Is

black hole patternpossible in Conway’s Game of Life that eats/clears the infinite universe plane ?Formally, is there a pattern that satifies following requirements?

- The pattern has finite size.
- The universe plane has infinite size.
- Each cell outside the pattern is initially alive or dead randomly. That is it can contain all possible patterns, including other black holes if they are possible.
- For any finite region surrounding the pattern, all cells inside this region will be dead after finite generations and never back to life.
Edit:

To describe this question more intuitively: Imagine there’s a circle that always expanding larger and larger and leaving the region inside it empty, no matter what patterns from the outside it collides.

Originally there was another rule: If two or more black holes collide then they will merge into a bigger black hole. It was deleted before posting because I think it’s already covered by the third and the fourth rule. In addition, it’s hard for me to define and describe the merging formally.

Thanks to @Vepir and @PM2Ring for mentioning related

superstable configuration question.~~I believe this question is an extended version of the~~superstable configuration questionwhich is an open problem. Because if such a black hole exists, it must be a supertable or become a supertable after finite generations.Edit: After a bit further thinking, the metioned

superstable configuration questioninspired a simple but flawed answer about this question. See it below.

**Answer**

^{PS: I am not good at English, any help or suggestions to improve the question or the answer would be greatly appreciated.}

Edit: Unfortunately, this answer has a flaw in that it applies Rule 4 on different generations of the black hole and the surrounding state. Thanks to @IlkkaTörmä for pointing this out. You may read the comments, @Yakk’s clearer but still flawed answer and @IlkkaTörmä’s explanation.

## Original answer

Thanks to @Vepir and @PM2Ring mentions the related *superstable configuration question* which inspired this simple answer.

Assuming that such a black hole exists, after finite generations, it makes a finite(but can be arbitrarily large) still empty region inside it and this region will be empty all the time.

Consider these two patterns:

*A*: a descendant of a black hole with a large still empty region inside, that means no cell in that empty region can be back to life.*B*: the same as A, except that a small*still lifes pattern*(eg. a 2×2 life block) is located at the centre of its large still empty region.

According to *the rule 4*, no cell in *B* surrounding the *still lifes pattern* can be back to life. Any *still lifes pattern* will keeps its state when it’s surrounded by an empty environment.

Then what if *A* collides with *B* since they are expanding larger and larger?

- If the
*still lifes pattern*in*B***can**be cleared in finite generations, that means one of the surrounding cells of the*still lifes pattern*must be back to life, which breaks*the rule 4*. - If the
*still lifes pattern*in*B***can not**be cleared in finite generations, it also breaks*the rule 4*.

So the answer is: such a black hole pattern is impossible.

### My attempt to explain (informally) why this is incorrect.

**Blackhole pattern** (**BHP**) is a pattern that satisfies OP’s rules. Roughly speaking, a BHP can clear any possible **initial** universe state.

**Descendant of black hole pattern** (**D-BHP**) is the pattern that derived from a BHP in any particular initial universe state(eg. an empty universe). **Note that a D-BHP is NOT necessary to be a BHP** because, by Rule 4, an N-generation-old D-BHP is only required to clear its own surrounding environment, which is an exact N-generation-old universe from the particular initial universe. For example, Garden of Eden can only occur in the initial universe so D-BHPs neither need to clear any orphans nor protect themselves from any orphans. A D-BHP aged enough finite generations will have an interior permanently dead region that can’t be populated any longer.

**Pseudo descendant of black hole pattern** (**PD-BHP**) is a pattern that looks like a D-BHP but it is in the initial universe or is a descendant of another PD-BHP. **Note that a PD-BHP is NOT a D-BHP** and can have junks(some alive cells) in its interior region. A PD-BHP doesn’t have to clear anything or protect itself in any universe aged any generations because it’s always in the wrong universe.

Now review the original answer, *A* is a D-BHP while *B* is a PD-BHP(or, more precisely, a descendant of another PD-BHP that looks like *A*). *Case 1* does not break *Rule 4* because *B*(with its still lifes inside) can be cleared and must be cleared by *A* by definition.

But how could that happen? Shouldn’t *A* and *B* have the same behaviour?

Note that when an initial universe with one BHP and one PD-BHP starts evolution, the PD-BHP is always some generations older than D-BHP by form. So they naturally can have different behaviour because of their different forms.

And if we put both *A* and *B* in any specified initial universe, both are PD-BHPs. In particular, with an empty surrounding universe, *A* have the same form as an N-generation-old D-BHP, so by *Rule 4* we may assume it can clear those patterns that can occur in an N-generation-old universe from outside. But *B* is NOT necessary to be such a pattern because of injected lives. *B* may be an orphan pattern that can only occur in initial states. so by *Rule 4*, *A* is not required to clear B, and *case 2* does not break *Rule 4*.

One potential possibility is that all D-BHPs are somehow counting their ages. When two potential D-BHPs meet, they merge as real D-BHPs if they have the same age, otherwise, the younger one eats and clears the other entirely because the older one must be a PD-BHP. I believe this is also impossible, but to prove it is far beyond my ability.

**Attribution***Source : Link , Question Author : VainMan , Answer Author : VainMan*