Hello All

I want to show you the first stage of my joint project with Oskar and Diogo Sousa.
Squashed Pentultimate puzzle. the idea came in parallel - me and Diogo.
http://twistypuzzles.com/forum/viewtopi ... =1&t=32548

Oskar came up with a rail mechanism and made blueprints. I printed it and assembled it.
Unfortunately, apparently, a mistake was made in the drawings and the puzzle cannot be scrambled.
I will continue to study this problem and try to finish this interesting puzzle!

Angle 1
Angle 2
Angle 3

Wow! That looks pretty surreal.

Why are the top and bottom faces not perfectly aligned with their pentagon pieces (perhaps due to having to make the mechanism fit inside)? What exactly is the geometry in its natural form?

Wow, that's weird but awesome. It reminds me the Squished Skewb

Good to see this puzzle materialise.

grigr wrote: ↑Sat May 16, 2020 10:13 amUnfortunately, apparently, a mistake was made in the drawings and the puzzle cannot be scrambled. I will continue to study this problem and try to finish this interesting puzzle!

My conjecture was that this simple-deep-cut puzzle would not scramble. So are you confirming my conjecture?

By the way, do we have a proper definition to determine whether a puzzle can be scrambled or not?

Oskar

Oskar wrote: ↑Sat May 16, 2020 11:14 am Good to see this puzzle materialise.

grigr wrote: ↑Sat May 16, 2020 10:13 amUnfortunately, apparently, a mistake was made in the drawings and the puzzle cannot be scrambled. I will continue to study this problem and try to finish this interesting puzzle!

My conjecture was that this simple-deep-cut puzzle would not scramble. So are you confirming my conjecture?
By the way, do we have a proper definition to determine whether a puzzle can be scrambled or not?

Unfortunately, I can only make two turns on this puzzle.
also it is a little different from my simulator version.
a little later I will try to fix it ...

Oskar wrote: ↑Sat May 16, 2020 11:14 am By the way, do we have a proper definition to determine whether a puzzle can be scrambled or not?

If there's a way of getting it back to the starting shape but with pieces permuted it can be scrambled.

Coincidentally I was just thinking about the skewb equivalent of this the other day. That one is most naturally shaped as block where the slice intersections meet at the centers of edges. Whether either of this can be scrambled I don't know, but the skewb one seems like it might have floating axes and need to be a coreless/minimalist mechanism.

The squished skewb isn't exactly equivalent, that one is distorted about a triangle, where this one is distorted about a non-triangle, as in the puzzle mentioned above (the squashed skewb?) But the squished skewb and the squashed pentultimate do have one thing in common in that there's an odd slice out which isn't moved, where in the squashed skewb every slice is moved the same amount.

The main mechanical problem with the squashed pentultimate is that it wants to have floating axes and the mechanism doesn't work well for those, while the squashed skewb can use simple male-female dovetails with no hidden pieces.

The squished pentultimate is an interesting idea. That one needs two different amounts of squish, and it isn't obvious how they interact, if at all. Probably also suffers from mechanical problems with floating axes.

that is so weird. i thought the picture was squashed.
nice puzzle!

Bram wrote: ↑Sat May 16, 2020 12:50 pm Coincidentally I was just thinking about the skewb equivalent of this the other day. That one is most naturally shaped as block where the slice intersections meet at the centers of edges. Whether either of this can be scrambled I don't know, but the skewb one seems like it might have floating axes and need to be a coreless/minimalist mechanism.

Bram, you seem to be describing the geometry of Almost-a-Skewb, am I correct? If not, where do they differ?

Oskar

Oskar wrote: ↑Fri May 22, 2020 3:56 am

Bram wrote: ↑Sat May 16, 2020 12:50 pm Coincidentally I was just thinking about the skewb equivalent of this the other day. That one is most naturally shaped as block where the slice intersections meet at the centers of edges. Whether either of this can be scrambled I don't know, but the skewb one seems like it might have floating axes and need to be a coreless/minimalist mechanism.

Bram, you seem to be describing the geometry of Almost-a-Skewb, am I correct? If not, where do they differ?

The almost-a-skewb is very different. The puzzle I was describing is in the shape of a square prism and has fourfold symmetry about the square faces, which look a lot like the individual faces of a skewb with a diagonal square pieces and triangular pieces on the corners, but the triangular pieces are pythagorean rather than equilateral. The puzzle only has the analogous 14 pieces to what a skewb has but has a lot of move restrictions similar to the melty cubes.

Bram wrote: ↑Sat May 30, 2020 11:59 pmThe puzzle I was describing is in the shape of a square prism and has fourfold symmetry about the square faces, which look a lot like the individual faces of a skewb with a diagonal square pieces and triangular pieces on the corners, but the triangular pieces are pythagorean rather than equilateral. The puzzle only has the analogous 14 pieces to what a skewb has but has a lot of move restrictions similar to the melty cubes.

So there are six square faces that are touching each other at the corners, correct? Do all six squares have the same size? If so, how can the triangles that their edges form not be equilateral?

Oskar wrote: ↑Sun May 31, 2020 4:39 am

Bram wrote: ↑Sat May 30, 2020 11:59 pmThe puzzle I was describing is in the shape of a square prism and has fourfold symmetry about the square faces, which look a lot like the individual faces of a skewb with a diagonal square pieces and triangular pieces on the corners, but the triangular pieces are pythagorean rather than equilateral. The puzzle only has the analogous 14 pieces to what a skewb has but has a lot of move restrictions similar to the melty cubes.

So there are six square faces that are touching each other at the corners, correct? Do all six squares have the same size? If so, how can the triangles that their edges form not be equilateral?

There are two square faces opposite each other and four rectangular faces orthogonal to them. All corners have the same shape of piece on them, but the square faces have square pieces at the centers and the rectangular faces have diamond shaped pieces at the centers.

Bram wrote: ↑Sun May 31, 2020 1:16 pmThere are two square faces opposite each other and four rectangular faces orthogonal to them. All corners have the same shape of piece on them, but the square faces have square pieces at the centers and the rectangular faces have diamond shaped pieces at the centers.

Like this? That would be a heavily bandaged version of PentaJumble, correct?

Oskar

Back in my conversation with Oskar and Evgeniy, I mentioned there was a specific height for the Squashed Pentultimate that would allow for extra jumbling interactions between the pieces. This is described in this post by Matt Galla where he introduces the concept of sweet-spots, and was later extended in this post when I derived the formulas for trapezohedral geometries. Back in 2017 I toyed around with this very concept and found a puzzle very similar to the one posted in the previous post by Oskar, that out of coincidence I started printing in FDM a couple of weeks ago (a non-fully-functional version). I call this puzzle the Skewboid, and it is based on the second sweet-spot of the tetragonal dipyramid (so basically, a distorted octahedron). Interestingly, this geometry happens to be a subset of the Rhombic Dodecahedron, and thus is not simply deep cut (hence why I'm not making a fully functional version in FDM, I'm not a madman).

To construct the geometry, you just need to make a bipyramid who's height from tip to tip is the edge length at the equator. For a deep cut puzzle like the Skewboid, you could also think of further reducing this geometry be deleting alternating faces. Below are some photos that illustrate the explanation. I believe this axis system would be interesting to explore, as it has a different behaviour from the Skewb, but still allows quite a few interactions that lead to good "scrableability".

On a sidenote, I find it fascinating how you can just think these geometries up in your head, Bram. Without aid from CAD and being able to look and manipulate things in front of me, I'd be useless at finding these odd geometries

Hi Diogo,

Thank you for identifying the rhombic-dodecahedron sweetspot for Bram's Squashed Skewb.

My conjecture is that unlike the regular Skewb, the Squashed Skewb is not simple deep cut. That is, one can build a regular Skewb with male+female rails without limiting its functionality, but not Squashed Skewb. Can anyone confirm my conjecture or prove it false, please?

I could not follow Evgeniy's analysis of Squashed Pentultimate prototype. The electronic simulation could be scrambled, whereas the physical sample with male+female rails could not. So I am wondering whether Squashed Pentultimate is also not simple deep cut. Can anyone confirm this?

Oskar

P.S. What is the opposite of "simple deep cut"? "Regular deep cut"? "Complex deep cut"? "Not-simple deep cut"?

Oskar wrote: ↑Mon Jun 01, 2020 2:12 am My conjecture is that unlike the regular Skewb, the Squashed Skewb is not simple deep cut. That is, one can build a regular Skewb with male+female rails without limiting its functionality, but not Squashed Skewb. Can anyone confirm my conjecture or prove it false, please?

If my understanding of this geometry is correct, it is not simple deep-cut, as there exist orthogonal cutting planes. The theoretical puzzle should allow 180 degree turns on a cut, but this prevents a simple mechanism from being functional.
EDIT: See here.

Oskar wrote: ↑Mon Jun 01, 2020 2:12 amP.S. What is the opposite of "simple deep cut"? "Regular deep cut"? "Complex deep cut"? "Not-simple deep cut"?

I believe the most common adopted term is "complex" deep-cut for non-jumbling geometries, and "hyper complex" for jumbling ones (as an internal rails mechanism with finitely many pieces cannot exist).

Yes, the thing I was describing is the 'skewboid' although I didn't see the sweet spot extra movement, and I can't see what it might have to do with the pentajumble. From the pictures and the description, it seems the extra movement is enabled by the slices intersecting at right angles in the middle of the short edges, which most definitely would make a simple male-female mechanism not work. It can work with floating hidden centers, if you attach the corners of one of the large square pieces to the centers. That should keep things from getting out of alignment.

The skewboid seems to be a fascinating puzzle both with and without the extra moves, for different reasons. I still don't know if it's scrambleable without the extra moves.

By the way I can't do all this sort of visualization totally in my head. The right angles in the skewboid are obvious in retrospect, but sweet spots in general are very hard to visualize and I often look at pictures of platonic or archimedean solids as reference.