(4,4), except

> you have to do each diagonal separately, whereas 2c(4,4) has all the pieces

> identical. The real problem comes with edges. From the previous logic, it

> can be seen that a single swap of two edges isn’t too bad as that is

> equivalent to swapping two pairs of pieces, which is even. So that can be

> fixed with a 3 cycle macro of edge pieces. But if you end up with a a

> single flipped 3c(5,4,4) edge, then bad luck for you: that is the only real

> parity problem that can occur in the length 4 pentagonal duoprism that can’t

> be fixed by the macros you already have from length 5! I played around a

> bunch before I finally stepped back to think about the problem and realized

> what I just stated, which is that it is impossible to fix with just my 3

> cycle. Once I did that, I spent a couple days of thinking about the problem

> in my free time without actually touching the puzzle, because I didn’t

> really know what to do. When I thought that it might be necessary to break

> everything up and rebuild, I had to then categorize the possible twists, to

> determine exactly which twist I need to do to apply an odd permutation to

> edge pieces so that I get the correct parity. Turns out there are only two

> possibilities when you think about it: twisting a pentagonal facet on a

> 2c(5,4) face / 3c(5,4,4) edge (equivalent), or doing the same twist while

> holding down ‘2’ on your keyboard. I’m not sure sure what that is called,

> so I’ll just call it twist2. Now, the first twist can be thrown out because

> it does something else that is odd: it also does an odd permutation (to be

> precise, a single pair swap) of the 2c(5,5) face central piece. This

> problem can only be fixed by the very same twist, which unfortunately means

> that no matter how you twist this way, you will have either your edges in

> odd parity, or your 2c(5,5) faces in odd parity. So the only possibility is

> the twist2. Now that I had this knowledge in hand, I proceeded first to try

> to find a better way to use this knowledge than just rebuilding from

> scratch. I won’t give the algorithm here, but turns out if you consider

> just the 1c pieces, you can use this twist2 along with some other carefully

> chosen twists to make a move sequence that changes parity of the edge

> pieces, and keeps your 1c pieces intact. You have to remember though that

> whatever your algorithm is, it ‘must’ have an odd number of these twist2s in

> order to put the cube in a solvable orientation. Turns out that the

> algorithm I used kept all edge blocks together, and only broke up a handful

> of the 2c(4,4) face blocks. This is not as good as an algorithm that just

> flips the edge I want, but it’s much better than the alternative of

> rebuilding everything! So I then proceeded to use my 2c(4,4) face 3 cycle

> macros to rebuild the faces, then solved as a length 3 puzzle, and voila!

> Victory!

>

> Well, it was quite an adventure, but I’m glad I followed this pentagonal

> duoprism road this far. There are still be length 6 or 7 puzzles left

> undone, and I’m quite sure that my solves were pretty inefficient and can be

> toppled in twist count too though :D. Not sure I have the time and effort

> to take on anything above length 5 though, but they are awesome achievements

> which I’m sure someone will one complete one day I’m quite sure.

>

> Not sure what I’ll attempt next, but one thing that is sorely missing from

> my hypercubing experience, is the 5D cube. I have avoided that monster for

> a long time! I like the 4D puzzles because the 3D projection that we can

> work with in MagicCube 4D makes it possible to use innate 3D visualization

> skills. Of course this is also why I have problems with really small 4D

> puzzles like the simplex, because they are so small that the 4D-ness is

> really important, whereas other puzzles are large enough that you can work

> in locally 3D regions effectively. The 5D cube is another beast entirely

> though, as you can’t rely on 3D instincts to guide you at all it feels.

> Nevertheless, one day I’ll will make a serious push to solve that one too I

> hope! Until then, I will enjoy this closer to 3D MagicCube 4D software as

> much as I can :)

>

> Chris

>