I was solving that room in Abyssian Fortress this morning when a line of insight popped out of nowhere. I immediately picked it up and worked on it thoroughly, which by then I figured out it's actually a fully workable theory on dealing with gels.
(The pics in this thread are badly drawn. If anyone knows of good tools to draw these like the ones on the tarstuff page, feel free to suggest.)
As we know, tar cutability is simple: it's a tiling of 2x1 tiles for the enclosed corners. Mud is even simpler. Gel, however, is so viscous
it resisted various angle of attack for years.
As a reminder, gel can only be cut on
inside corners.
Before we begin, perhaps we can look at the situation in a different perspective: Tar cutability is easy to analyse because cutting tar is essentially a clean cut inside our analysis scheme: for each cut, we're taking away
exactly one piece of 1x2 enclosed corners, and doesn't change anything else. This implies that the analysis doesn't depend on
the order of evaluation.
It is important because if not so, things may happen differently just because one part is evaluated before another, and everyone knows that it's down the path of "
so horrible I won't touch at all"
. An example would be horde movement order tracking, such as mass wraithwing manipulation.
In this sense, L-tromino tiling and enclosed corner analysis are both bad:
L-tromino tiling, even if it can be made to work, would be bad because the two side tiles of the L piece only get taken away some of the time, depending on the surrounding tile connectivity.
Enclosed corner analysis is bad for a different reason: the amount of cuttable enclosed corner formations could increase instead, so it fluctuates between being helpful and being unhelpful. We don't want to get more gel when we cut gel.
The new theory of gel cutability goes as follows:
Instead of using the old tar way (tiling and enclosed corners), we'll consider the tiles themselves instead, which we'll call them
nodes:
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Some example gel formations. Every blob besides the square is cutable.
I've marked the bright and dark tiles in red and green dots. They are in different colors because they are nodes of different parity.
Next, we'll connect two diagonally adjacent tile together, if this diagonal has both dots of opposite color present on the sides (Or, in other words, cross out all 2x2 blocks). This forms a network of diagonals. We'll call the connected lines
links:
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From this, we can immediately see that a tile is cutable if and only if that tile is at the center of an
exposed straight line segments of the length of 3 nodes.
It can be of T shape or I shape. The primitive gel formation for such T shape and I shape are the 8-tile and 7-tile gel, respectively, shown in the bottom right corner.
Cutting gel in this setup equates to this algorithm:
1. A node with the above-mentioned property (connected with adjacent nodes by T or I shape) is cut and destroyed.
2. Links connected with this node is destroyed.
3. All links of the opposite color, which cross over any links that was destroyed in step 2, is also destroyed.
And then we repeat the progress with the new, modified network, until nothing can be cut.
This is an improvement on the enclosed corner analysis: We've found a pair of quantities which is guarantee to decrease with each cut. This is also an improvement over the L-tromino tiling, because in this case it's exact: every step cuts specific nodes, and nothing else; it's similar to how tar are cut by taking away exactly 2 adjacent enclosed corners at a time.
Note that
singled red/green nodes will be formed in the progress. We don't destroy them because they become gel babies. Singled nodes cannot have any stable form (for a 2x2 formation you need 2 red/green nodes which is connected). Incidentally, this provides a way to calculate the amount of gel babies produced by any particular sequences of cuts.
What about the deadly gel formation? We know that some gels are cutable, but Beethro can't cut the gel himself because it'll end up in a lethal situation. This analysis also gives the exact condition of such lethal cuts:
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Two images taken from the tarstuff page, with the analysis.
Basically, two gel babies form next to Beethro's sword if and only if these two corner (say, green) nodes are singled as a result of the central node of opposite color (red) being destroyed. In order to make this lethal, Beethro must have no space to retreat, which means there are another two corner (green) nodes blocking your way. This corresponds to
one I-shaped empty node formation of the same color. The following example gives a clear view at the sufficient and necessary condition:
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Two cuttable red segments is sandwiching the blue segment. Hence, Beethro cannot cut the gel NE and SW while he's standing on the middle of the blue segment.
Onto gel cutability. Gel cutability problem with a single color only is simple, as the problem reduces to a
graph traversing problem for that color.
Since cutting gel requires each step landing on a node with two existing nodes perpendicular to your direction of travel, this means
all sections of a path cannot be directly adjacent from any other path segment of the same color, as that closes the path. It
can't also be a piece adjacent to empty space. A rule of thumb is you need to stay
at least 4 tiles (2 diagonals) away from any path you've drawn of the same color to continue your gel-cutting path.
Additionally, you can't begin from a node which is at the middle of a I-shaped empty node.
After all red links are destroyed, green links are destroyed as well, and so the gel is gone.
When the links are all destroyed, all remaining uncut nodes will become singled.
Example:
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An example. We cut along the red path, then the three yellow path. At this point, all red links are gone, and the gel is no more.
Gel babies produced: 34
This also explains why a corner with different parity cannot be revealed:
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The corner is in green. By our rules that we can only cut at the middle of 2-link long connected nodes, we can't cut the 2 red nodes surrounding that corner green node. Therefore, a 2x2 blob must remain at the corners.
Gel cutability problem with both colors involved only becomes slightly more complicated, because by our construction any
two paths of different color cannot touch each other. They must stay at least
3 tiles away from all other paths of opposite color. (If the paths cuts all gel, they form 2 diagonal lines of gel babies.)
For a gel to be cuttable, the paths of opposite color must always stay 3 tiles away from each other. Otherwise there is a gap of 2x2 blob.
(By the way, once an entrance is selected, everything along the path opposite to the color of this entrance is suppressed, so you can safely ignore them as you extend this path.)
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Two minimal examples.
If you are evil enough, you can use this information to make a gel formation with 2 colored paths spiraling each other while only barely able to stay 3 tiles away from them. I can already see this coming.
EDIT: Now it's done:
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Note that gel cut solutions are not unique!
In fact, in general, a larger gel blob gives you more possibility in walking the paths along the nodes.
The following is a recreation of an example from the threads linked above:
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A minimalist example of a gel formation that has two solutions.
Also, note that because of the lethal cut condition, gel cutting paths cannot branch from a straight path. If you want to make a 3-way or 4-way branch, make the 90 degrees turn first.
Mimics breaks tar invariant when two swords stabs two adjacent tar, reducing the number of enclosed corners by 3 instead of 2 or 4.
Similarly, with mimic's help, some unbreakable gel formations are breakable. This is done via
forcing the creation of two paths of opposite color to be exactly 1 tile from each other:
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Another minimalist example taken from Gigantic Jewel Lost. After Beethro turns his sword, two paths that's 1 tile from each other (normally forbidden) are simultaneously created, which exactly turns everything into gel babies.
So, now you've mastered the theory of gel-cutting! Feel free to brag at your fellow delvers the ability of eyeball gel cutability just as easy as tar and mud.
(I might consider making a little gel cutability quiz level soon. Stay tuned in this thread.)
[Last edited by uncopy2002 at 02-05-2016 12:49 PM]