I've spent the majority of the yesterday searching the forum for information of this puzzle out of curiosity and boredom.

I've learned a fair bit of interesting information.

First, I found a very interesting quote from Aleksey from August 2007


viewtopic.php?p=70369#70369

The last part of Aleksey's quote is referring to the following thread, where Robert Webb illustrated several shape modifications that could be done to this deep cut puzzle.

viewtopic.php?t=1609

His Rhombic Triacontahedron ("Rhombicultimate") is the only one that would not change shape. Also included is the cubic "Cubultimate", the octahedral "Octultimate", and the tetrahedral "Pyrultimate".

While we are discussing shape modifications, I am almost 100% certain that this applet from Gelatin Brain's page is the Icosahedron transformation of the Pentultimate. For lack of a better name, we could call it the Icosaimate.
http://users.skynet.be/gelatinbrain/App ... osa_v4.htm

Bram has provided us two very plausible and most likely achievable solutions on how it could be built.

First was mentioned way back in March 2003.
viewtopic.php?t=1583

With this model, he has discussed a puzzle held together by a gear system that would cause it to stay taut because of it's rounded structure. With this structure, the puzzle can be any size. I believe if someone were to experiment with this sort of mechanism, to attempt to apply it to the skewb first and try to make a spherical skewb that is hollow before moving onto the Pentultimate

The second was discussed in January 2006.
viewtopic.php?t=4229

Here he mentions how to build one out of a Megaminx that has already been transformed into a Pyraminx Crystal. It's very interesting. First he bandages essentially half of the the internal Megaminx, so that in a certain axis, on once face is turnable. From there he extends the puzzle out in a certain way. This method would cause the puzzle to rather large (A puzzle in a puzzle in a puzzle) and I think would rather fragile. The Pyraminx Crystal part could and should be spherical.

There have been a few patents.

Ky Thierry from France filed a patent for it in 1995. He used an interesting Star shaped wire to hold the pieces in, but to not block each other.
http://v3.espacenet.com/textdoc?DB=EPOD ... 714298&F=0

Ten years prior, William L. Alford filed a patent for several twisty puzzles including a Pyraminx with the trivial tips non existent (one of Thomas' mods), and a Skewb Diamond. There is only one picture of the Pentultimate, with no drawings showing the internal mechanism. The mechanism is a metal ball with magnetic pieces stuck onto it.

Going off that idea, while some people considering using magnets cheating (myself somewhat included), but for the sake of having this puzzle 'functional' (or at least tangible) in a sense, I wouldn't mind seeing someone use something like a 1" or a 25mm steel ball and the pieces are magnetic. This way you could have a fairly nice hand held puzzle. It could even be the same dimensions as the Meffert's Megaminx.

Now, as for a solution, I believe the first person to derive one was Doug Li, known as UMichSpeedCubist here on the forum. This is the quick solution he showed me when I met him for the first time.

http://users.skynet.be/gelatinbrain/App ... eca_f6.htm

Notation: The algorithms I will be using only turn 3 of the faces of the puzzle. It will be required to orient the puzzle differently in order to solve of course. (Apply algorithms to different areas)

The 4 faces will be L, U, and R. Since each turn can end in 4 new positions, I will be using the following notation.

X+ : One position clockwise
X++: Two positions clockwise
X- : One position counter/anti clockwise
X--: Two positions counter/anti clockwise



Step 0) Take a screenshot of the solved puzzle

-This is done to help you find the color scheme. If you consider this cheating, feel free to figure out the color scheme out using the corners as a guide.

Step 1) Solve/Permute the centers

-Using the algorithm [R+ L- R- L+] it creates the 2 swap shown here.



You can more or less finish the first 6 centers (one half) of the puzzle without having to use this algorithm. You can apply this algorithm from different angles to achieve the needed 3 cycles. You can tell which pieces belong where by either using the screenshot you took before having the puzzle scrambled, or if you decided against doing so, or you somehow have a tangible puzzle, you will have to use the corners to find the correct color scheme.

Step 2) Permute the corners.

- The main algorithm you will be using here is (R++ L-- R-- L++)x3 which will achieve the following on the puzzle.



Now rotating the face around the red ring, we continue to swap the two pieces in the back, and the piece at the top of the screen with the piece at the bottom of the face facing us (U). Repeating the 12 move algorithm and using U face turns will allow you to permute the pieces in different ways. But after an odd number of times applying the algorithm, you must then return the piece that doesn't belong adjacent to the U center back where it belongs. This limits you to either a 3 cycle, a 5 cycle, or two 2 cycles.

Here is an example of a 3 cycle.



The following algorithm was used.

(R++ L-- R-- L++)x3 U+ (R++ L-- R-- L++)x3 U+ (R++ L-- R-- L++)x3 U-- (R++ L-- R-- L++)x3

One could use a 5 cycle, as shown in the following image to solve the rest of the puzzle only using the following algorithm if they so desired. I believe this would be far from effective though, and you would end up having to apply the same 150 move algorithm many times (depending on the move count it would take to solve an Impossiball.


The algorithm demonstrated is [(R++ L-- R-- L++)x3 U+]x6

Note: The U+ in the previous algorithm could be replaced with U++, U- or U-- to achieve similar effects on the puzzle.

Step 3) Orient the corners.

We will be using a similar algorithm to the first one used (Step 1).

[R+ L- R- L+]x4 creates an eight corner orienting 'algorithm'. Taking it further, you can use set up moves and use it multiple times.

It seems a very useful algorithm can be applied to twist only 2 pieces.

[R+ L- R- L+]x4 U++ [R+ L- R- L+]x4 U-- [R+ L- R- L+]x4



You can use this how ever many times it takes to solve the rest of the puzzle using few set up moves, or you can use an algorithm that changes 4 corners to speed up the process. This however, will require more set up moves to finish the corners faster and using the algorithm efficiently.



This is achieved by changing the U turns in the previous algorithm.

[R+ L- R- L+]x4 U+ [R+ L- R- L+]x4 U- [R+ L- R- L+]x4

Experimentation could be useful here to apple different types of corner turning algorithms, but from this point in the solution the puzzle should be solvable.

Step 4) Celebrate on being one of handful of people who have completed this challenge.


And with this, my write up of the Pentultimate is done. I hope you enjoyed reading, and hopefully you learned something.


Regards, and Happy Puzzling,
Noah Hevey

The gearing concept is one which hasn't been experimented with much in practice. In my list of things to work on there's a vastly simpler puzzle which makes use of sort of the same gearing trick as an actual part of the puzzle, rather than just an internal mechanism. My list of projects is kinda long though.

The build-up of pyraminx crystal mechanism is a lot less fragile than you think. The lack of open interior spaces within the puzzle tends to shove everything into place and prevent jostling. It is rather complex though.

I know of at least two people actively (and credibly) working on a pyraminx crystal. Given that plus the two (!) high quality gigaminxes which have been created it's only a matter of time before a pentultimate is actually made.

With the pyraminx crystal about to be a mass produced puzzle from mefferts, that is definitely a possibility. I'll take a position as a bystander however as the creation of such a puzzle does not really play to my strengths as a modder.

Oh yea, very nice writeup, Noah. You're turning out to be quite an archivist.

I agree, the time is coming closer and closer that one of these will be a reality. Right now I'm just waiting for someone to reveal this puzzle. I would love to have the ability to play with one in real life.

Also, care to discuss any of your puzzle ideas you have on your list? I'm really curious about it.


And the reason I assumed the Megaminx/Pyraminx Crystal method wouldn't provide a stable puzzle is because the internal puzzles have to have a bit of tolerance to be able to move. From there, as you build out, you have more and more leverage, which would tend to make it less taut, or so I would assume.


By any chance, do you know of anyone directly working on the Pentultimate? Since we have Katsuhiko Okamoto, Aleh Hladzilin and Uwe Meffert are all, or have at one point to time made Pyraminx Crystals, the last of which is working on mass producing them, which should be a great opertunity for this puzzle to see the light of day in the next couple years.


@Jin: Thank you very much! That means a lot to me.

Noah, I'll post about my designs when they're ready (they tend to mutate a lot in the development process, and they're kinda hard to describe in general), but for those who might be interested here are some of my ideas for the distinctly non-twisty puzzle genre of puzzle rings:

http://bramcohen.livejournal.com/47641.html
http://bramcohen.livejournal.com/47101.html
http://bramcohen.livejournal.com/46477.html
http://bramcohen.livejournal.com/46086.html
http://bramcohen.livejournal.com/45239.html
http://bramcohen.livejournal.com/40699.html
http://bramcohen.livejournal.com/37297.html

I hold the small honor of creating a name for this puzzle that stuck. I probably did so sometime in the late 80's, but the name did not appear on the Internet until 1993. At that time I wrote what was probably the first computer simulation of the Pentultimate for IBM OS/2. A later version, ported for Windows, can be found here:

http://www.chrisandkori.us/_filelib/Fil ... e=pent.zip

The program is largely unfinished (no scramble feature!), but it did allow me to play with the puzzle I had envisioned.

I believe Rubik himself may have been the first to envision the puzzle. It is mentioned in the 1987 book Rubik's Cubic Compendium (page 210) but no name is given for it, nor are there any pictures of it.

I think these puzzles are "discovered" more than they are "invented." It's not a great mental leap to generalize the idea behind the Skewb to a dodecahedron. The mechanism has been the tricky part, though-- that does require invention.

There are a few other simulations linked here on my original Pentultimate page:

http://www.chrisandkori.us/fw/main/Pent ... -1422.html

Chris

Yes, of course. I hope it didn't seem like I was bugging you or demanding you to tell us. I was just curious if there were any ready to share with the public. Thank you though.

Thanks Noah!
You put an enormous amount of time and effort in your "information" posts. You should have your own Wikipuzzler site.

[I've moved the content of this post (alternative solution details for the Pentultimate) to a more appropriate thread: SpeedSolving - Gelatin Brain's Applet Solutions Discussion Thread.]

I solved the pentultimate last night using this solution. A big thank you to Doug and Noah!

Wow, really great writeup. Reminds me of your "History of the Pyraminx Crystal" post. Who coined 'Pentultimate'?

I did.

Chris

Glad you guys liked it. I really ought to make more posts like this. But which puzzles... Hmm...