I've always been fascinated Conway's Game of Life, but one thing that always bugged me was the idea that a cell could live forever. There are alternate rulesets (e.g., Generations) which address the aging issue, but they address it by making a cell's lifespan even longer; much like Shakespeare's coward, cells must die many times before their death. I wanted to implement a ruleset where death from aging, independent of cell crowding, is possible.
So I did. In this alternate cellular automaton, cells may be in three states: young, old, or inactive. At each stage:
- If a young cell has two or three neighbors, it becomes old; otherwise it dies and becomes inactive.
- If an inactive cell has three neighbors, it becomes young; otherwise it stays inactive.
- If an old cell has three neighbors, it becomes young; otherwise it dies and becomes inactive. **
There is some very Life-like behavior in this cellular automaton, but the biggest experimental difference is the lack of stable patterns ("still lifes"). You still have the "block", a 2x2 group of cells that are all alive, but the aging constraint seems to force this automaton to be like the desert...some oases, but a lot of things on the move. These include "oscillators", patterns that repeat after some number of generations, and "spaceships" that recreate a translated copy of themselves periodically.
To get started, I ran some random simulations ("soups") and sifted the leftovers ("ash") for persistent remnants. The vast majority of persistent remnants fall into 4 types:
- block: if uninterfered with, a 2x2 block of live cells will live forever, with cells alternating between young and old. Any combination of young and old amongst the four cells is possible and is stable.
- glider: as in Life, a glider is a set of five cells that translate themselves diagonally. Only certain combinations of young and old cells work, though.
- racer: a racer is an even simpler "spaceship" than the glider and appears much more commonly in this automata (about 4 times more often than gliders). It does not work in Life. As pictured below (yellow = YOUNG, orange = OLD), the racer moves 1 cell to the right every two generations.
- swimmer: rarer than racers or even gliders, but still relatively common (about 4 times less often than gliders), they move at the same speed as racers
Some stuff from Life does translate over to this automaton, for example the idea of a phoenix; since no cell lasts for more than one generation, these will behave the exact same as in Life, so Phoenix 1 is a bounded period 2 "oscillator".
So where's the puzzle here? There is quite a bit more to find. I've found some things in a few days of searching (some automated, some by hand), so I'll pose those as questions for you to focus on. I've set up an answer as a Community Wiki in case multiple people are interested.
Find a period 2 oscillator that is not a phoenix. (There are at least two.)
Find a period 4 oscillator.
Find a period 6 oscillator.
Find a period 16 oscillator.
In Life, the "stator" of an oscillator is the set of cells that are alive in all periods. Show that any period 3 oscillator in Life with an empty stator can be implemented in this cellular automaton with a proper assignment of young and old states to the live cells. Create such an assignment of states to the statorless p3 oscillator to create a period 3 oscillator in this cellular automaton.
Find a period 3 oscillator with a non-empty stator.
Find a "reaction" where two racers collide "head-on" to form a single racer moving in an orthogonal direction.
Find a reaction where two racers collide to form a block.
Find a "tagalong" for a racer, that is, a set of blocks that a racer can pull behind it without affecting its motion.
This cellular automata cannot have a true still-life, since cells age, but we can define a still-life to be a figure where the same set of cells is always live (young or old), and it never grows new cells. Show that the only finite still-lifes in this cellular automaton are the union of non-interfering blocks.
Find an infinite still-life in this cellular automaton that is but two cells high.
** Regarding the transition rule for old cells, my initial thought was to transition all old cells to inactive. But, this made the automaton pretty uninteresting...there are some persistent patterns, but most every random starting point transitions to extinction very quickly. Upon further thought, the current rule does make sense; I think of this as the current occupant dying, but its neighbors reproducing into the vacated space...they just don't wait a whole generation to do it, since life is pretty short.
I have had some angst about whether this is really a puzzle or not, but I enjoyed developing it. It is enough like Life that there shouldn't be a steep learning curve, but it doesn't seem to have been well-studied. At any rate, I hope it will let other puzzlers (and certainly it let me) have the fun of a green field