INTRODUCTION

This post describes the processes needed to solve a fun twisty puzzle, the Emperor Pyraminx (aka "7th-order tetrahedral twisty puzzle"). This twisty tetrahedron seems to have been ignored for the most part since it was introduced. Why was it shunned - it's complexity? The marketing? It's inherent instability? (puzzle pieces falling out if all layers aren’t kept perfectly aligned during turns) All apply, to some degree. The complexity aspect should become better-defined from the discussion below.

Specific solve details aren't included herein. Those can be found on this twistypuzzles.com site, in a longish thread initiated by me, and in a handful of YouTube tutorials (none of them are mine though). 

Before attempting the Emperor solve, it should be obvious  that the other Pyraminxes should be mastered first, to learn  the tools necessary to proceed. Conquering the Royal Pyraminx (6-layer) presumes mastery of the Professor Pyraminx (5-layer) and so forth for the 4- and 3-layer versions. Indeed, that is how I progressed. In fact, when ruminating over the Professor Pyraminx as a part-time hobby over a year or two, I wasn’t much aware of the Royal or Emperor versions. But eventually figuring out the Professor gave me confidence to push on to those. 

DISCUSSION

Why was the Emperor shunned? Initially, back in 2011 when released, it was hailed as a breakthrough. Puzzlers purchased it to add to  their collections, as it must have been quite a novelty - well maybe just a handful of people did. Originally, there was a nice lightweight version, one of which I was able to obtain. Nowadays a heavier version is sold, and the quality is, well, different from the original. 

(1) First, the expense, at over $300 including shipping cost from overseas (Germany originally and more recently, Russia).

(2) Next, the puzzle’s instability caused center pieces to easily fall out. Not designed to be a speed-puzzle, to say the least. 

(3) Third, the solve was much more complex compared to earlier versions - due to the extra layer, which increased the number of center- and edge-pieces. 

(4) Last, the puzzle had to be special-ordered, made, assembled and sold through a reseller. 

The above factors made this a unique puzzle - not to say that even higher-order Pyraminxes don’t exist, as they do. 

- For example, there is a nicely-done Youtube tutorial of an 8-layer Pyraminx solve on a puzzle simulator. 

- A 13-layer Pyraminx, created by Corenpuzzle, can be viewed on YouTube in several videos. However, it wouldn’t be practical or realistic to solve it and I'm not aware of any tutorials that cover it. The same restrictions as described herein, apply to that puzzle as well, with several orders of magnitude more complexity added (especially for the center-solve; but there may be a workaround for that).  Anyway...

PROCESSES

I counted up to 11 processes for a typical solve (labelled below as 1a, 1b, 1c, 2a, 2b, 3a, 3b, 4, 4a, 5, 5a). Summing each of the processes, the entire solve should take 1.5 hours ideally. Below is a list of the processes, in order: 

1. Build the Centers. There are 16 pieces/center x 4 centers total center-pieces.

1a. Solve 2/4 (2 out of the 4) centers, each center region containing 16 center-pieces. The steps are intuitive, meaning no special knowledge is necessary. 

1b.  Solve the last 2/4 centers. This is especially Emperor-specific and not based on either Professor or Royal. An order of magnitude more complex than Royal. 

1c. Optionally, an experienced user can choose instead to separate colors such that 2 sides contain only 2 colors, and the other 2 sides only have the other 2 colors. For example, 2 sides are yellow and red pieces and the other 2 sides are blue and green pieces only. Then,  pattern-matching can be performed on both sets, for a fun solve. Also, a computer algorithm (created for solving the Last 2 Centers) can be used twice, to solve each set of 2-faces separately. 

This (1b) phase is the most difficult one, whichever solving method is used.

It has also become one of the most fun parts of the solve. I've back-engineered a large group of center-piece patterns that can be used to recognize patterns and relate them to solves. To automate this process, a perl script was created to develop solutions based on specific patterns. Also, commutators can be used instead. So many options, what a great puzzle!

2. Build the Edges. There are 4 wings/edge x 6 edges total wing pieces. Wing pieces are also called inner-edge pieces or inner-edges.

2a. Build 4/6 edges (4 out of the 6), one at a time. A single algorithm is used, which protects the solved parts as you go along. There's one tricky part, used to hide solved edges, that's very easy to overlook. Same technique as used for the Professor.

2b. Hardest is the last 2/6 edges, also know as "Last Two Edges". Emperor-specific, but based on the technique used for the Professor. 

Note: Yes you could solve the edges first and then the centers. However, this would be hugely impractical. The centers would need to be solved using commutators dozens of times, at around 10 moves per commutator - extremely impractical, but in theory it could be done (and I've done it). Such as where you want to practice commutators for some reason (such as executing them from memory). For Royal, I can see a use case for this, but there’s not a good use case for Emperor.

This second phase (phase 2) is the next-hardest to solve, but for me it was also one of the most fun to figure out, as there came to be 4 types of patterns to recognize. Once those patterns became apparent after many solve attempts, it became all the more fun to match up my Last Two Edges pattern to patterns from previous solves. Currently, I'm trying to figure out how to simplify this "pattern cheat-sheet" so it's easier to understand.

Most outer-edge pieces can, optionally, be solved during this phase. Why do this? Simply to make the edges easier to recognize, as the inner-edge piece faces are smaller than the outer-edge piece faces. This is best illustrated rather than described. In the below image, the edge to the left is yellow-green. However, on the green side of the edge (4 inner edges are green), more yellow is seen (on the outer edge pieces) than green. And on the yellow side of the edge, more blue/orange/green than yellow is seen. So, rearranging these outer edge pieces to correspond to the inner edge pieces would be a big plus. Not essential - but very helpful.

3. Match Whole Edges. Match whole edge colors, one at a time, three color-matching edges to each face (edge swap)(edge reverse). TIBPS ("This is basic Pyraminx Stuff"). The center colors are ignored in this step.

3a. Swap Edges, if needed. This is basic Professor Pyraminx stuff or "TIBPPS". The need to swap edges occurs more often than not - and using commutators here is a "necessary evil". There are 22 moves to do this double-3-cycle procedure (two separate 3-cycles for the upper and lower inner-edges needing to be swapped). Yes, the two sets of 11 moves are somewhat similar, which you'd think might make them easier to memorize. For Emperor, about half of the 22 moves need to be slightly modified (this is easily overlooked because it's not written down anywhere that I know of, except here of course - and maybe some of my other writings). 

This is the most error-prone process, with the 22 moves. You can try  to memorize them all; however, even just performing the moves without making a mistake is challenging (and fun) on its own. 

Another error-prone area is holding the puzzle while making these 3-cycle moves - i.e., should the Last 2 Edges be facing left or right? Many times I've goofed here, but it is fun to recover back to where you were, and re-try. Just perform the 22 moves backwards! For me, this takes some concentration and makes the procedure that much more error-prone. But in the end, it is good practice!

3b. Reverse Edges, if needed. TIBPS. The main challenge is  remembering how to hold the puzzle while executing the needed algorithm - with the edges needing reversal facing either to the left or to the right. Using the wrong direction doesn't cost much time to correct, but it is nonetheless a step backwards. There is just a single algorithm needed.

4. Match the Colors. Match colors of (solved) centers with (solved) edge colors - TIBPS. There is a single algorithm for this, which needs to be executed 3 times. The starting point is where none of the centers match the edges on any of the 4 faces. 

4a. Match Color Parity. There is a parity to be aware of, where the color of just one of the faces (solved center and solved edges) is already matched, but the other 3 faces don’t contain such a match. TIBPS. The parity occurs more often than not. One algorithm is needed to correct this. And this algorithm ends up un-matching each of the 4 faces' centers and edges - so then, the algorithm mentioned in #4 above can be executed 3 times. 

5. Cleanup. Solve the outer-edge pieces - TIBPS. Although this is one of the easiest tasks to complete, it's not the quickest as there are usually many out-of-place pieces to resolve, on all edges. However, this step is also lots of fun, and I don't like to hurry it along too much...resolve a few, put the puzzle down for a few hours or days...resolve a few more, etc. Similar as with the other steps.

Be aware that this step could be completed earlier, like in step 3, or even started during step 2. Why would you do this? Well it makes sense, for more easily recognizing the color of inner-edge pieces. Having outer-edge pieces matching inner-edge pieces is definitely a plus as it's less confusing. Again - not essential early in the solve, but a big plus to try it sometimes.

5a. Cleanup Parity Reversal. There is a parity to be aware of, where pieces on 2 edges are in their correct positions, just reversed. TIBPS. This situation doesn't occur on every solve, but maybe 25% of the time. I usually end up solving this intuitively, instead of using a set algorithm although I'm very sure one exists. I just don't care to use one (yet), that's all.

NOTES

A thought might have occurred to the reader, as it occurred to me. Namely, this puzzle has less algorithms to master than Rubik's cube - so why hasn't the solve been timed competitively? Yes I agree, it definitely should take less time and effort to memorize the algorithms than Rubik's. However, the puzzle's instability and cost are likely the main roadblocks. 

A final side note - regarding video tutorials. Sometimes parities are mentioned in the videos...sometimes 3-cycles are mentioned...sometimes a new way of solving the inner edges is  covered...or a different way of solving the center-pieces. But generally, the user doesn't get the entire picture about multiple parities or commutators or 3-cycles from any single solve covered by such videos. I recall many times, after watching a tutorial, feeling like I missed a lot of content - as the content I was interested in, simply wasn't covered. Plus, in every tutorial I viewed, it was extremely difficult to follow moves - slowing the speed and magnifying the image didn't help. This gave the feeling I was merely watching a demo instead of a genuine tutorial. All the same, even in light of the above, I feel that videos on the higher-order Pyraminxes were helpful - they provided a feeling that, yes, even I could solve these puzzles because someone else solved them. But unfortunately they only go so far...

WHAT'S NEXT

At the top of my to-do list is to create another script - one that quickly and efficiently scrambles the Last-2-Centers using random moves that don't scramble the center pieces of the other 2 centers.  Why bother with this, over manually scrambling? Automated scramble would be more:

(1) Random

(2) Thorough

It also requires takeapart of the Last-2-Centers' pieces, to  reassemble in the random pattern. That helps the user familiarize with puzzle center-pieces. 

To avoid this, simulation can instead be used. However, using the physical Emperor puzzle, instability and all, is quite fun.

Update1

Just made some progress (as of 3/24/25). Namely, a much better shuffling method of the 16x2=32 Last 2 Centers pieces (better than a manual shuffle that is). Avoiding the use of a simulator (for increased flexibility), I instead coded my own shuffling algorithm, which shuffles the 32 pieces with random (and repeated) use of the 22 moves available to the user. The end result is a 32-digit binary which can then be fed into the solving algorithm. Can it be solved though, like the manually shuffled pieces? 

After correcting a long-existing coding error in the solving algorithm itself (and only then!), I found it will indeed work nicely solving the shuffled Last 2 Centers. Just as before when the puzzle pieces were only being shuffled manually with a real physical Emperor puzzle.

As a reminder, all of the 22 moves being simulated here, are  variations of the following paradigm, in order:

- A counterclockwise slice turn on the vertex facing the user. And this can be one slice, or two slices, etc. This is where most of the variability comes from.

- A face move. Only using the L-face, and this turn will be either clockwise or counterclockwise.

- A clockwise slice turn, the exact reverse of the counterclockwise move just mentioned above.