Unproven conjectures

[Gamedziner]

UPDATE: This one's been proven. Its bounding box is smaller in area than 50*50, and even just barely (31*79 = 2449 = 2500 - 51)

Technically, it doesn't actually fit in a 50*50 box, though.

Either way, the part in italics should now be updated IMO.

[Scorbie]

Conjecture: No glider eater can be constructed with a recovery time of three ticks or less.
(All known glider eaters take at least four ticks to recover to their original state after eating a glider.)

I think this one is somewhat easier to tackle than others; Is it proved?

The definition of "recovery time" of a catalyst isn't very well-defined, I think...
1) Traditional catalyst searchers / gencols etc. define them as "having a common neighbor".
e.g. two blinkers in a row are "reacting" every 2 generations in this sense.
2) Personally I think it's appropriate to define as the following:

Code: Select all

Pattern p, c; // Pattern and Catalyst
reaction_gen(p, c) = min{i | i∈ℕ, p[i] ∪ c[i] != (p∪c)[i]};
recovery_gen(p, c) = min{i | i∈ℕ, i >= first_reaction_gen(p, c), c[i] ⊆ (p∪c)[i]};
recovery_time(p, c) = recovery_gen(p, c) - reaction_gen(p, c);


That makes the eater1's recovery time 3 ticks, which smakes sense in a catalyst-program's perspective, but not so much in Real Life(TM), I guess.

[Hdjensofjfnen]

Is the smallest possible Garden of Eden known?

[dvgrn]

Is the smallest possible Garden of Eden known?

Probably not. There are no Gardens of Eden inside 6x6, as proved by exhaustive search. Extending that search to 6x7 or 7x7 will be quite a computational feat -- roughly 30 times or 8000 times as much work, respectively, as the 6x6 case.

My guess would be that all 7x7 patterns have at least one predecessor, and quite possibly all 8x8 patterns also. There might be a Garden of Eden hiding inside 9x9. Steven Eker has almost made it down to 9x10, but it appears to be very difficult to get rid of those last few cells sticking out into 9x11.

[Hdjensofjfnen]

What is the smallest pattern that has exactly one father?