Message #1550

From: schuma <mananself@gmail.com>
Subject: Re: Magic Tile {8,3} 6 Colors, 3 Layers, Slicing factor = 1.15 solved
Date: Sun, 13 Mar 2011 09:22:12 -0000

Hi Andrey,

I went through many puzzle with size-3. I believe size-3 is the "first-order" problems that need to be solved. Here is what I found so far.

Bullet "-" means not interesting because it is (a) too easy to solve or (b) directly related to familiar existing puzzles.
Bullet "+" means interesting case because it is (a) not too easy and not too hard and (b) has new features
Bullet "*" means not interesting because it is too complicated and tedious to solve (using my patience as a reference).

• {n,3} 3 colors: 1 < factor < 1.2. When it turns, it seems to be broken, because each edge is moved by two circles into two contradicting directions. But the end effect of either direction is the same. Therefore neglecting the animation, I still consider it a working puzzle. But anyway, solving is trivial. It’s solvable in 3~5 moves.

• {6,3} 4 colors: factor = 1.2. It can be solved in the same way as factor = 1.0. Although it has a new type of pieces, the new pieces will be automatically solved once the old ones are solved.

• {6,3} 4 colors: factor = 1.3. As Andrey said, the pattern is pretty neat, but solving is trivial. It can always be solved within at most 4 moves.
  
  
• {6,3} 4 colors: factor = 1.4. Just like {n,3} 3 colors, during turning the pieces are "broken". But neglecting the animation, I think of it as a working puzzle. Again solving is trivial.

I think the above three cases covers all the {6,3} 4-color puzzles.

• {6,3} 16 and 25 colors: factor = 1.3. Neat pattern. The edges are like the edges in pyraminx or pyraminx crystal, which can be 3-cycled using a four-move commutator. Therefore solving is nontrivial but not hard.

• {6,3} 16 and 25 colors: factor = 1.2. Just a combination of the pieces from factor = 1.0 and 1.3. Solving is a little bit harder than factor = 1.0.

* {6,3} 16 and 25 colors: factor = 1.4. Too many small pieces. Although I can see the algorithms to solve it, I won’t enjoy the solving experience unless I can use macros.

• {6,3} 9 colors: factor = 1.3. The edge pieces cannot be easily solved like in 16 and 25 colors. I don’t understand the behavior of them. This seems to be an interesting puzzle.

• {6,3} 9 colors: factor = 1.2. It should be similar to the 16/25-color case, except the edge pieces should be treated as in factor = 1.3.

* {6,3} 9 colors: factor = 1.4. Too many small pieces. Seems even harder than the 16/25-color case.

• {7,3} 24 colors (Klein’s quartic): factor = 1.2. Similar to {6,3} 25 colors. A combination of Klein’s quartic with factor = 1.0 and pyraminx edges.

* {7,3} 24 colors (Klein’s quartic): factor = 1.35 (1.30 is not a sweet spot). The situation is similar to {6,3} 25 colors factor = 1.4. Too many small pieces to solve.

• {8,3} 6 colors: factor = 1.15. It’s the puzzle I talked about earlier. Regular Rubik’s cube plus a special type of edges. A compact and challenging puzzle.

• {8,3} 6 colors: factor = 1.29. This is really a weird looking puzzle. Neglecting glitchy animation, I find it an interesting one. When it’s scrambled, it looks weird, weird, weird. At the center of each circle there is a daisy with eight petals. In principle it is equivalent to Rubik’s cube but you always turn two faces (e.g. R and MR, it is equivalent to turning L’ and reorient the cube). The checkerboard pattern on the Rubik’s cube looks more like a "daisy" pattern in this puzzle. A snapshot of such pattern can be found here: http://f1.grp.yahoofs.com/v1/gHl8TQM7KVekRGSLFshClhWBghiXnOq6JjY0g8rfXHDlDthzbbJpTTY1pDV0ByHELCAP8ji9dppnRq_hs2-aduIUCLoPGpnP/Nan%20Ma/83_6colors_size3_129.PNG

• {8,3} 12 colors: factor = 1.15. The edges behave like {6,3} 9 colors factor = 1.2. I don’t understand but seems interesting.

* {8,3} 12 colors: factor = 1.29. Similar to {6,3} 9 colors factor = 1.4. Too many small pieces.

Puzzles {n,3} for n>9 are similar to the smaller counterparts with the same number of colors.

• Dodecahedron 12 colors. As Brandon pointed out earlier, the face-turning dodecahedron with cuts of different depths has been well studied: <http://twistypuzzles.com/forum/viewtopic.php?p=179381&#p179381>. I sampled several values for the factor and the puzzles are always covered by Gelatinbrain 1.1.1 ~ 1.1.5. Therefore although this is a very rich family of nice puzzles, it’s not unique in Magic Tile.

• Hemi-Dodecahedron 6 colors. Another rich family of puzzles, whereas this family is unique in Magic Tile. I interpret them as slice-only face-turning dodecahedra. For example, if factor = 1.75, it’s a slice-only Starminx. Unfortunately I cannot find the sweet spot to make a slice-only pyraminx crystal. The tiny pieces are always around.

For size-5 puzzles, I tried to analyze {8,3} 6 colors factor 1.15. It seems to be a pretty hard puzzle. I don’t think I’m patient enough to solve it without using macros. Generally speaking I think size-5 is too complicated.

Nan

— In 4D_Cubing@yahoogroups.com, "Andrey" <andreyastrelin@…> wrote:
>
> Hi Nan,
> I think it’s good idea. I think that any puzzle of this kind deserves a line in records page. Just change "Size" column to "Size, factor" and add a line with "3, 1.15" there.
> Strange thing is that, say, for {8,3}, 12 colors you can play with factors 1.15 and 1.4, but not with 1.25. Probably it’s a bug in the program.
> One of nice puzzles is {6,3}, factor=1.3 - where circles meet in centers. With 4 colors I solved it in 2 twists, but 16 or 25 colors may be funny.
>
> Andrey
>
> — In 4D_Cubing@yahoogroups.com, "schuma" <mananself@> wrote:
> >
> > Hi,
> >
> > The puzzle I’m talking is illustrated here:
> >
> > http://wwwmwww.com/Puzzle/MagicTile/3x3x3UDRF.png
> >
> > I came to this puzzle because Carl talked about it in the Twistypuzzles forum <http://twistypuzzles.com/forum/viewtopic.php?f=1&t=20697&p=251131#p251131>, and I just learned that I could change the "slicing circles expansion factor" to make deeper cuts. I set it to 1.15 and solved the puzzle. The log file has been posted here: <http://wiki.superliminal.com/wiki/User:Schuma#Octa_6col_length3_115>.
> >
> > This puzzle is essentially Gelatinbrain 3.1.31, except the centers are normal 3x3x3 centers. Here the edge pieces are quite special. They cannot be found in regular 3x3x3. Solving them is quite challenging for me. This is a nice, compact and hard puzzle to solve.
> >
> > Note that: for expansion factor = 1, {8,3} 6 colors is equivalent to Rubik’s cube. However, for expansion factor = 1.15, {8,3} 6 colors is not equivalent to Rubik’s cube with any expansion factor, because of different geometries.
> >
> > I think some puzzles with slicing expansion factor>1 are quite neat (as long as they turn properly) and can be regarded as standard challenges. What are the other nice puzzles with special challenges? We may put some to the wiki records page.
> >
> > Nan
> >
>