Miscellaneous topics in Conway's Game of Life -- unfinished projects of all kinds and conditions

15 October 2016

14-, 15-, and 16-bit still life syntheses

A week or so ago, a better recipe was found for the last still life on Mark Niemiec's list of expensive 14-bit syntheses. Now all 14-bit still lifes can be constructed with less than 14 gliders -- less than 1 glider per bit, as the old saying goes.

Catagolue results continue to be very useful in finding new recipes.


Code: Select all
#C 12-glider synthesis for the last 14-bit still life,
#C snake bridge snake / 12.105, which had previously cost at least
#C one glider per bit.
#C Goldtiger997, 6 October 2016, optimized by Mark Niemiec on 7 October.
x = 79, y = 71, rule = LifeHistory
7.A$.A6.A$2.A3.3A$3A2$16.A$14.A.A$15.2A6$36.A$34.A.A$35.2A8$30.3A$32.
A$31.A4$31.3A$33.A11.2D.D$32.A12.D.2D$43.2D$39.2D.D$39.D.2D6$52.A$51.
2A$20.2A5.3A21.A.A$21.2A6.A$20.A7.A22$3.3A$5.A70.2A$4.A4.2A65.A.A$10.
2A64.A$9.A!
#C [[ AUTOFIT AUTOSTART GPS 25 LOOP 150 ]]

UPDATE: The next challenge along these lines was to similarly reduce 15-bit still life costs to below 1 glider per bit. The process started later in the same forum thread, and was completed on November 19, 2016, with the following 14-glider synthesis:

Code: Select all
#C 14-glider synthesis for the last 15-bit still life
#C which had previously cost at least one glider per bit.
#C Extrementhusiast, 19 November 2016
x = 48, y = 38, rule = B3/S23
17bobo$17b2o$18bo$4bobo$5b2o$5bo$18bo$18bobo$18b2o2$obo$b2o39b2o$bo40b
o3b2o$20b3o21bo2bo$20bo22b2obo$21bo6bo16bo$8b2o18bobo14bobo$7bobo18b2o
16b2o$9bo2$5b2o$4bobo$6bo9b2o$10b2o3bobo$11b2o4bo$10bo4$8b3o$7bo2bo$
10bo$6bo3bo$10bo$7bobo$32b3o$32bo$33bo!
#C [[ AUTOFIT AUTOSTART GPS 25 LOOP 150 ]]

UPDATE 2: The next project involved a similar reduction for 16-bit still life recipes. The official project kickoff was on December 16, 2016, when 443 of the 3,286 16-bit still lifes had no synthesis in less than 16 gliders in Chris Cain's database. It concluded successfully on May 24, 2017.

12 June 2016

New spaceship speed: 3c/7

Tim Coe has found a symmetrical spaceship with a new speed, 3c/7 (left, below) after a series of searches that took a total of "one or two months". At 29 cells wide, it is the minimum width odd symmetric spaceship -- an exhaustive width 27 search was run some time ago by Paul Tooke. The author seems to have officially chosen a name of "Spaghetti Monster" for the new 3c/7 spaceship.

Matthias Merzenich has pointed out that two of these spaceships can support a known 3c/7 wave (right, below).


Code: Select all
#C 3c/7 FSM spaceship: Tim Coe, 11 June 2016
#C Period-28 3c/7 wave found by Stephen Silver on Feb. 2, 2000
x = 187, y = 139, rule = B3/S23
10bo7bo65bo7bo$8b2ob2o3b2ob2o61b2ob2o3b2ob2o$8b2ob2o3b2ob2o61b2ob2o3b
2ob2o73bo7bo$11b2o3b2o67b2o3b2o74b2ob2o3b2ob2o$7bo5b3o5bo59bo5b3o5bo
70b2ob2o3b2ob2o$7bo13bo59bo13bo73b2o3b2o$8bo11bo61bo11bo70bo5b3o5bo$9b
2o7b2o63b2o7b2o71bo13bo$6bobobobo3bobobobo57bobobobo3bobobobo69bo11bo$
6bobob2o5b2obobo57bobob2o5b2obobo70b2o7b2o$6bobo11bobo57bobo11bobo67bo
bobobo3bobobobo$164bobob2o5b2obobo$11bo5bo67bo5bo72bobo11bobo$10b2o5b
2o65b2o5b2o$8b2o9b2o61b2o9b2o74bo5bo$8bo3bo3bo3bo61bo3bo3bo3bo73b2o5b
2o$10bo2bobo2bo65bo2bobo2bo73b2o9b2o$10bobo3bobo65bobo3bobo73bo3bo3bo
3bo$9bo9bo63bo9bo74bo2bobo2bo$7bo3bo5bo3bo59bo3bo5bo3bo72bobo3bobo$6b
4o9b4o57b4o9b4o70bo9bo$4b2obo2bo7bo2bob2o53b2obo2bo7bo2bob2o66bo3bo5bo
3bo$4b2o2b3o7b3o2b2o53b2o2b3o7b3o2b2o65b4o9b4o$7bobo2bo3bo2bobo59bobo
2bo3bo2bobo66b2obo2bo7bo2bob2o$5bob3o2bo3bo2b3obo55bob3o2bo3bo2b3obo
64b2o2b3o7b3o2b2o$5bo4bo7bo4bo55bo4bo7bo4bo67bobo2bo3bo2bobo$163bob3o
2bo3bo2b3obo$6bo15bo57bo15bo66bo4bo7bo4bo$6b2obo9bob2o57b2obo9bob2o$5b
o3b2o7b2o3bo55bo3b2o7b2o3bo66bo15bo$164b2obo9bob2o$5b2o4bo5bo4b2o55b2o
4bo5bo4b2o65bo3b2o7b2o3bo2$8b2ob2o3b2ob2o61b2ob2o3b2ob2o68b2o4bo5bo4b
2o$2bo5b2o3bobo3b2o5bo49bo5b2o3bobo3b2o5bo$bob2o5bobo3bobo5b2obo47bob
2o5bobo3bobo5b2obo64b2ob2o3b2ob2o$2o2bo3b2obo5bob2o3bo2b2o45b2o2bo3b2o
bo5bob2o3bo2b2o57bo5b2o3bobo3b2o5bo$2bob2ob6o3b6ob2obo49bob2ob6o3b6ob
2obo58bob2o5bobo3bobo5b2obo$7bo2bobo3bobo2bo59bo2bobo3bobo2bo62b2o2bo
3b2obo5bob2o3bo2b2o$4bobo2bo9bo2bobo53bobo2bo9bo2bobo61bob2ob6o3b6ob2o
bo$2b3o3bo11bo3b3o49b3o3bo11bo3b3o64bo2bobo3bobo2bo$2b3obobo11bobob3o
49b3obobo11bobob3o61bobo2bo9bo2bobo$3b3o17b3o51b3o17b3o60b3o3bo11bo3b
3o$160b3obobo11bobob3o$4bo19bo53bo19bo62b3o17b3o$2b2o21b2o49b2o21b2o$b
obo21bobo47bobo21bobo60bo19bo$b3o21b3o47b3o21b3o58b2o21b2o$159bobo21bo
bo$b2o23b2o47b2o23b2o57b3o21b3o$b3o21b3o47b3o21b3o$4bo4b3o5b3o4bo53bo
4b3o5b3o4bo60b2o23b2o$9bo2bo3bo2bo63bo2bo3bo2bo65b3o21b3o$2bobo4bo9bo
4bobo49bobo4bo9bo4bobo61bo4b3o5b3o4bo$3bo7b2o3b2o7bo51bo7b2o3b2o7bo67b
o2bo3bo2bo$6b5o7b5o57b5o7b5o63bobo4bo9bo4bobo$5b4o11b4o55b4o11b4o63bo
7b2o3b2o7bo$4b2o17b2o53b2o17b2o65b5o7b5o$6bob2o9b2obo57bob2o9b2obo66b
4o11b4o$5bob2obo7bob2obo55bob2obo7bob2obo64b2o17b2o$7b5ob3ob5o59b5ob3o
b5o68bob2o9b2obo$2b3o2b2o2b2o3b2o2b2o2b3o49b3o2b2o2b2o3b2o2b2o2b3o62bo
b2obo7bob2obo$4bo2b2o2b2obob2o2b2o2bo53bo2b2o2b2obob2o2b2o2bo66b5ob3ob
5o$3bo3b2o2b3ob3o2b2o3bo51bo3b2o2b3ob3o2b2o3bo60b3o2b2o2b2o3b2o2b2o2b
3o$3bo5bobob3obobo5bo51bo5bobob3obobo5bo62bo2b2o2b2obob2o2b2o2bo$3bo3b
5o5b5o3bo51bo3b5o5b5o3bo61bo3b2o2b3ob3o2b2o3bo$4bo3b2o9b2o3bo53bo3b2o
9b2o3bo62bo5bobob3obobo5bo$11bo2bo2bo67bo2bo2bo69bo3b5o5b5o3bo$11b2o3b
2o67b2o3b2o70bo3b2o9b2o3bo$13bobo71bobo79bo2bo2bo$169b2o3b2o$8b3o7b3o
61b3o7b3o76bobo$7bo3b2o3b2o3bo59bo3b2o3b2o3bo$8bo11bo61bo11bo71b3o7b3o
$8bo4bobo4bo61bo4bobo4bo70bo3b2o3b2o3bo$7bobobo5bobobo59bobobo5bobobo
70bo11bo$7bo3bo5bo3bo59bo3bo5bo3bo70bo4bobo4bo$7b2o3bo3bo3b2o59b2o3bo
3bo3b2o69bobobo5bobobo$11bo5bo67bo5bo73bo3bo5bo3bo$9bo9bo63bo9bo71b2o
3bo3bo3b2o$9b2o7b2o63b2o7b2o75bo5bo$10bo7bo65bo7bo74bo9bo$167b2o7b2o$
168bo7bo$9b3o5b3o63b3o5b3o$9b2o7b2o63b2o7b2o$8bo3bo3bo3bo61bo3bo3bo3bo
72b3o5b3o$9bo3bobo3bo63bo3bobo3bo73b2o7b2o$13bobo71bobo76bo3bo3bo3bo$
11bo5bo67bo5bo75bo3bobo3bo$171bobo$11b3ob3o67b3ob3o77bo5bo$11b2obob2o
67b2obob2o$9bobo5bobo63bobo5bobo75b3ob3o$8bob2o5b2obo61bob2o5b2obo74b
2obob2o$8bo11bo61bo11bo72bobo5bobo$7bo2b2o5b2o2bo59bo2b2o5b2o2bo70bob
2o5b2obo$8b2o9b2o61b2o9b2o71bo11bo$7bob2o7b2obo59bob2o7b2obo69bo2b2o5b
2o2bo$9b2o7b2o63b2o7b2o72b2o9b2o$6bo15bo57bo15bo68bob2o7b2obo$6b2o3bo
5bo3b2o57b2o3bo5bo3b2o70b2o7b2o$6b3o2bo5bo2b3o57b3o2bo5bo2b3o67bo15bo$
7bo13bo59bo13bo68b2o3bo5bo3b2o$9b2ob2ob2ob2o63b2ob2ob2ob2o70b3o2bo5bo
2b3o$10bob2ob2obo65bob2ob2obo72bo13bo$9b2ob2ob2ob2o63b2ob2ob2ob2o73b2o
b2ob2ob2o$10bo7bo65bo7bo75bob2ob2obo$10bobobobobo65bobobobobo74b2ob2ob
2ob2o$10bo7bo65bo7bo75bo7bo$168bobobobobo$8bo4bobo4bo61bo4bobo4bo7bo7b
o7bo7bo41bo7bo$8bo3bo3bo3bo61bo3bo3bo3bo6b3o5b3o5b3o5b3o$7b2obo7bob2o
59b2obo7bob2o4bo2b2o3b2o2bo3bo2b2o3b2o2bo5bo7bo7bo7bo7bo4bobo4bo$8bob
2o5b2obo61bob2o5b2obo4b2o2b2o3b2o2b2ob2o2b2o3b2o2b2o3b3o5b3o5b3o5b3o6b
o3bo3bo3bo$6bob3o7b3obo57bob3o7b3obo2b2o2b3ob3o2b2ob2o2b3ob3o2b2o2bo2b
2o3b2o2bo3bo2b2o3b2o2bo4b2obo7bob2o$5bo17bo55bo17bob3o9b3ob3o9b3ob2o2b
2o3b2o2b2ob2o2b2o3b2o2b2o4bob2o5b2obo$12bo3bo69bo3bo9b2o9b2o3b2o9b2o2b
2o2b3ob3o2b2ob2o2b3ob3o2b2o2bob3o7b3obo$11bobobobo67bobobobo39b3o9b3ob
3o9b3obo17bo$7b3o3bobo3b3o59b3o3bobo3b3o36b2o9b2o3b2o9b2o9bo3bo$7b4obo
3bob4o59b4obo3bob4o73bobobobo$9b2o7b2o63b2o7b2o71b3o3bobo3b3o$7bob2o7b
2obo59bob2o7b2obo69b4obo3bob4o$6b2ob2o3bo3b2ob2o57b2ob2o3bo3b2ob2o70b
2o7b2o$5b2o2b2o2bobo2b2o2b2o55b2o2b2o2bobo2b2o2b2o67bob2o7b2obo$8b3obo
3bob3o61b3obo3bob3o69b2ob2o3bo3b2ob2o$5b5o2bo3bo2b5o55b5o2bo3bo2b5o65b
2o2b2o2bobo2b2o2b2o$4bo7b2ob2o7bo53bo7b2ob2o7bo67b3obo3bob3o$4bo3b2o3b
obo3b2o3bo53bo3b2o3bobo3b2o3bo64b5o2bo3bo2b5o$4bobo2bo3bobo3bo2bobo53b
obo2bo3bobo3bo2bobo63bo7b2ob2o7bo$11b3ob3o67b3ob3o70bo3b2o3bobo3b2o3bo
$7b3ob3ob3ob3o59b3ob3ob3ob3o66bobo2bo3bobo3bo2bobo$13b3o71b3o79b3ob3o$
13b3o71b3o75b3ob3ob3ob3o$171b3o$11bo5bo67bo5bo79b3o$11bobobobo67bobobo
bo$169bo5bo$169bobobobo!
#C [[ AUTOFIT AUTOSTART GPS 4 ]]

This is the twenty-second spaceship velocity constructed in Conway's Life -- counting each of the four infinite families of spaceships (Gemini, HBK, Demonoid, Caterloopillar) as one velocity each.

10 March 2016

New c/10 "copperhead" spaceship

Reposted with permission from Alexey Nigin's blog:

The day before yesterday (March 6, 2016) ConwayLife.com forums saw a new member named zdr. When we the lifenthusiasts meet a newcomer, we expect to see things like “brand new” 30-cell 700-gen methuselah and then have to explain why it is not notable. However, what zdr showed us made our jaws drop.

It was a 28-cell c/10 orthogonal spaceship:

An animated image of the spaceship

To explain why this is such a groundbreaking discovery, I should first tell you that Life spaceships can be loosely divided into two categories. Engineered ships are the ones that consist of various small components. They often have adjustable speed. However, the population of tens of thousands to millions of cells causes these spaceships to have no practical value.

There is much more incentive in hunting for elementary spaceships, which can be used for complex constructions. They are found using programs such as gfind or WLS. The algorithms behind these programs are beyond the scope of my article, but the important thing is that the search time goes up exponentially as the period of the ship grows. It is most interesting to find spaceships of new speeds, and the number of speeds that low-period ships can have is unfortunately limited:

Orthogonal Diagonal
c/2 Yes Impossible
c/3 Yes Impossible
c/4 Yes Yes
c/5 Yes Yes
2c/5 Yes Impossible
c/6 Yes Yes
c/7 Yes Yes
2c/7 Yes Impossible
3c/7 No Impossible
c/8 No No
3c/8 No Impossible
This table does not include oblique speeds, which causes little inconvenience because no elementary oblique ships are known.

The table above shows that ships exist for most of possible speeds, and it seems obvious that the speeds for which there are no ships have been searched by numerous people with good knowledge of search programs, powerful computers and lots of patience. As for higher periods, even the smallest searches would take years on modern computers. It appears that low-hanging fruit have been harvested clean during the 46 years of Life research… or, more precisely, it appeared so before zdr’s post.

The idea we all missed is that if the ship is really microscopic, it can be found in reasonable time despite its high period. After zdr boldly went where no man has gone before, Josh Ball set up the corresponding search in gfind and refound the spaceship in a little over an hour. zdr said that their program found it in a matter of 19 seconds.

To be frank, similar event did happen before when the aforementioned Josh Ball pulled loafer out of a hat. However, zdr’s spaceship (which is now called Copperhead, as proposed by muzik) is much more awesome for a number of reasons:
  • Loafer is not so mind-bogglingly high-period.
  • Copperhead was much easier to find, so it is more surprising that nobody found it before.
  • Copperhead’s tail is relatively strong and can interact with other objects without breaking down.

The discovery of a new spaceship speed immediately opened a few new areas of research, which are being explored now.

Synthesis

Aidan F. Pierce came up with a Copperhead synthesis only 10 hours after the completion of the spaceship. The synthesis was inefficient, and a few people discovered better ones. The current best synthesis, made by Chris Cain, requires only 22 gliders. Its repeat time is 375 ticks, which means that a gun can start constructing the second spaceship 375 ticks after the first one. There is a 23-glider synthesis with a better repeat time of 373 ticks.

Incremental 22-glider synthesis of the copperhead

The synthesis can be substantially improved if we find this spaceship crawling out of a random soup. Adam P. Goucher has written a wonderful program called apgsearch, which is perfectly suited for this task. While the current version may be too slow to find a soup in reasonable time, highly anticipated version 3.0 can probably do the trick. Once it is found, it will appear here.

Guns

Once the synthesis was complete, building a gun was nothing but corollary-sniping. The first copperhead gun was created by myself, and a video of it is available here. It was put together in a hurry and is therefore extremely inefficient. In particular, skilled gun builders can spot a silly mistake in the Northeast.

gmc_nxtman then made another gun with an almost optimal period of 376 ticks.

Eaters

simeks found two eaters for this ship, the better of which is shown below:

A copperhead eater

It is now time to search for a good copperhead-to-something-useful converter. The only existing one is clumsy and slow.

Sawtooths

Sawtooths often work by sending a flotilla of fast ships towards a slower ship. The more is the difference in speed, the less is the expansion factor of a sawtooth. Since expansion factor is proportional to how boring the sawtooth is, increasing the speed difference is a good thing. Dean Hickerson collided c/2 standard spaceships with c/10 copperhead to get a sawtooth with expansion factor 6:

Hickerson's sawtooth

He then made another sawtooth with expansion factor 10/3.

Puffers and rakes

Suppose a c/10 flotilla is hit by a glider. The glider turns into loads of mess, but all copperheads somehow survive and move on. The mess releases a glider, which flies into strategically placed second flotilla that is identical to the first one. Gliders continue to bounce back and forth between flotillas leaving mess behind them, and a c/10 puffer is complete!

Unfortunately, this cool technique doesn’t work out easily in our case. There are no interesting interactions between a glider and a single copperhead, and it is unclear how one can place two or more copperheads so close to each other that a glider interacts with all of them. Assuming we figure it out, we can try to make a rake by perturbing the mess with copperheads so that it evolves into gliders, but that seems even less likely.

However, all this hand-waving can be turned it real puffers if we find…

Tagalongs

Tagalongs are small things that are attached to the back of a spaceship and move with it. Here is an example tagalong, called the Schick engine:

Schick engine pulled by two LWSSs

Finding a tagalong for the copperhead (or two copperheads) will be really nice. We can also try searching for pushalongs, but they are generally rarer.

Other patterns

There are a few other areas of Life exploration where the copperhead can be useful. For example, universal constructors often need to create an elbow still life very far away. It can be done by producing a copperhead, waiting for some time, and then shooting the copperhead down with a LWSS. At the moment I do not see why the copperhead can be better than the loafer in this aspect, but who knows?