Methuselah Lidka improved

[Mats]

Today I found a way to add another 45 generations to David Bell's version ov Lidka:

Code: Select all

x = 13, y = 12, rule = B3/S23
bo$obo$bo6$8b2ob2o$8bobobo$12bo$11bo!

The lifespan is 29100 generations. Is it the longest lifespan known for a methuselah with 13 or fewer cells?

[calcyman]

Is it the longest lifespan known for a methuselah with 13 or fewer cells?

That is indeed the longest-lived 13-cell methuselah in a 13*13 box. (An 8-cell methuselah in a 7500*7500 bounding box could last longer, for instance.)


Anyway, that's cause for celebration!


As for the longest-lived x-cell methuselahs in a x*x box, I believe the records are:

1 1
2 1
3 2
4 10
5 1105
6 1108
7 5206
8 7468
9 17410
10 17423
11 17465
12 23334
13 29100

This is a weakly-increasing uncomputable function, and thus there is no O() bound based on computable functions. It should be possible to impose a limit of O(BB(f(x))), where BB() denotes Rado's 'Busy Beaver' function, but this limit will grow much faster than the actual function for small values of x.

[dvgrn]

As for the longest-lived x-cell methuselahs in a x*x box, I believe the records are:

1 1
2 1
3 2
4 10
...

o$obo$o! lasts for 11 ticks, I believe; the classic T-tetromino lasts 9. Is there a four-cell traffic-light predecessor that lasts exactly 10 generations?

[Lewis]

Is there a four-cell traffic-light predecessor that lasts exactly 10 generations?

This 4-cell predecessor lasts 10 generations:

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x = 3, y = 3, rule = B3/S23
o$b2o$o!

[Sokwe]

These two predecessors increase the lifespan by one generation while staying within the constraints in Calcyman's post:

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x = 25, y = 5, rule = B3/S23
4bo$2obo16b2ob2o$o3bo15bo3bo$2bobo17bo$3bo19b2o!

These two one-generation predecessors result in a smaller bounding box (14*10):

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x = 28, y = 4, rule = B3/S23
3bo19bo$4bo19bo$obobo17bobo$2bo2b3o12bobo2b3o!

I did some limited searching for 2-generation and 3-generation predecessors but was unsuccessful.

[Mats]

These two one-generation predecessors result in a smaller bounding box (14*10)

Or you can arrange the tub and either one of them in an even smaller bounding box (4*20):

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x = 4, y = 20, rule = B3/S23
bo$obo$bo10$3bo2$2b2o$o$b2o$3bo$3bo$3bo!

[137ben]

The following 3 by 4 pattern that I found a few years ago lasts 1104 generations:

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x = 3, y = 4, rule = B3/S23
b2o$obo$2bo$bo!

So f(4) is substantially higher than Calcyman indicated.

[Mats]

That's a 6-cell pattern so it's a contestant for f(6), not f(4).

[137ben]

Ah, I thought we were just looking at bounding box, now I see that the pattern must have x cells AND be contained in a x by x box...never mind