# HalfLife: A Cellular Automaton

This puzzle is part of the Monthly Topic Challenge #8: Cellular Automata.

Conway's Game of Life has been creating patterns to beguile amateurs and professionals alike for more than 50 years now, and there are tons of variations. But the ones I tend to like are the ones that are easily explicable in terms of the original model of how "human" populations evolve. I looked at the lack of aging in Middle-Aged Life, and I propose another modification, namely to account for the fact that in the standard 8-cell neighborhood, the corner cells are further away from the center, and so they should have a diminished effect on the population of the center cell.

So my idea was to rescale so that when counting neighbors, corners only count as half. So for example, if a cell has one live orthogonal neighbor and one live diagonal neighbor, it is considered to have 1.5 neighbors, as a cell is not its own neighbor. Similar to Life, in each generation, a live cell with 1.5 or 2 neighbors continues living, while all other live cells die. A dead cell which has 2 neighbors is spawned as a live cell.

I've run a bunch of random searches with this CA, and you can see many of the possible patterns at Catagolue. But the focus of this puzzle is the two spaceships that have been found:

An eater is a stable pattern that destroys spaceships that run into it (from a fixed direction and offset). The 4-long spaceship has a very simple eater:

Your task? Find an eater for the 5-long spaceship. I hope you enjoy!

#### Solver Notes

Regarding the design of the rules, because the total possible numbers of neighbors has changed from Life, it seemed wise to scale the neighbor requirements for transition as well. Being discrete, I could not get the percentages perfect, but I tried to keep these ratios as close to the original game as possible.

For those who use Golly, a rule file for HalfLife is available at https://pastebin.com/tkPLTb2f. For those who wish to use other tools, Hensel notation for this automaton is B2ei3inqy4c/S2-cn3cinqy4c.

Turns out it's pretty simple — all you need to do is take your simple eater for the 4-wide spaceship and…

…double it:

Here's the pattern in RLE format, suitable e.g. for copy-pasting into Golly:

x = 6, y = 6, rule = B2ei3inqy4c/S2-cn3cinqy4c5bo$$2o2b2o$$bo$$bo$$bo3bo$2o2b2o! FWIW, I found this pattern using Logic Life Search. It took me a few minutes to set it up and to write a suitable search pattern, but after that LLS spat out the solution in about 0.2 seconds (literally — it prints the total solver time). As a side effect, LLS also proved that this is the only solution that fits the constraints of my search pattern, which required the pattern to be non-oscillating, to fit in the 4×6 cell box shown in green below, to stabilize in at most 7 generations and to never expand outside the 7×6 cell interior area of the picture below: Ps. It's interesting to note that the eater above can also eat a spaceship arriving one row higher: x = 6, y = 6, rule = B2ei3inqy4c/S2-cn3cinqy4c2o3bo$$bo2b2o$$bo$$bo$$2o3bo$4b2o!

However, whereas a spaceship arriving on the lower track gets eaten very cleanly in just four generations, eating a spaceship arriving on the upper track takes seven generations and emits a bunch of sparks in the process. (In fact it's these sparks that prevent this version of the eater — or rather its mirror image — from being found by the LLS search pattern I linked above, since they extend too far vertically to fit inside the allowed search area.)

Pps. Here's a couple more eaters with the same "front end" column:

x = 9, y = 5, rule = B2ei3inqy4c/S2-cn3cinqy4c2o2b2ob2o$$bo3bo2bo$$bo4b2o$$bo3bo2bo$$2o2b2ob2o!
x = 9, y = 5, rule = B2ei3inqy4c/S2-cn3cinqy4c2o2b2ob2o$$bo3bobo$$bo4bo$$bo3bobo$$2o2b2ob2o!
x = 10, y = 5, rule = B2ei3inqy4c/S2-cn3cinqy4c2o2b2o$$bo3bo$$bo4b2obo$$bo3bo2b2o$$2o2b2o!
x = 10, y = 5, rule = B2ei3inqy4c/S2-cn3cinqy4c2o2b2o2b2o$$bo3bo2bo$$bo4b2o$$bo3bo2bo$$2o2b2o2b2o!

These eaters are all one row narrower (but several columns longer) than the first one above, and unlike the first eater they're fully connected strict still lifes. Plenty of other versions with different back ends can be constructed too. Based on their comment below, I assume one of the two "X-shaped" versions above is what the OP originally had in mind.

Alas, unlike the first eater above, these eaters cannot eat a spaceship arriving one row higher or lower and remain intact — the spaceship does get destroyed, but one of the sparks interacts with the eater and destroys it too.

Ppps. The period-3 oscillating eater I had originally included here doesn't actually work, because I made a mistake. :( Yes, it can technically eat the spaceship, if the ship is placed right in front of it in the right phase, but there's no way to actually get the ship into that position from far away without triggering an unwanted cell birth on the previous step that destroys the eater. My bad, I should've checked.

• Now, THAT is an answer :-) Super details, and great information. For the record, the one I had in mind was bigger...same basic idea, but an X shape. Awesome job...hope you enjoyed! Commented Mar 19, 2023 at 21:33
• @JeremyDover: Thanks! BTW, I think I found your original eater too; I'm just not 100% sure which of the two X-shaped ones I found it was. Commented Mar 20, 2023 at 15:29
• Yes, you did find my original...the 12-cell with the single lit cell in the middle. I actually constructed it by hand. Somewhat embarrassed I didn't see your smallest example...I tried two L-triominos, but with both tails bent the same direction, and built the back structure to stabilize...never occurred to me to swap the orientation of just one :-) Really appreciate your detailed answer! Commented Mar 20, 2023 at 22:55