Conway's Game of Life
Abstract. I describe my findings on the Game of Life. I also ask a few questions on them.
Findings
Figure 1: A Methuselah of fourteen cells, evolving into ten stable units and two gliders after 270 generations.
Questions
Connected diehards
Consider a pattern that start with a fixed number of starting cells and vanishes eventually. What is the maximum number of steps it can take before vanishing?
Note that we need some kind of requirement on connectivity. Without that, using eight cells, you can create a glider and a beacon. Place them far away enough and have them collide into each other. We define a weak form of connectivity called "Conway-connectivity": A pattern is Conway-connected if there exists at least one empty cell that would become alive in the next generation given the current pattern, but this same empty cell would remain dead if any connected component of the current pattern were removed.
Since the Game of Life is turing complete, this question is similar to the busy beaver problem. Here I give some basic results.
| Starting cells | Steps to halt |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 1 |
| 3 | 1 |
| 4 | 5 |
| 5 | ? |