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

Aleksey wrote:

There are a number of people who are seriously working on making the Pentultimate a reality. Just wait. This is indeed the puzzle that I do want a lot. I just hope that the makers will make it in a way that different initial puzzle shapes can be easily made, as designed by Robert Webb. That includes cubic, triagonal, sphere, and some other shapes.

viewtopic.php?p=70369#70369The 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=1609His 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.htmBram 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=1583With 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=4229Here 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=0Ten 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.htmNotation: 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