3c/7 othogonal and 2c/9 diagonal spaceships

[Saka]

I think this is the the time to look for 3c/7 orthogonal spaceships and/or 2c/9 diagonal spaceships, since computers are constantly getting more powerful and faster.
I haven't had time to look for partials put I'll post them when they are found!







P.S. I'm using wls, I don't get how jls is faster.

[velcrorex]

Paul Tooke has searched for 3c/7 and his search will not be easily surpassed.

To quote from here:

To add to the 3c/7 results posted earlier I have since completed friutless gfind searches with o7n3l182uvw and o7n3l196u. That last search took nearly a year to complete, so I won't be banging my head against this particular wall any time soon!

For the benefit of folks unfamiliar with gfinds options these correspond to width 26 even, 27 odd and 27 odd with a central column of dead cells. That last search would also have found any width 13 asymmetric spaceships.

[A for awesome]

Thanks for the serendipitous thread opening, Saka! I just found what I think is a promising 2c/9 diagonal front end:

Code: Select all

x = 11, y = 11, rule = B3/S23
4bo$5b2o$5bobo$7bo$o5bo2bo$b2o4bo2bo$bo2bo5bo$2b2obo2b3o$7b2o$4bo2bo$
5b3o!

I think there is a good chance (I.E. maybe not very likely, but the best so far for a 2c/9 partial) that it can be completed.

[Saka]

Thanks for the serendipitous thread opening, Saka! I just found what I think is a promising 2c/9 diagonal front end:

Code: Select all

x = 11, y = 11, rule = B3/S23
4bo$5b2o$5bobo$7bo$o5bo2bo$b2o4bo2bo$bo2bo5bo$2b2obo2b3o$7b2o$4bo2bo$
5b3o!

I think there is a good chance (I.E. maybe not very likely, but the best so far for a 2c/9 partial) that it can be completed.

Hmm... Looks promising enough to me, maybe I can find something, also, what about 2c/9 orthogonal?
Anyway, two 3c/7 partials from mniemiec's website:

Code: Select all

x = 23, y = 48, rule = B3/S23
7bo7bo$6b3o5b3o$5bo2b2o3b2o2bo$4b2ob9ob2o$5bob2o5b2obo$4b3ob2o3b2ob3o$
4b3o4bo4b3o$7bo7bo$4bo2bo7bo2bo$3bobo11bobo$4b2obo7bob2o$6b2o7b2o$7bo
7bo$6bobo5bobo$5bo3bo3bo3bo$5bo2bo5bo2bo$5bo2b2o3b2o2bo$5b2ob2o3b2ob2o
$6bo9bo$5bo11bo$3bobo11bobo$2b2obo2bo5bo2bob2o$bobo15bobo$o6b2o5b2o6bo
$b2obobo2bo3bo2bobob2o$2bobobobo5bobobobo$4b3o9b3o$3bobo11bobo$2bo3bo
9bo3bo$2bo4bo7bo4bo$8bo5bo$2b3o2bo7bo2b3o$8bo5bo$7b2obobob2o$7b2obobob
2o$5b2obobobobob2o$4b2o11b2o$3b2obo9bob2o$2bo2bobo7bobo2bo$2b3o13b3o$
3b2o2bobo3bobo2b2o$6bo3bobo3bo$10bobo$9bo3bo$5bo3bo3bo3bo$5bo2bo5bo2bo
$2b2o4b2o3b2o4b2o$2o19b2o!

Code: Select all

x = 25, y = 69, rule = B3/S23
8bo7bo$7b3o5b3o$6bo2b2o3b2o2bo$5b2ob9ob2o$6bob2o5b2obo$5b3ob2o3b2ob3o$
5b3o4bo4b3o$8bo7bo$5bo2bo7bo2bo$4bobo11bobo$5b2obo7bob2o$7b2o7b2o$8bo
7bo$7bobo5bobo$6bo3bo3bo3bo$6bo2bo5bo2bo$6bo2b2o3b2o2bo$6b2ob2o3b2ob2o
$7bo9bo$6bo11bo$4bobo11bobo$3b2obo2bo5bo2bob2o$2bobo15bobo$bo6b2o5b2o
6bo$2b2obobo2bo3bo2bobob2o$3bobobobo5bobobobo$5b3o9b3o$4bobo11bobo$3bo
3bo9bo3bo$3bo4bo7bo4bo$9bo5bo$3b3o2bo7bo2b3o$9bo5bo$8b2obobob2o$8b2obo
bob2o$6b2obobobobob2o$5b2o11b2o$5bo13bo$5b2o11b2o$6b4o2bo2b4o$9bobobob
o$7bo3bobo3bo$3b3o4bo3bo4b3o$2bobo2bobo5bobo2bobo$2bob2ob3o5b3ob2obo$
3b2o3bo7bo3b2o$2bobo2bo3b3o3bo2bobo$2b2ob3o2bobobo2b3ob2o$2obo5b2obob
2o5bob2o$2bo2bo2b4ob4o2bo2bo$2o3b2ob3o3b3ob2o3b2o$2bo3bob3o3b3obo3bo$
10bo3bo$4bo15bo$4bo4b2o3b2o4bo$4b2o5bobo5b2o$7b2ob2ob2ob2o$3bo2bob3o3b
3obo2bo$b3o3b2obo3bob2o3b3o$6b4o5b4o$o3b5obo3bob5o3bo$o7b3o3b3o7bo$5b
2o2bobobobo2b2o$o5bo2bo5bo2bo5bo$b2obob3o7b3obob2o$4b2ob2o7b2ob2o$3b2o
15b2o$8b3obob3o$10bobobo!

They look somewhat promising, to be honest.

[Saka]

After wls-ing around for the day, it found (still a partial):

Code: Select all

x = 22, y = 22, rule = B3/S23
14bo6bo$8bob4o2bo4bo$3b2o2b2o2b3o2b2o$2b2obo6bob2o2bo$2bob2obobo2bo$3b
3o6bo3bo$7b3o3b3o$2bobobobo3bobo$b2o3b2obo2bo$4bobobo$bo8b2o$b2o7bo$b
5ob2o$b2o3bo$o2bo2b2o$3bo2bo$b2o2bo$2bo$3bo3$2o!

[Sphenocorona]

After wls-ing around for the day, it found (still a partial):

That isn't a very promising partial, since the partial keeps getting wider going in the opposite direction of its movement.

[HartmutHolzwart]

Coming from about 25 years of experience with wls and lifesrc:

I like that new people with new enthusiasm join the field. It's a pity that we most liely already harvested most of the low hanging fruit.

Concerning 3c7: Current search programs are good at searching either long narrow or short wide ships. Otherwise they suffer from combinatorial explosion. From previous searches it seems as if there is no narrow 3c/7 ship, neither short nor long. So the obvious approach is to try a T-shaped search:

- First start with some promising short wide start partial result. Then try to complete that with a narrow long search. That's what I tried with moderate success for c/6:

Code: Select all

x = 84, y = 54, rule = B3/S23
6$14b3o10b3o17b3o14b3o$9b3obo7b2o7bob3o7b3obo20bob3o$13bo3bo2bo2bo2bo
3bo15bo3bo12bo3bo$13bo5bo4bo5bo15bo4bo10bo4bo$19b2o2b2o27bo8bo$16bo3bo
2bo3bo20bo7b2o7bo$16bobo6bobo21b2ob2ob4ob2ob2o$17b10o26bobo2bobo$19bo
4bo$17bo8bo$16bo10bo$17bo8bo26bo6bo$52bobob2obobo$52b2o2b2o2b2o$56b2o
2$54b2o2b2o$54b2o2b2o$55b4o$55bo2bo$54bo4bo$54bo4bo$54bob2obo$55bo2bo$
55bo2bo2$49bo14bo$49b2o4bo2bo4b2o$49bo5b4o5bo$50b2o4b2o4b2o$51bo10bo$
49bobo10bobo$50b5o4b5o$52bo2b4o2bo$51bo4b2o4bo$55bo2bo$56b2o$53b2ob2ob
2o2$52bo8bo$51b3o6b3o$51bo2bo4bo2bo$50b2o10b2o!

It still will be a lot of try and error without any guarantee of success. But at least it's more promising than pure brute force.

[dvgrn]

Coming from about 25 years of experience with wls and lifesrc:

I like that new people with new enthusiasm join the field. It's a pity that we most li[k]ely already harvested most of the low hanging fruit.

Yes, lifesrc has been a great tool, but it does still succumb to combinatorial explosions without really allowing for any good workarounds.

The only exception I can think of (there are probably others, but I'm not enough of a lifesrc expert to know) is the "Combine Solutions" option in WLS/JLS, which allows for a new filter to be set up partway through a search. Basically it amounts to "Okay, now show me something that isn't like anything I've seen so far."

Anyway, it's very clear from watching lifesrc/WLS/JLS behavior that if you set up a search badly, it will happily retread the exact same ground blindly for untold millions of cycles. For example, if one corner of the search space has solutions A0 through A99, and another corner has independent solutions B0 through B99, then lifesrc will run through each A(n) solution and effectively re-run the exact same B search a hundred times.

I'm pretty sure this is, to say the least, behind the times in terms of the best available algorithms these days.

Something I've been wanting to follow up on for several months now is Donald Knuth's discussion, in his upcoming Volume 4B of the Art of Computer Programming, of SAT solvers.

The .ps.gz file extension in Knuth's posting of his pre-fascicle 6a -- updated less than a week ago, I'm excited to see! -- might be puzzling to some people. Based on a previous request from Bill Gosper to rebroadcast this to a wider audience, I'm venturing to post a temporary copy of the file here, converted to PDF format:

fasc6a.zip

9/15/2015 preview PDF of Art of Computer Programming section on SAT solvers

(3.38 MiB) Downloaded 535 times

The Life-related material starts on page 17, and there's a lot more of it scattered through the text. See the discussion of "mobile paths" in Life on page 19, for example. And in the answer section starting on page 198, among the problems being solved efficiently by SAT solvers are minimal-sized oscillators of various periods, minimal eaters for spaceships and gliders, and solutions to other sample Life-related problems given as exercises earlier in the text.

It's an open question what other Life problems this SAT-solver toolkit could be applied to, but it will certainly be interesting to see how much overlap there is with problems that have been traditionally tackled by lifesrc -- and whether there ends up being any significant improvement in search speed (!)

EDIT: According to Knuth's Exercise #85 and the discoverers' paper, SAT-solver techniques were used to find Garden of Eden #6. I wonder if they tried anything besides 90-degree rotational symmetry... ah, yes, they did -- see the table on page 13 of the Symmetry in Gardens of Eden paper.

[Kazyan]

Since 3c/7 orthogonal seemed to be the main hunt in terms of new spaceship speeds, I started looking into 3c/8 orthogonal a while back, and have picked that up again recently. Here's a derivative of a partial I keep seeing:

Code: Select all

x = 17, y = 16, rule = LifeHistory
8.2A$8.A.3A$4.3C6.A$2.2C7.2A$2.2C10.3A$4.C4.A6.A$2.C6.A.2A.A.A$2C3.C
3.A$2C3.C3.A$2.C6.A.2A.A.A$4.C4.A6.A$2.2C10.3A$2.2C7.2A$4.3C6.A$8.A.
3A$8.2A!

The back parts of that aren't important; mostly it's the surprisingly resilient front end in white. For some configurations, I've seen the front end keep going for 6 or more cycles before the dysfunctions behind it catch up, and none of them were nearly as long as the 3c/7 partials on Mark Niemec's site.

There's also an interesting asymmetric front end that has a nice side spark in some of its versions--they come out looking something like this:

Code: Select all

x = 8, y = 14, rule = B3/S23
6bo$6b2o$bo4bo$2b4o$2b2o2bo$3b3o$3bo2bo$2b2o$bo2b3o$bob4o$2o$b2ob3o$2b
3o$3bo!

[Sokwe]

Yes, lifesrc has been a great tool, but it does still succumb to combinatorial explosions without really allowing for any good workarounds.

The only exception I can think of (there are probably others, but I'm not enough of a lifesrc expert to know) is the "Combine Solutions" option in WLS/JLS, which allows for a new filter to be set up partway through a search. Basically it amounts to "Okay, now show me something that isn't like anything I've seen so far."

The variant of this idea that I've been trying to get a handle on for years, is a hash-enabled WLS/JLS. The key would be successful partitioning of the problem: if there were a way to monitor the search and recognize that a lot of searches were being duplicated because Section A was actually completely independent of Section B, then it would be possible to run a Section A search just once, and a Section B search just once.

I think Nicolay Beluchenko worked in this direction somewhat. He modified WLS to detect solutions soon after they appeared, without having to search all of the remaining cells in the search area. It would then store the completed patterns in a hash table, and would back up whenever any of those objects were detected again. What it didn't do (I think) was store any partial patterns. I think the main utility of Nicolay's modification was to search for c/4 diagonal spaceships without ending up with large fields of gliders. His very large collection of c/4 diagonal spaceships can be found here.

I think people in this topic are ignoring possibly the most glaring omission in our collection of low-period ships: a period-6 knightship.

[HartmutHolzwart]

It seems as if symmetric searches have somewhat more options for a success.

Does someone have a copy of Nicolay's latest wls version? I had one, but that was deleted by my virus checker without further notice. In addition to the early recognition feature it also had support for searches on skewed tori, which is extremely useful e.g. for finding grey ship parts or more general space ship families.

Is there a version of wls that supports symmetric torus searches with odd symmetry? I had one hacked together years ago, but lost everything in a computer crash (together with the drive and energy to do it again). I remember having found some 2c/7 parts back in the early 2000s, but I'm unable to recover them now.

Of course, the p6 knightship is the holy grail of low period space ship searches!

But finding one example is only the start. It's amazing how elusive the weekender has been for years. Except for a small tagalong, there was no progress over decades until finally Ivan Fomichev came up with his wonderful high period rakes. A masterpiece of engineering, all based on the unique 2c/7 space ship available! So how about finding another base ship in 2c/7 or even finding a family of ships?

The situation is only slightly better in the other >p5 areas. In my view, it will turn out fruitful to first test new ideas for search algorithms in areas more accessible.

[Sokwe]

Does someone have a copy of Nicolay's latest wls version?

The latest version that I know of is the one available here. A warning: it often crashes when I save the search state, but it still successfully creates the file.

[A for awesome]

I started looking into 3c/8 orthogonal a while back, and have picked that up again recently. Here's a derivative of a partial I keep seeing:

Code: Select all

x = 17, y = 16, rule = LifeHistory
8.2A$8.A.3A$4.3C6.A$2.2C7.2A$2.2C10.3A$4.C4.A6.A$2.C6.A.2A.A.A$2C3.C
3.A$2C3.C3.A$2.C6.A.2A.A.A$4.C4.A6.A$2.2C10.3A$2.2C7.2A$4.3C6.A$8.A.
3A$8.2A!

I think that front end is quite promising. I am searching it right now and I keep getting long partials like this one:

Code: Select all

x = 25, y = 20, rule = B3/S23
o$4obo$o5b2o7b2o$5b2o5b3obo$3ob2o5bo6b3o$2b2o8b2o7b2o$bo4bob3o10b2o$bo
4bobo6bo4bo$2bo2b2obobob2obo6bo$15bo3bo3b2o$15bo3bo3b2o$2bo2b2obobob2o
bo6bo$bo4bobo6bo4bo$bo4bob3o10b2o$2b2o8b2o7b2o$3ob2o5bo6b3o$5b2o5b3obo
$o5b2o7b2o$4obo$o!

Edit: Search finished, 0 solutions found. I may not have used a large enough area.

[Kazyan]

Just reporting a negative result: gfind with the input of b3/s23/l176/o8/n3/a gives no solutions, which means that there's no 3/8 orthogonal ship that's at most 11 cells wide in all phases. The search took over a week. I'm now running even bilateral symmetry up to 22 cells wide, hoping that the symmetric front end will pull through.

[codeholic]

I think it would make totally sense to make another table on the Game of Life Status page in the wiki to keep track of gfind/afind searches, their results, including negative, and in the comments below who performed it, when, and using which tool. This way we could avoid doing double work.

Another concern I've got is that actually no one has ever proved whether gfind and afind work properly, because neither has even a test suite. If you remember, CatForce found a conduit with 3 catalyst block, 2 of which were transparent, whereas ptbsearch hadn't, though I'm pretty sure Paul Callahan did a search with 2 transparent blocks. I noticed other peculiarities about ptbsearch (e. g. I was not able to find the catalysis that leads to a half of p18 honey farm hassler), so I'm pretty sure ptbsearch has bugs.

Yet another concern is that the code was written for a different architecture, for a different version of a C compiler and that it doesn't work (at least, afind) without code modifications. And there is no canonical version of either, that would be well-maintained by the community. Shall we change that?

I've made repos for gfind and afind. Please contribute (if you can't write code, create issues, for instance, when you can't compile code for a certain platform).

[Sokwe]

Another concern I've got is that actually no one has ever proved whether gfind and afind work properly

In this post, Paul Tooke seems to indicate that if a standard gfind search finds no spaceships, then no spaceships of that width exist. Of course, this doesn't necessarily mean that he's right.

gfind does do some duplicate elimination, which will cause it to miss some ships unless the 'h0' command is given. For example, consider the following spaceships:

Code: Select all

x = 37, y = 24, rule = B3/S23
b2o3b2obo$o3b4obo22bo$b3ob2o2bo21bo$5bo25b3o$9bo$6b3o22b2o$7b2o$6bo24b
o2$5bobo23bobo$2b2o5bo18b2o5bo$5bo3bo21bo3bo$2b2obob2o19b2obob2o$3bobo
23bobo$5b2o24b2o$3bobo23bobo$5bo25bo$2b3o23b3o$o25bo$o7b2o16bo7b2o$o4b
o4bo15bo4bo4bo$bobo2b4o17bobo2b4o$2b2ob2o21b2ob2o$2b2o24b2o!

The following search will only find the first of these ships (given a suitably high 'f' value):

Code: Select all

B3/S23/o3/n1/l72/a

This might be because the ships have the same front ends, but I'm not sure. Adding 'h0' causes the search to find both ships. This issue might not affect a negative result.

[Scorbie]

I've made repos for gfind and afind. Please contribute (if you can't write code, create issues, for instance, when you can't compile code for a certain platform).

Here's a short comment of mine about compiling the original afind source on windows:
viewtopic.php?f=9&t=1759&p=21665#p21665

(e. g. I was not able to find the catalysis that leads to a half of p18 honey farm hassler)

How did you run the search? Shouldn't you also use gencols to search such a thing?

[codeholic]

Here's a short comment of mine about compiling the original afind source on windows:
viewtopic.php?f=9&t=1759&p=21665#p21665

I had a different problem when I compiled it for Mac. That's why I think it's important to have community support for search tools.

(e. g. I was not able to find the catalysis that leads to a half of p18 honey farm hassler)

How did you run the search? Shouldn't you also use gencols to search such a thing?

As I wrote, I tried to find a half of p18 hassler.

[Scorbie]

As I wrote, I tried to find a half of p18 hassler.

If you added two eaters to a honeyfarm and that didn't come out, I think this is a serious bug... Exactly what params did you run the search with?

EDIT: Whoops, it's three...

[chris_c]

As I wrote, I tried to find a half of p18 hassler.

If you added two eaters to a honeyfarm and that didn't come out, I think this is a serious bug... Exactly what params did you run the search with?

I checked it out and it is because of the low survival time of the catalysts when only half of the oscillator is present.

The first eater reacts at generation 5 and survives for a good time on its own. No problems there.

The second eater reacts at generation 7 but without the third eater it would be killed at generation 19. Therefore the value of 15 on this line needs to be 11 or lower to get the first two eater placements.

We would get the three eater placements OK with just this change, but then there is another problem... The third eater reacts at generation 9 but there are just 9 generations (specifically generations 12 to 20) where all of the eaters are completely clean. Therefore the value of 10 on this line needs to be 9 or lower to output those three eater placements.

In other news, I have begun hacking some symmetry options into ptbsearch and I will post when I get something vaguely usable.

EDIT: I should have made it clear that by generation 0 I mean half of the standard form according to LifeWiki:

Code: Select all

#N P18 honey farm hassler
#O Nico Brown
#C A period 18 oscillator found in January 2015.
#C www.conwaylife.com/wiki/index.php?title=P18_honey_farm_hassler
x = 30, y = 18, rule = B3/S23
26bo$3bo20b3o$3b3o17bo$6bo16b2o$5b2o21b2o$28bo$8b2o16bobo$8bo17b2o$20b
3o$7b3o$2b2o17bo$bobo16b2o$bo$2o21b2o$5b2o16bo$6bo17b3o$3b3o20bo$3bo!

[Bullet51]

Off-topic: What about c/5 orthogonal glide symmetric spaceships?
Here are two partials:

Code: Select all

x = 44, y = 25, rule = B3/S23
37bo$6b2o28bobo$5b2o29bobo$6bo32bo$7bo28b2ob2o$8bo26bo4b2o$34bob3obo$
6b2o28bo3bo$4b3o$4b2o28b2o5b2o$34b3o4b2o$5b4o24b2obobob3o$6bobo28b3o$
8bo24b2ob3o$9bo23bo2b2o2b4o$2b3ob2ob2o23bob2ob2ob2o$2obobobo2bo26bobo$
3bobobob2o$2o5bo$4bobob2o$2ob2o2bo2bo$4b2o3bob2o$2ob2ob3o$3bo$3bo5bo!

[codeholic]

I think it would make totally sense to make another table on the Game of Life Status page in the wiki to keep track of gfind/afind searches, their results, including negative, and in the comments below who performed it, when, and using which tool. This way we could avoid doing double work.

I propose the following format: http://www.conwaylife.com/wiki/User:Codeholic/Sandbox

Can you suggest any improvements?

[Sokwe]

I think it would make totally sense to make another table on the Game of Life Status page in the wiki to keep track of gfind/afind searches, their results, including negative, and in the comments below who performed it, when, and using which tool. This way we could avoid doing double work.

I propose the following format: http://www.conwaylife.com/wiki/User:Codeholic/Sandbox

At (3,0)c/6, odd-symmetry, width-21 you should have at least found the following ship:

Code: Select all

x = 21, y = 32, rule = B3/S23
8bo3bo$7bo5bo$8bo3bo$8b2ob2o2$5b2o7b2o$5b5ob5o$6bob2ob2obo$2b2o4b2ob2o
4b2o$3b4o7b4o$3b2ob2o5b2ob2o$6b2o5b2o$bo2bo11bo2bo$o3b2o9b2o3bo$b2obo
11bob2o$4bo11bo$2bo15bo$3bobo9bobo$4bo11bo$4b2o9b2o$2o3bo2b2ob2o2bo3b
2o$4bob3o3b3obo$o3bob2obobob2obo3bo$b2obo4bobo4bob2o$2bobobo2bobo2bobo
bo$3b2obob2ob2obob2o$4bobo7bobo$7bo5bo$5bo9bo$5bo3bobo3bo$6bo2bobo2bo$
7b3ob3o!

Edit: I hope you don't mind that I updated your table to include the results that I known about. I also added a "gutter" column for bilateral symmetry with an empty column down the center of the spaceship. (1,0)c/6 and (2,0)c/6 have question marks because I think they are minimal, but I haven't yet been able to show it. All other results were checked by me personally with gfind except for (1,0)c/7 asymmetric (checked by Josh Ball using WLS) and the entire (3,0)c/7 row (checked by Paul Tooke using gfind). I also double checked the results for periods 1-4 and (2,0)c/5 with WLS.

I also added a 'Known spaceships by height' section, which gives the minimum-possible or maximum-searched height for ships of periods < 8. These were all checked by myself using WLS.

[codeholic]

At (3,0)c/6, odd-symmetry, width-21 you should have at least found the following ship:

That is what I was afraid of. It seems that my afind build has a bug.

Edit: I hope you don't mind that I updated your table to include the results that I known about.

Of course, not!

I also added a "gutter" column for bilateral symmetry with an empty column down the center of the spaceship.

Did you do search of this type with WLS?

(1,0)c/6 and (2,0)c/6 have question marks because I think they are minimal, but I haven't yet been able to show it. All other results were checked by me personally with gfind except for (1,0)c/7 asymmetric (checked by Josh Ball using WLS) and the entire (3,0)c/7 row (checked by Paul Tooke using gfind). I also double checked the results for periods 1-4 and (2,0)c/5 with WLS.

I also added a 'Known spaceships by height' section, which gives the minimum-possible or maximum-searched height for ships of periods < 8. These were all checked by myself using WLS.

Great! Thank you for your contribution! I think after adding references we can move this table onto the Game of Life Status page.

[Sokwe]

I also added a "gutter" column for bilateral symmetry with an empty column down the center of the spaceship.

Did you do search of this type with WLS?

I mostly used gfind to check the table. There is a mode in gfind that allows this type of search (the "w" symmetry). For example, "B3/S23/o5/n2/l70/w" searches for a width-15 2c/5 ship with a "gutter".

Note that all of the "thin" results can be checked in at most a few hours, except for (1,0)c/5 asymmetric, (1,0)c/6 gutter, (1,0)c/7 asymmetric, and the entire (3,0)c/7 row.