The Jumble Prism has drawn quite a lot of interest in the community and the recent discussion in the solving thread inspired me to bust out the ol' Jumble Explorer code and give it a whirl on the surprisingly difficult Jumble Prism. These topics typically count any shape where all grips are in a predefined stop, including shapes where only a single grip is available, but that has always bothered me because you could argue shapes with only a single available grip shouldn't really count as shapes at all. So this time I am only counting shapes that have at least two grips available. I will constantly remind you of this by calling them essential shapes instead of just shapes. Please enjoy these results

The Jumble Prism can reach 495 essential shapes!
This number takes into account all symmetries of each shape, including mirror images, which means every pair of chiral shapes is counted only once. For this first post, all colorings are ignored; we are only looking at shape. The furthest essential shape from solved takes 21 moves to return to the solved shape, and there is only 1 distinct, asymmetric essential shape at this depth (and its enantimorph). Here is the distribution of essential shapes by depth:

0:1
1:2
2:7
3:12
4:18
5:21
6:33
7:25
8:24
9:31
10:65
11:81
12:68
13:46
14:33
15:14
16:6
17:3
18:2
19:1
20:1
21:1
total: 495

If you instead wish to count every chiral pair of shapes twice (once for the right-handed version and once for the left-handed version), the distribution becomes:
0:1
1:4
2:14
3:24
4:36
5:42
6:64
7:49
8:48
9:61
10:128
11:155
12:133
13:89
14:64
15:28
16:12
17:6
18:3
19:2
20:2
21:2
total: 967

By comparing these two lists, you can correctly deduce that there are 23 essential mirror symmetric shapes and 472 pairs of essential chiral shapes. By subtracting the first distribution from the second, you can compute exactly how many pairs of essential chiral shapes occur at each depth. Keep in mind, this list does not include shapes with only 1 grip available whereas I normally do on these topics, but even if I throw them in, the total only climbs to 1,847 (3,671 if you're counting chiral pairs as 2 shapes each). This is not a very big shape space compared to other jumbling puzzles like the Helicopter Cube (14,098) the Geranium (37,730) or the Jumble Trap (553,478) and yet there is a surprising amount of trickiness in the Jumble Prism shape space and it can be quite easy to get lost. My search has revealed why this is the case! This shape space is quite sparse and has TWO extreme chokepoints between large clusters of shapes. This shape space is about as difficult to navigate as possible with this few shapes. Still, 495 essential shapes is just simply not enough shapes to be lost forever, so wandering deep into the Jumble Prism shape space is not something to fear. Be brave and get lost! You are statistically nearly guaranteed to eventually stumble back upon solved if you just keep at it for a bit.



Each grip on the Jumble Prism has 5 Stops
The Jumble Prism implements the first of only 2 layers of jumbling present on the Triangular Bipyramid geometry. As shown in my Jumbling Geometries Catalog, all triangular bipyramid puzzles have the same 5 stops since they are all implemented by the very first jumbling layer:

stop #0: 0°
stop #1: arccos(1/8)≈82.8°
stop #2: arccos(-3/4) ≈138.6°
stop #3: -arccos(-3/4) ≈221.4°
stop #4: -arccos(1/8)≈277.2°

I prefer the grip names I used in my More Madness Shape Space thread, but for everyone's benefit, I will use the same grip names introduced by Konrad in the Jumble Prism solving thread:

(although I am just now realizing as I post this that I actually used the hand-drawn notation but the computer generated color scheme Sorry, Konrad, I can't win... My colors match the pCubes screenshot in the solving thread, but that is actually upside-down compared to this picture... )



Shorthand Notation
I noticed some people in the solving thread struggling with my notation. I stand by writing both the starting stop and ending stop as the best way to unambiguously capture the information contained in a jumbling move (and it makes it especially easy to translate into the inverse sequence!). But I do admit that it is verbose. A few people have tried to introduce shorthand notation that is much easier to read and I am open to writing some of the more important sequences in this topic down in shorthand so it can be easier to follow, but I want to caution about the limits of this shorthand.

Most moves on the Jumble Prism are the same rotation angle of ~138.6°. This rotation moves CW from 0->2, 1->3, 2->4, and 3->0. Many conjugates and commutators can be written with only this angle and it is similar enough to 90° that it feels good as a "standard rotation" move. So the shorthand move F can be used in place of F(0->2) and F(3->0), and that covers about 80-85% of all the moves you ever need. However, other rotation angles are possible and sometimes needed, especially if you're going for optimal algorithms. A smaller rotation of ~82.8° appears from time to time, moving CW from 0->1, 2->3, or 4->1. I have seen a plus or minus sign used here to indicate a rotation less than the standard angle we defined above. So F+ can be used in place of F(0->1). So far so good, and probably around 98% of all move sequences can now be written in shorthand.

The problem is there are a few other possible rotation angles left that we haven't covered. If you ever need to move from 1->2 or 3->4, that is a rotation angle of ~55.8°, even smaller than the ~82.8° rotation angle we encountered above. One possible solution here is to define the +/- signs to indicate "a small rotation to the next stop" and this actually works since only one angle or the other is ever possible from a given stop. The only downside is you may notice that the + move you just did felt a bit bigger than the last + move you did, and that's because maybe it was! The final hiccup is that a wide ~165.6° sweep rotation moving from 4->1 is also possible, and this specifically skips over stop 0. This feels most similar to a standard ~138.6° turn so someone naively trying to notate move on a physical puzzle may mistakenly identify this as a standard move. And this is where the shorthand gets dangerous because a properly sized standard move actually puts you somewhere in between stops 0 and 1 and someone trying to follow the notation might turn a bit less than standard and convince themselves that stop 0 is correct when it really isn't! You may be tempted to call this a "standard turn and a half", so maybe F F+, but it would be more accurate mathematically to write F+ F+ or perhaps F++, since it's literally the smaller ~82.8° rotation applied twice. The good news is, this turn is extremely rare (It did not come up in this post), so by and large we can get away with shorthand notation - just be aware of the possibilities of ambiguity! I will include shorthand notation for the more interesting algorithms in this topic


Number of Piece Locations
There are 4 types of pieces on the Jumble Prism. Only 3 of these piecetypes are visible on the surface, but for my shape space search, I accounted for the internal pieces as well. The number of locations for each piecetype is as follows:

A. Internal Roots - 6 pieces, 30 locations (literally 6 grips×5 stops)
B. Edges - 3 pieces, 33 locations
C. Wide Triangles - 6 pieces, 66 locations (interchangeable with edges, but have orientation, so 33×2)
D. End Caps - 2 pieces, 38 locations



Symmetry Groups
The Jumble Prism can reach shapes that come in 6 different symmetry groups. There is an interesting question of whether it's possible for a nonessential shape to express symmetry. This would require the axis and/or plane of symmetry to intersect the only available grip as otherwise the symmetry itself would imply the existence of multiple free grips. This greatly limits the possibilities for a symmetric nonessential shape, but it is difficult to prove one cannot necessarily exist at a defined stop. In any case, I checked and there are no symmetric nonessential shapes on the Jumble Prism. Therefore with the exception of the final symmetry group, this list is complete for both essential and nonessential shapes. I will give the Schoenflies Notation name for each symmetry group, as well as try to briefly describe the symmetry for those less familiar with Schoenflies.


1. Symmetry Group D3h (order 12)
Total number of occurrences: 1

Full Mirror Triangular Bipyramid symmetry: Only the solved shape expresses this symmetry. My simple simulator can't handle the curved stickers of the mass produced Jumble Prism, so I used the sticker design from the original Jumble Prism prototype instead (updated to the mass-produced colors). I hope they are similar enough



2. Symmetry Group C2v (order 4)
Total number of occurrences: 2

The 'C' stands for cyclic. C2 indicates that this symmetry class can be rotated about some axis by 180°. The 'v' stands for vertical and indicates that in addition to the rotational symmetry, this group exhibits mirror symmetry as well and specifically that mirror is coincident with the rotation axis (which is always set to vertical for consistency). Both shapes that express this symmetry in the Jumble Prism Shape Space are very interesting, so I want to give them each a chance in the spotlight.

The first C2v shape in the Jumble Prism shape space occurs at 11 moves from solved. This confusing shape appears to be perfectly solved except an for an edge that has apparently twisted 90°. However, such a 90° twist is not supported by the mechanism at all, and indeed the reality is that the protruding edge is the only piece in the correct place! Every other piece has been reassembled into what appears to be the solved shape but twisted 90° relative to the core. Although it takes on a different look for this puzzle, this shape is objectively analogous to the original jumbling wonder of the Helicopter Cube Meson Shape! This is one of the best secrets waiting to be discovered in the Jumble Prism Shape Space

B(0->2), R(0->2), F(0->3), R(2->0), F(3->0),
B(2->0), R(0->3), BL(0->1), BR(0->2), BL(1->3),
L(0->2)

Shorthand:
B R F' R' F B' R' BL+ BR BL L
which can be condensed into [B:[R,F']] R' BL+ BR BL L

The only other example of this symmetry is also the deepest symmetry in the Jumble Prism. This is the only time (so far) I have discovered that the deepest symmetry in a shape space is actually MORE symmetric than the second deepest. It occurs 18 moves deep and is in fact along the same branch as the deepest shape possible. The deepest shape is just 4 moves beyond this shape (4 instead of 3 because one move can be saved by skipping over this shape).

L(0->2), BR(0->2), BL(0->3), BR(2->0), BL(3->0),
L(2->0), BR(0->3), F(0->1), R(0->4), F(1->3),
R(4->2), F(3->0), BR(3->0), R(2->0), L(0->3),
BL(0->2), L(3->0), BL(2->0)

Shorthand:
L BR BL' BR' BL L' BR' F+ R- F R' F BR R' L' BL L BL'
which can be further condensed to [L:[BR,BL']] BR' F+ R- F R' F BR R' [L',BL]

The path to both of these shapes are highlighted in red on the shape space map below!


3. Symmetry Group C2 (order 2)
Total number of occurrences: 5
*Chiral

Just like the previous group, this symmetry group can rotate it 180° but it is NOT mirror symmetric. There are exactly 5 shapes that express this symmetry and they occur at depths 2, 6, 8, 11, and 16:

1.L(0->2), BR(0->2)

2.L(0->3), F(0->2), L(3->0), F(2->0), B(0->3), F(0->3)

3.L(0->2), R(0->3), L(2->0), BL(0->3), R(3->0), BR(0->2), BL(3->0), L(0->2)

4.B(0->3), L(0->3), F(0->2), L(3->0), F(2->0), B(3->0), L(0->2), BR(0->4), BL(0->3), BR(4->2),
BL(3->0)

5.F(0->3), BR(0->3), B(0->2), BR(3->0), B(2->0), F(3->0), BR(0->2), L(0->4), R(0->3), L(4->0),
BL(0->3), BR(2->0), L(0->2), R(3->0), BR(0->2), BL(3->0)

All 5 of these shapes are highlighted in yellow on the shape space map below!


4. Symmetry Group Cs vertical (order 2)
Total number of occurrences: 10

Cs is Schoenflies Notation for mirror symmetry across a plane. I have added the word "vertical" to distinguish it from the next symmetry group since this group refers to mirror symmetry across a vertical plane. The 10 shapes that express this symmetry are located at depths 6, 7, 10, three at 11, 12, 13, and two at 14:


1.L(0->2), R(0->3), L(2->0), R(3->0), BR(0->2), BL(0->3)

2.L(0->2), R(0->2), BR(0->3), R(2->0), BR(3->2), L(2->0), BL(0->3)

3.F(0->3), BR(0->3), B(0->2), BR(3->0), B(2->0), F(3->0), BR(0->2), L(0->2), BL(0->3), L(2->0)

4.F(0->3), R(0->2), F(3->1), R(2->0), F(1->0), BR(0->2), L(0->2), BL(0->3), BR(2->0), BL(3->0),
L(2->0)

5.BR(0->3), B(0->2), BR(3->1), B(2->0), BR(1->0), F(0->2), L(0->3), F(2->0), L(3->0), BR(0->2),
BL(0->3)

6.F(0->3), R(0->2), F(3->1), R(2->0), F(1->0), BR(0->2), L(0->2), BL(0->3), L(2->0), BL(3->1),
BR(2->4)

7.B(0->2), BR(0->3), B(2->4), BR(3->0), B(4->0), R(0->3), L(0->3), BL(0->3), L(3->0), R(3->2),
BR(0->2), R(2->0)

8.F(0->2), L(0->3), F(2->4), L(3->0), F(4->0), BL(0->3), R(0->3), BR(0->2), L(0->3), BL(3->0),
L(3->0), R(3->0), BL(0->3)

9.F(0->2), BL(0->2), B(0->3), BL(2->0), B(3->0), F(2->0), BL(0->3), R(0->3), BR(0->2), L(0->2),
R(3->0), L(2->0), BL(3->1), BR(2->4)

10.F(0->3), R(0->2), F(3->1), R(2->0), F(1->0), BR(0->2), L(0->4), R(0->3), L(4->3), BL(0->3),
L(3->0), R(3->0), L(0->2), R(0->3)

All 10 of these shapes are highlighted in green on the shape space map below!


5. Symmetry Group Cs horizontal (order 2)
Total number of occurrences: 10

This symmetry group also expresses mirror symmetry across a plane, but this time the plane is horizontal. The 10 shapes that express this symmetry are located at depths 6, 9, 10, three at 11, two at 12, and two more at 13:


1.BL(0->3), L(0->2), R(0->2), BR(0->3), R(2->0), BR(3->0)

2.F(0->2), L(0->3), F(2->4), L(3->0), F(4->0), B(0->3), F(0->2), L(0->2), BL(0->3)

3.F(0->3), R(0->2), F(3->1), R(2->0), F(1->0), BR(0->2), L(0->2), BL(0->1), BR(2->0), BL(1->3)

4.L(0->2), R(0->3), L(2->4), R(3->0), L(4->0), BR(0->3), B(0->2), BR(3->0), B(2->0), L(0->2),
BL(0->3)

5.R(0->3), L(0->2), R(3->1), L(2->0), R(1->0), BL(0->2), F(0->2), B(0->3), BL(2->0), L(0->2),
BL(0->3)

6.F(0->3), R(0->2), F(3->1), R(2->0), F(1->0), BR(0->2), L(0->2), BR(2->0), R(0->3), BL(0->3),
R(3->0)

7.F(0->2), BL(0->2), B(0->3), BL(2->0), B(3->0), F(2->0), BL(0->3), R(0->1), L(0->4), R(1->3),
L(4->2), R(3->0)

8.R(0->2), F(0->3), R(2->4), F(3->0), R(4->0), B(0->3), BL(0->4), B(3->0), BL(4->2), L(0->3),
BL(2->0), L(3->0)

9.F(0->3), BR(0->3), B(0->2), BR(3->0), B(2->0), F(3->0), BR(0->2), L(0->4), R(0->1), L(4->2),
R(1->0), BL(0->3), BR(2->0)

10.L(0->3), F(0->2), L(3->1), F(2->0), L(1->0), B(0->2), BR(0->3), B(2->0), BR(3->0), BL(0->3),
BR(0->2), BL(3->0), BR(2->0)

All 10 of these shapes are highlighted in blue on the shape space map below!


5. Symmetry Group C1 (order 1)
Total number of occurrences: 467
*Chiral

And the "Everything-Else" group. This group represents all shapes that have no symmetry, the Asymmetry group. Keep in mind these 467 shapes are only the essential asymmetric shapes. All 1352 nonessential shapes my program found would also belong to this group had I included them. The depths of the asymmetric shapes range from the very first move all the way down to this shape at the very bottom of the essential shape space, 21 moves away from the solved shape:

BL(0->3), R(0->3), L(0->2), R(3->0), L(2->0), BL(3->0), R(0->2), B(0->4), BR(0->1), B(4->2),
BR(1->3), B(2->0), R(2->0), BR(3->0), BL(0->2), L(0->3), BL(2->0), L(3->2), R(0->2), BR(0->3),
R(2->0)

Shorthand:
BL' R' L R L' BL R B- BR+ B' BR B' R' BR BL L' BL' L- R BR' R'
which can be further condensed to [BL':[R',L]] R B- BR+ B' BR B' R' BR [BL:L'] L- [R:BR']
*Notice the first 17 moves are just the mirror image through the horizontal plane of the first 17 moves of the second C2v symmetry above!*



Bram's Legendary Shape
In the solving thread, Bram shared this story with us

Bram wrote: ↑Sat Feb 13, 2021 6:33 pm At the risk of starting a legend here's a story and a question.

Many years ago, back when dinosaurs walked the earth and 3d printing was still a new thing, I got to play with an early unstickered prototype of this puzzle. It wasn't *exactly* the same thing, in that the shape was slightly different so it's possible that which moves were blocked was slightly different, but I don't think so. The thing I found playing with it was that once in a great while it would get into a jumbled state where it seemed like one of the two equilateral triangle pieces was swapped with a piece immediately next to it. This was extremely rare of a state to get into and equally hard to get back out of. Does anybody know a move sequence for getting into/out of this state?

This shape is indeed possible and in fact has already been mentioned in this post! This shape expresses Cs (vertical) symmetry and is actually the 4th shape in the 4th symmetry class described above. Swapping a wide triangle and an end cap piece produces this tooth shape that is not unlike the tooth shapes that appear on a Helicopter Cube when a corner is swapped with a triangle. Except on the Helicopter Cube, these teeth always come in pairs, so it might be quite surprising that it's possible to create just a single tooth on the Jumble Prism. It's too late to stop the legend, Bram, but at least the Legend of the Single Tooth is a true story! This shape can be reached optimally in 11 moves:

F(0->3), R(0->2), F(3->1), R(2->0), F(1->0),
BR(0->2), L(0->2), BL(0->3), BR(2->0), BL(3->0),
L(2->0)

Shorthand:
F' R F' R' F- BR L BL' BR' BL L'
which can be further condensed to F' [R:F'] F- BR [L:[BL':BR']]

As for whether or not this shape should truly be considered difficult to stumble upon and get back out of again, let's take a look at the full Shape Space pictured below, where the path to this shape is highlighted in green!



Jumble Prism Essential Shape Space
495 shapes is definitely small enough to graph, so let me now present the Jumble Prism Essential Shape Space in its full glory:

(click the image for a slightly bigger version!)
The Jumble Prism Shape Space is beautifully organized into 3 nearly equal clusters with only a single move chokepoint separating each cluster from the next! The solved shape is in the center cluster, Bram's Legendary Shape is in the left cluster, and both the Meson shape as well as the deepest shape are in the right cluster. I therefore propose the clusters be named from left to right: Legend, Solved, and Deep clusters. Or LSD for short

• The solved shape is the white diamond in the Center cluster.
• The 2 C2v shapes (the Meson Shape and the Deepest Symmetry) and the paths to them are both shown in red (the Meson Shape is the shorter path).
• The path to Bram's Legendary Shape is shown in green.
• The 5 C2 shapes are marked in yellow (3 in the Solved cluster, 2 in the Deep cluster).
• The 10 Cs vertical shapes (including the Legendary Shape) are marked in green (6 in the Legend cluster, 2 in the Solved cluster, 2 in the Deep cluster).
• The 10 Cs horizontal shapes are marked in blue (also 6 in the Legend cluster, 2 in the Solved cluster, 2 in the Deep cluster).
• The deepest shape is not marked in a special color, but easily identifiable as the rightmost node in the map.

Keep in mind only shapes with at least 2 free grips are pictured if you are trying to follow along. The nonessential shapes are all exactly 1 move away from some node in this map, so they don't add any structure (hence the name nonessential!)



Jumble Prism Shape Space Clusters
The fact that the Jumble Prism Shape Space divides so cleanly into 3 distinct clusters was another surprise waiting to be discovered about this puzzle. Both bridges between cluster consist of only a single bridge between 2 specific shapes, so these are as restricted of choke points as you can ever have! The existence of these chokepoints is what makes wandering around the Jumble Prism Shape Space so unexpectedly challenging sometimes and also why subsequent jumbling expeditions can feel like totally different spaces. Even if you feel like you have given the Jumble Prism a thorough scramble, you may have been stuck in only one cluster the whole time! And even if you DO find yourself in the Legend or Deep clusters, there's a decent chance your solution will never visit the other non-Solved cluster. I want to briefly look at each cluster.


Solved Cluster
The Solved cluster contains only 124 essential shapes, making it the smallest of the 3 clusters. In addition to containing the most symmetric shape (solved), it also contains 3 of the 5 C2 shapes, so another name for it could be the Spiral cluster (C2 symmetry often being demonstrated by a spiral). 2 examples of each simple mirror symmetry can also be found here, but neither of the C2v shapes live in the Solved Cluster


Deep Cluster
The Deep cluster is a bit cleaner, so let's discuss it next. The entrance to the Deep cluster lies 7 moves away from solved and contains 142 essential shapes. I made this gif to demonstrate the path into the Deep cluster:

R(0->2), B(0->2), BR(0->3), B(2->0), BR(3->0), R(2->0), B(0->3)

Shorthand:
R B BR' B' BR R' B'
which can be further condensed to [R:[B,BR']] B'

The final move of this sequence crosses the bridge from the Solved cluster into the Deep cluster. From here, grip L is available, and the journey into the Deep cluster begin with either L+ which frees up F, or L', which frees up BL. The deep cluster has exactly 2 copies of each symmetry group except for solved: The Meson Shape and the Deepest Symmetry (the only 2 C2v shapes), the remaining 2 C2 shapes, as well as 2 shapes from each Cs group can all be found in this cluster. Finally, this cluster contains the deepest shape in the entire space, giving it its name


Legend Cluster
The bridge to the Legend cluster is not quite as clean cut as its Deep counterpart. There is still a single move connecting the Legend cluster to the Solved Cluster, but the grip that gives this move can stop at 2 other stops, leading almost immediately to dead-ends and resulting in 3 essential shapes that are orphaned somewhere in between the Legend and Solved clusters. Excluding those 3 shapes, the Legend cluster contains 226 shapes, making it the largest of the 3 clusters by about 59%. The entrance to the Legend cluster lies only 6 moves away from solved and I have again made a gif to demonstrate the path into the Legend cluster:

R(0->2), F(0->3), R(2->4), F(3->0), R(4->0), B(0->3)

Shorthand:
R F' R F R+ B'
which can be further condensed to R [F':R] R+ B'

The final move of this sequence crosses the bridge from the Solved cluster into the Legend cluster. From here, a ton of moves are available to lead you into the Legend cluster:

• L+ frees up F
• L' frees up F, B, and BL
• BL frees up L and B
• BL' frees up L
• BL- frees up B

I think it could be argued that this gathering of free grips right at the chokepoint makes the Legend cluster harder to escape than any other cluster. Even if you do wander your way to this chokepoint shape, you have to choose the correct grip of the 3 available to cross the bridge back to the Solved cluster. The other 2 grips both lead you back into the Legend cluster. Another potential name for the Legend cluster could be the Mirror cluster, because this cluster contains 60% of both types of Cs symmetry groups, triple the density of Cs shapes of any other cluster. Hey, it was a tough call between Legend and Mirror - in the end I gave the honor to Bram, but you can call it what you like. There is a certain ominousness to "entering the Mirror cluster"


Those 3 orphaned shapes that lie somewhere between the Solved and Legend clusters can be reached by replacing the final B' in the Legend chokepoint sequence above with B+ or B BR. The former frees up BL and the latter frees up R, but in both cases no stops free up any additional grips so you are forced to undo moves.



Finally, I have attached an Excel spreadsheet to this post that breaks down the instances of each symmetry group by depth as well as some statistics on the number of blocked grips throughout the shape space. There isn't too much to talk about regarding blocked grips on this puzzle. I didn't track shapes with only 1 free grip, so you won't see any in the excel document. I suppose it's also worth mentioning that there are no shapes that block only 1 grip, though this is a far from unique property for jumbling puzzles. The full list is in the excel document if anyone would like to see!

Hopefully you have enjoyed this thread so far but we aren't done! The shape space of the Jumble Prism is small enough that I can explore deep into the State Space, where I DO account for the stickers quite quickly. This valuable tool allows me to fully analyze the impact all of these jumbling moves have on the possible configurations of the Jumble Prism! But let's make a new post for that discussion, because this one is getting a bit lengthy

-Matt Galla

Computing the possible solved shape configurations for a jumbling puzzle is a daunting task, because the traditional method of tracing out orbits and determining parity dependencies kinda falls apart when the puzzle doesn't stay in shape. Nevertheless, I've done extensive testing on a few jumbling puzzles now and I am confident in explaining the nature of jumbling puzzles in the following way:


Even Permutations and Twists
If there is enough freedom in the shape space of a jumbling puzzle, then just like all other twistypuzzles, a given piecetype can AT LEAST reach all even permutations, and orientations can AT LEAST be twisted in pairs in opposite directions. What exactly constitutes "enough freedom" is extremely difficult to quantify and a method for determining that may have to be unique for each jumbling puzzle, but I can confirm that the Jumble Prism DOES have "enough freedom".

As a reminder, the Jumble Prism has 3 edges with 2 orientations each, 6 Wide Triangles that cannot be oriented, and 2 End Caps with 3 orientations each.
The 3 edges (B pieces) can at least reach the 3!/2 even permutations and the 2^3/2 orientations reachable by twisting them in pairs.
The 6 wide triangles (C pieces) can at least reach the 6!/2 even permutations
And the 2 end caps (D pieces) can at least reach the 2!/2 even permutations (which is only the solved permutation, so this statement is not very enlightening) and the 3^2/3 orientations of twisting them together in opposite directions.

Having a piecetype that consists of only 2 elements is a bit odd for a twisty puzzle; there is only one even permutation of two elements - the solved permutation! And if you twist one end cap one way, the other end cap twists the other way, so the two end caps will always align colors on opposite ends of the puzzle. Nevertheless, this is the way I think it makes the most sense to consider the configurations of a jumbling puzzle. We start with the basic operations and then look for the extra properties that break these parity assumptions


Grip Parity
I want to briefly make an observation about the usual culprit of parity on doctrinaire twisty puzzles - the cumulative rotation on each grip. On a Rubik's Cube the parity of both the edge and corner permutations toggle every time a face is rotated 90°. This is why the total number of twists on a Rubik's Cube from solved to scrambled and back to scrambled must always be an even multiple of 90°. In many puzzles from the tetrahedral and octahedral family when the orientation is visible, it's possible to have a single corner piece twisted and that's because the underlying grip that aligns with that piece has not been twisted a multiple of 360° (this happens on other puzzles too, it's just tetrahedral and octahedral puzzles seem to always be the puzzles that people first encounter this). Each 180° turn of a Helicopter Cube grip toggles the parity of the corners and 2 of the 6 triangle orbits, so careful selection of grips can lead to the surprising situation of having all triangles correct, but the corners in an odd permutation. There are hundreds of examples of this across nearly every doctrinaire puzzle in existence.

However, this is one area where jumbling puzzles are actually simpler than doctrinaire puzzles. On a pure jumbling puzzle, any grip that has not been rotated a multiple of 360° will not be in the solved shape. That's one of the most fundamental properties of a jumbling puzzle after all! Puzzles with both jumbling and non-jumbling moves can only ever be in the solved shape if every grip is in one of the non-jumbling stops. The point is jumbling puzzles can never experience parity errors due to residual twist on the grips. All parity errors must come from other sources!
It so happens that for the Jumble Prism, there are two sources of parity errors:


Jumble Parity
Like other puzzles that mix different sized pieces together (for example, the Square-1), it is possible to trigger a parity toggle while the puzzle is out of shape and find yourself in an otherwise impossible situation once you are back in shape. This might sound like an incredibly complex phenomenon to understand and identify in the space of a jumbling puzzle, but I actually have a pretty clean explanation of what happens.

On the Jumble Prism an edge piece (B) is mechanically equivalent to a Wide Triangle joined to an End Cap (C+D). Assuming no bandaging from the rest of the pieces, a Wide Triangle and an End Cap can nestle together in the space left behind by an Edge, and that Edge can fill up the space previously taken by the Wide Triangle and End Cap. Even though they are different piecetypes, they can temporarily behave like each other long enough to swap places. Another way to view this phenomenon is to notice that there are two types of (layer 1) jumbling interactions - the interaction between two grips around the same end cap (same hemisphere) and the interaction between two grips on either side of an edge (across the equator). These two interactions are also exchangeable with each other, and the pieces affected by each type of interaction are exactly the pieces that are mechanically equivalent.

Now, once we are in the solved shape again, every piecetype must have returned back to one of its native locations and there's none of this mechanically equivalent piece-swapping happening, but during jumbling moves, these pieces behave like elements of a single orbit of pieces. The argument against grip parity given above still applies to this combined orbit of pieces, so the overall parity must remain even, but the combined orbit can make a 2-2 swap of like groups of pieces, leading to an overall even permutation in the combined orbit, but with the B pieces and C+D pieces individually in odd permutations. There are many possible paths through the shape space of the Jumble Prism that can result in this parity, but I will present this one for its simplicity:

R(0->2), F(0->3), R(2->0), F(3->0),
B(0->2), BR(0->3), B(2->0), BR(3->0),
R(0->2), F(0->3), R(2->0), F(3->0)

Shorthand:
R F' R' F B BR' B' BR R F' R' F
which can be condensed into [R,F'] [B,BR'] [R,F']

During this sequence, 2 Edges and 2 groups of Wide Triangles and End Caps perform a 2-2 swap, overall toggling the parity for all 3 pieces. This is what I am naming a Jumble Parity (Error).


Global Orientation Parity
On a Rubik's Cube, the center stickers tell you what color each face is supposed to be. On a Jumble Prism, you have no hints as to what color is supposed to end up where, so you are forced to guess. In general, if you guess the wrong global orientation for a puzzle, there's a chance you may run into parity problems. For the Void Cube, this can lead to a situation where the edges and corners apparently have different parity. We need to analyze the effects of global orientation on the Jumble Prism to figure out what type of parity situations can arise if we guess the wrong global orientation.

A triangular prism has 6 possible orientations in space, and unlike cubes or other platonic solids, these can't be generated by a single type of global twist, we will have to analyze both global twist types. Fortunately, one type of twist has no impact!

Performing a global rotation of 120° about the axis perpendicular to the triangle face results in a 3-cycle of Edge pieces and two 3-cycles of Wide Triangles, while also adding both a positive and negative twist to the two End Cap pieces. Both of these permutations are still even and the induced twists on the End Caps are already possible anyway, so this type of global orientation has no effect on either the parities or cumulative twists of any piecetype. Indeed it is possible to solve the Jumble Prism into any of the 3 global orientations generated by this rotation (Remember this for later!)

Performing a global rotation of 180° about the axis perpendicular to the square face however is a different story. This results in a swap of two Edges, 3 swaps of pairs of Wide Triangles, and a swap of the two End Caps, while also adding a twist to one Edge. All three permutations are odd, so if you try to solve the Jumble Prism into the upside-down configuration of where it started, you will find that all 3 piecetypes will have odd parity as well as only a single edge needing to flip.


Flipping a Single Edge
But wait a second, both the Jumble Parity and the Global Orientation Parity errors result in the parity of all 3 piecetype permutations toggling. Don't these cancel out? Yes! The Jumble Parity completely negates the effects flipping the puzzle upside-down has on permutation parities. But not the single flipped edge! The Global Orientation Parity toggles the parity of all 3 piecetypes and flips one Edge, while the Jumble Parity only toggles the parity of all 3 piecetypes without flipping any edges. So if you combine the effects of these two sources of parity errors, you can reach a state that is quite rare in the twisty puzzle universe: a single flipped edge! Below is one of many optimal 26 jumbling move sequences to flip only a single edge (in this case the L/BL edge):

R(0->2), B(0->2), BR(0->2), R(2->0), BR(2->0), R(0->2), F(0->2), B(2->0), F(2->0), B(0->2),
BR(0->2), R(2->3), BR(2->3), R(3->2), B(2->0), R(2->0), F(0->2), B(0->3), F(2->3), B(3->0),
BR(3->0), R(0->3), BR(0->3), R(3->0), F(3->0), BR(3->0)

Shorthand:
R B BR R' BR' R F B' F' B BR R+ BR+
R- B' R' F B' F+ B BR R' BR' R F BR
There are a few commutators that could be used to condense the sequence a bit:
R B [BR,R'] [F,B'] BR R+ BR+ R- B' R' F B' F+ B [BR,R'] F BR
And if you accept jumbling conjugates, you can do a bit better, but I think this sequence is just fundamentally messy at heart:
R B [BR,R'] [F,B'] BR [R+:BR+] B' R' F [B':F+] [BR,R'] F BR

I showed this animation at a slightly different angle than all of the other animations to emphasize the fact that L and BL are never used in this sequence. As a result, the L/BL edge, which we are supposedly flipping, never actually moves. Instead all of the other pieces move around it. This also highlights the fact that we had to flip our definition of the solved configuration upside-down to achieve the flipped edge appearance, thereby inducing the Global Orientation parity


The Number of Jumble Prism Configurations
So bringing everything together.... In the solved shape, the Edge pieces can reach both odd and even permutations by implementing the Jumble Parity. Each edge can be twisted independently by implementing the Global Orientation Parity. After that, the parity of the Wide Triangle permutation must match the edges. And the parity of the End Caps must match as well, leading to only one possibility. The orientation of the End Caps must sum to 0 modulo 3, meaning one must always align with the other. Finally, the Jumble Prism can be solved into 3 of its 6 global orientations, so unless we have some visual indicator of which of these 3 orientations we are in, we must divide the overall product by 3.

This leads to
(3!*2^3 × 6!/2 × 2!/2*3^2/3)/3 =
17,280 solved shape configurations.

To find the total number of configurations including all essential shapes, we need to multiply this number by the sum of the symmetry indexes of each shape:
17,280 × (1×1 + 2×3 + 5×6 + 10×6 + 10×6 + 467×12) =
99,550,080 essential configurations

This number doesn't include any configuration where only one grip is free, but honestly, I don't think it should. These are the configurations you care about where a choice can be made This is somewhere between the number of configurations for a Rubik's Ufo (39,916,800) and a 2x3x3 Domino (406,425,600). Anyone who has played with the Jumble Prism knows that this puzzle can be very difficult to navigate, which is remarkable considering how relatively few positions it can really reach (the space is over 400 billion times smaller than the Rubik's Cube)!

Thank you for coming to my Ted Talk
-Matt Galla

PS: Let me know if there are any configurations you would like to know the optimal solution to! I can search for anything now!

These analyses are one of my favorite things to read on the forum, even though I don’t even own a jumble prism

Thank you very much Matt for this detailed analysis!
I have to come back to read it more thoroughly, but I admire the length and depth of these two posts above.

I'm sure that you understand that I never intended criticism of your 5 stop notation.
I looked at some details in the Solving Jumble Prism, when a fellow member asked me for help, because he had difficulties to understand the used notation.
(Obviously, it was misunderstood even by people who made valid contributions.)

The names of the grips were already invented and introduced by ironesp and alacoume.
(I would prefer D,DL,DR instead of B,BL,BR, but - as you - I did want to use the names that were in use, already. )


I introduced, though, what you are calling the "shorthand" notation.
In the specific case of this puzzle, I'm not able to recognize the "0" state of a grip, easily.
Therefore, I feel that I do not lose much using the shorter notation compared with your precise "5 stop notation"
In case of a different jumbling puzzle like the 4 Corners Cube, I use myself a shorter version of your notation by shortening something like R(0->3) just to R03.

Thank you again!

This is one of the best articles I have seen on studying a cube. It is the most complete thing that can be done.
Having studied jumble prism a lot almost a year ago, I know how hard this cube is and how long it takes to solve it.
Bravo!!! Very good Allagem

Good job!

I managed to get a mass produced version of the Jumble Prism into the legendary shape as well. This time I got it out of it reasonably quickly by leaving it in my living room, after which it got magically put into the solved shape. I suspect Briar had something to do with this.

I love reading these !
Thank you

@Allagem Would you by any chance, be willing to share the adjencency list / table for the Jumble prism?

Shape# - Shape#
0-0
0-1
1-1
1-2
1-3
1-4
...

(like you did for the helicopter cube?)