Middle-Aged Life

Question

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.

PUZZLER NOTES

** 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

Answer 1

Info

A .rule file, for Golly users, provided by AxiomaticSystem

Further features of this cellular automaton beyond the questions posed here, including new spaceships, a puffer and rakes that generate gliders, racers, and swimmers, are in this thread at the ConwayLife forums

Oscillators

period 2

Phoenix 1: (added by Jeremy Dover)

Beacon and Clock: (AxiomaticSystem)

[JeremyDover: These are the two I had. I found clock via random searching, beacon by trying common Life patterns. In a set of 512x512 toroidal runs, beacons occur about one out of every 450 trials, clocks about one out of every 25 trials.]

Clock Variation: (Jeremy Dover, appeared from random soup)

period 3

6P3: (by AxiomaticSystem)

[JeremyDover: In a set of 512x512 toroidal runs, this guy occurs about one out of every 50 trials. This is the p3 I had without a stator. I called it firecracker, but open to alternatives.]

period 4

[JeremyDover: I found the first of these through random searching, but not the second. Nice find!]

8P4.1, 8P4.2: (by ConwayLife user Rocknlol)

period 5 (!)

[Jeremy Dover: This is REALLY interesting, since the corresponding pattern goes extinct after 8 generations in Life. The aging seems to clear out some overcrowding that dooms certain patterns. GREAT catch!]

12P5: (by AxiomaticSystem)

period 6

This pattern oscillates with period 3 in regular Life: the eight interior cells, however, switch between young and old with period 6. (AxiomaticSystem)

[JeremyDover: WOW! This is not the period 6 I had. This one evolves in a really interesting pattern.]

[JeremyDover: These two patterns were the ones I had found.]

6P6.1 and 6P6.2: (Rocknlol)

period 16

Spaceships

Racer: (added by Jeremy Dover)

Swimmer: (added by Jeremy Dover)

Glider: (added by Jeremy Dover)

Fish(es): This spaceship can have its tail lengthened by one or two cells (but not three or more), just like Life's lightweight spaceship. The B-like frontend is also a very familiar sight. (Rocknlol)

There's also a spaceship that moves one cell every four ticks (Rocknlol)

Reactions/Synthesis

Riffing on Rocknlol's 8P4.2, adding a pair of L-triominoes at right evolves into a racer travelling left and a swimmer travelling right.

Adding the symmetric triominoes to the left as well yields two swimmers.

Your interesting findings here!

Answer 2

Some more answers:

Find a reaction where two racers collide to form a block.

Find a reaction where two racers collide head-on to form a single racer moving in an orthogonal direction.

Answer 3

Answers to some questions:

Coloring statorless p3's:

Just initialize them in the young state and let them run! After the initial generation, cells in the envelope of the oscillator will have three kinds, corresponding to the three possible states: dead, born-this-tick (young), and born-last-tick (old). Every cell has to be of one of these types, because otherwise it'd be a stator cell.
(As an aside, period-3 oscillators with stators are technically impossible under the given rules: any stator cell switches between old and young every tick and thus constitutes a period-2 background as long as it is alive. This is why the Life p3 I posted is filed under p6.)

Two-cell still life?

An infinite chain of siamese snakes:

Two-Racer collisions:

As luck would have it, the two answers are closely related: the left pair makes the block, advancing either racer shoots off another racer away from it.