## Rules with small adjustable spaceships

For discussion of other cellular automata.

### Rules with small adjustable spaceships

Last edited by muzik on January 5th, 2018, 1:47 pm, edited 2 times in total.
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3300
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Rules with small adjustable spaceships

muzik,

Your post would be more useful if each of the listed rules were pointers to the example patterns. Readers should never be required to search the Forum for referenced material.

Brian Prentice
bprentice

Posts: 548
Joined: September 10th, 2009, 6:20 pm
Location: Coos Bay, Oregon

### Re: Rules with small adjustable spaceships

that's that done
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3300
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Rules with small adjustable spaceships

http://www.conwaylife.com/forums/viewtopic.php?f=11&t=803&start=850#p51682
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3300
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Rules with small adjustable spaceships

All speeds of the form 2c/(4n+2), n>3:
`x = 98, y = 24, rule = 1e2-en3aejnr4ir5y7c/2c3a4ak6ack7e/3.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A\$A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A\$ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA\$2.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$7.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$12.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$17.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$22.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$27.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$32.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$37.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$42.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$47.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$52.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$57.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$62.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$67.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$72.A2.A.A2.A.A2.A.A2.A.A2.A.A\$77.A2.A.A2.A.A2.A.A2.A.A\$82.A2.A.A2.A.A2.A.A\$87.A2.A.A2.A.A\$92.A2.A.A\$97.A![[ THEME Blues ]]`

Generations feels like it has a lot more potential for adjustable-speed technology, with the whole one-way-signal mechanic.
Last edited by muzik on February 10th, 2019, 5:26 am, edited 1 time in total.
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3300
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Rules with small adjustable spaceships

Found another one by myself. 2c/(4n+2), n>1:
`x = 39, y = 4, rule = B2c3acjr4y5ci/S01e2ac3en4n5ij6a3b2o7b2o8b2o9b2o\$3b2o7bo9bo10bo\$3b3o6b2obo6b2o2bo6b2o3bo\$o3bo4bo9bo10bo!`

Can we get more people looking for these?
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3300
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Rules with small adjustable spaceships

If you're the person that uploaded to Sakagolue illegally, please PM me.
`x = 17, y = 10, rule = B3/S23b2ob2obo5b2o\$11b4obo\$2bob3o2bo2b3o\$bo3b2o4b2o\$o2bo2bob2o3b4o\$bob2obo5bo2b2o\$2b2o4bobo2b3o\$bo3b5ob2obobo\$2bo5bob2o\$4bob2o2bobobo!`

(Check gen 2)

Saka

Posts: 2837
Joined: June 19th, 2015, 8:50 pm
Location: In the kingdom of Sultan Hamengkubuwono X

### Re: Rules with small adjustable spaceships

`x = 10, y = 28, rule = 1e23-ai4ain5jy/2c3-in6ack7/35.4A\$5.A.B2A\$7.2A3\$4.5A\$4.A2.B2A\$6.3A3\$3.6A\$3.A3.B2A\$5.4A3\$2.7A\$2.A4.B2A\$4.5A3\$.8A\$.A5.B2A\$3.6A3\$9A\$A6.B2A\$2.7A![[ THEME Blues ]]`
Last edited by muzik on February 10th, 2019, 5:26 am, edited 1 time in total.
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3300
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Rules with small adjustable spaceships

`x = 10, y = 28, rule = 1e23-ai4ain5jy/2c3-in6ack7/35.4A\$5.A.B2A\$7.2A3\$4.5A\$4.A2.B2A\$6.3A3\$3.6A\$3.A3.B2A\$5.4A3\$2.7A\$2.A4.B2A\$4.5A3\$.8A\$.A5.B2A\$3.6A3\$9A\$A6.B2A\$2.7A!`

WE MUST APGSEARCH

Majestas32

Posts: 524
Joined: November 20th, 2017, 12:22 pm
Location: 'Merica

### Re: Rules with small adjustable spaceships

Majestas32 wrote:WE MUST APGSEARCH

Sorry but apgluxe doesnt support nontot generations yet.

Also, muzik, you haven't addrd the 3 wick-rules I found. I'll link them later when I'm on my computer and not on this tiny screen.
If you're the person that uploaded to Sakagolue illegally, please PM me.
`x = 17, y = 10, rule = B3/S23b2ob2obo5b2o\$11b4obo\$2bob3o2bo2b3o\$bo3b2o4b2o\$o2bo2bob2o3b4o\$bob2obo5bo2b2o\$2b2o4bobo2b3o\$bo3b5ob2obobo\$2bo5bob2o\$4bob2o2bobobo!`

(Check gen 2)

Saka

Posts: 2837
Joined: June 19th, 2015, 8:50 pm
Location: In the kingdom of Sultan Hamengkubuwono X

### Re: Rules with small adjustable spaceships

2c/(2n+1) p(2n+1) spaceships (n>=4):
`x = 14, y = 39, rule = B2c3aejr4ijr5i7e/S12-en3einr4eijt5ry6e7e86bo\$2bo4bo\$o2b6o\$b2o4bo\$6bo3\$7bo\$2bo6bo\$o2b7o\$b2o6bo\$7bo4\$2bo7bo\$o2b8o\$b2o7bo5\$2bo8bo\$o2b9o\$b2o8bo5\$2bo9bo\$o2b10o\$b2o9bo5\$2bo10bo\$o2b11o\$b2o10bo!`
Iteration of sigma(n)+tau(n)-n [sigma(n)+tau(n)-n : OEIS A163163] (e.g. 16,20,28,34,24,44,46,30,50,49,11,3,3, ...) :
965808 is period 336 (max = 207085118608).
AbhpzTa

Posts: 460
Joined: April 13th, 2016, 9:40 am
Location: Ishikawa Prefecture, Japan

### Re: Rules with small adjustable spaceships

So that's all the 2c/n speeds proven possible then?
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
muzik

Posts: 3300
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

### Re: Rules with small adjustable spaceships

All the diagonal adjustable wickships, because you requested it.
22+66n, n >= 0

62+6n, n >= 0

29+6n, n >= 0
If you're the person that uploaded to Sakagolue illegally, please PM me.
`x = 17, y = 10, rule = B3/S23b2ob2obo5b2o\$11b4obo\$2bob3o2bo2b3o\$bo3b2o4b2o\$o2bo2bob2o3b4o\$bob2obo5bo2b2o\$2b2o4bobo2b3o\$bo3b5ob2obobo\$2bo5bob2o\$4bob2o2bobobo!`

(Check gen 2)

Saka

Posts: 2837
Joined: June 19th, 2015, 8:50 pm
Location: In the kingdom of Sultan Hamengkubuwono X

### Re: Rules with small adjustable spaceships

c/(2n+1) diagonal for n>=3
`x = 41, y = 35, rule = B3aeijq4awyz5jr/S1c2in3ackqr4q5a6e30b3o2bo\$34bo\$33bobo\$36bo\$37bo\$38bo\$39bo\$40bo3\$20b3o2bo\$24bo\$23bobo\$26bo\$27bo\$28bo\$29bo4\$10b3o2bo\$14bo\$13bobo\$16bo\$17bo\$18bo5\$3o2bo\$4bo\$3bobo\$6bo\$7bo!`

A variation on the above (same allowed speeds)
`x = 39, y = 37, rule = B3aeijk4az5j/S1c2ein3ack4q5a6ae30bo\$30bo\$30bo2bo\$32bo\$31bobo\$34bo\$35bo\$36bo\$37bo\$38bo\$20bo\$20bo\$20bo2bo\$22bo\$21bobo\$24bo\$25bo\$26bo\$27bo2\$10bo\$10bo\$10bo2bo\$12bo\$11bobo\$14bo\$15bo\$16bo3\$o\$o\$o2bo\$2bo\$bobo\$4bo\$5bo!`

Another variation but with a stable tail-end (same allowed speeds)
`x = 40, y = 35, rule = B3aeijk4a/S1c2en3ackqr4q5aq6e31bo2bo\$30bo2bo\$32bobo\$35bo\$36bo\$37bo\$38bo\$39bo3\$21bo2bo\$20bo2bo\$22bobo\$25bo\$26bo\$27bo\$28bo4\$11bo2bo\$10bo2bo\$12bobo\$15bo\$16bo\$17bo5\$bo2bo\$o2bo\$2bobo\$5bo\$6bo!`
wildmyron

Posts: 1073
Joined: August 9th, 2013, 12:45 am

### Re: Rules with small adjustable spaceships

(1,0)c/(3n+2) p(3n+2) spaceships (n>=3):
`x = 13, y = 54, rule = B2c3aj4k5cnr/S2-n3-ajky4j5jnbo\$ob5o\$bo5bo\$6b2o7\$bo\$ob6o\$bo6bo\$7b2o7\$bo\$ob7o\$bo7bo\$8b2o7\$bo\$ob8o\$bo8bo\$9b2o7\$bo\$ob9o\$bo9bo\$10b2o7\$bo\$ob10o\$bo10bo\$11b2o!`

EDIT:
(2,1)c/3n p3n spaceships (n>=5):
`x = 12, y = 38, rule = B2cin3aej4kr5cnr/S1c2-n3enr4jr5anq6i7co8b2o\$9o2bo\$9b2o5\$o7b2o\$8o2bo\$8b2o5\$o6b2o\$7o2bo\$7b2o5\$o5b2o\$6o2bo\$6b2o5\$o4b2o\$5o2bo\$5b2o5\$o3b2o\$4o2bo\$4b2o!`
Iteration of sigma(n)+tau(n)-n [sigma(n)+tau(n)-n : OEIS A163163] (e.g. 16,20,28,34,24,44,46,30,50,49,11,3,3, ...) :
965808 is period 336 (max = 207085118608).
AbhpzTa

Posts: 460
Joined: April 13th, 2016, 9:40 am
Location: Ishikawa Prefecture, Japan

### Re: Rules with small adjustable spaceships

AbhpzTa wrote:(2,1)c/3n p3n spaceships (n>=5):
`x = 12, y = 38, rule = B2cin3aej4kr5cnr/S1c2-n3enr4jr5anq6i7co8b2o\$9o2bo\$9b2o5\$o7b2o\$8o2bo\$8b2o5\$o6b2o\$7o2bo\$7b2o5\$o5b2o\$6o2bo\$6b2o5\$o4b2o\$5o2bo\$5b2o5\$o3b2o\$4o2bo\$4b2o!`

Nice! I somehow didn't see this until now.
C/32+4n for n>0:
`x = 23, y = 23, rule = B2cik3aik4cir5ekqr6i7e8/S02aen3eij4cinwz5aeijr6ei7e2bobo13bobo8\$2bobo13bobo\$2b3o13b3o3\$3bo\$b5o13bo\$bo3bo11b5o\$2obob2o10bo3bo\$bo3bo10b2obob2o\$b5o11bo3bo\$3bo13b5o\$19bo2\$3bo\$19bo!`

This means that all speeds C/40+4n for n>0 have max population of 29:
`x = 21, y = 25, rule = B2cik3aik4cir5ekqr6i7e8/S02aen3eij4cinwz5aeijr6ei7e2bobo11bobo7\$2bobo11bobo\$2b3o11b3o6\$3bo\$b5o11bo\$bo3bo9b5o\$2obob2o8bo3bo\$bo3bo8b2obob2o\$b5o9bo3bo\$3bo11b5o\$17bo2\$3bo\$17bo!`
Last edited by AforAmpere on February 17th, 2018, 9:26 am, edited 2 times in total.
Things to work on:
- Find a (7,1)c/8 ship in a Non-totalistic rule (someone please search the rules)
- Find a C/10 in JustFriends
- Find a C/10 in Day and Night
AforAmpere

Posts: 986
Joined: July 1st, 2016, 3:58 pm

### Re: Rules with small adjustable spaceships

AbhpzTa wrote:(2,1)c/3n p3n spaceships (n>=5):

What a great rule! It has this small c/8 spaceship:
`x = 2, y = 3, rule = B2cin3aej4kr5cnr/S1c2-n3enr4jr5anq6i7co\$2o\$o!`

A c/2 spaceship too:
`x = 4, y = 8, rule = B2cin3aej4kr5cnr/S1c2-n3enr4jr5anq6i7c2bo\$ob2o\$b2o3\$b2o\$ob2o\$2bo!`

Last but not least, the wickstrecher:
`x = 9, y = 4, rule = B2cin3aej4kr5cnr/S1c2-n3enr4jr5anq6i7c7bo\$6ob2o\$o5b2o\$6bo!`

I will apgsearch soon.
she/they // Please stop using my full name. Refer to me as dani.

"I'm always on duty, even when I'm off duty." -Cody Kolodziejzyk, Ph.D.

danny

Posts: 937
Joined: October 27th, 2017, 3:43 pm
Location: New Jersey, USA

### Re: Rules with small adjustable spaceships

Another 2c/4n, this time, for n>=3:
`x = 38, y = 4, rule = B2e3eijr4aijknrw6akn7e/S02aci3anr4ijrty5ey6e7c5bo14bo16bo\$o2bo10bo2bo12bo2bo\$2bo13bo15bo\$3bo13bo15bo!`

EDIT, 2c/76+8n for n>=0:
`x = 22, y = 25, rule = B2ci3ai4cir6kn7c8/S02ae3aeij4ciz5akqr6ai7e3bo14bo\$b5o10b5o\$bo3bo10bo3bo\$2obob2o8b2obob2o\$bo3bo10bo3bo\$b5o10b5o\$3bo14bo3\$2b3o12b3o\$2bobo12bobo4\$3bo\$b5o12bo\$bo3bo10b5o\$2obob2o9bo3bo\$bo3bo9b2obob2o\$b5o10bo3bo\$3bo12b5o\$18bo2\$3bo\$18bo!`
Things to work on:
- Find a (7,1)c/8 ship in a Non-totalistic rule (someone please search the rules)
- Find a C/10 in JustFriends
- Find a C/10 in Day and Night
AforAmpere

Posts: 986
Joined: July 1st, 2016, 3:58 pm

### Re: Rules with small adjustable spaceships

danny wrote:What a great rule! It has this small c/8 spaceship:
`x = 2, y = 3, rule = B2cin3aej4kr5cnr/S1c2-n3enr4jr5anq6i7co\$2o\$o!`

*c/4
`x = 81, y = 96, rule = LifeHistory58.2A\$58.2A3\$59.2A17.2A\$59.2A17.2A3\$79.2A\$79.2A2\$57.A\$56.A\$56.3A4\$27.A\$27.A.A\$27.2A21\$3.2A\$3.2A2.2A\$7.2A18\$7.2A\$7.2A2.2A\$11.2A11\$2A\$2A2.2A\$4.2A18\$4.2A\$4.2A2.2A\$8.2A!`
Gamedziner

Posts: 678
Joined: May 30th, 2016, 8:47 pm
Location: Milky Way Galaxy: Planet Earth

### Re: Rules with small adjustable spaceships

AforAmpere wrote:This means that all speeds C/40+4n for n>0 have max population of 29:

This limit can be reduced to 5 for all speeds c/n (given 3 cell ships for most speeds c/n where n<100)

(1,0)c/4n, p(4n), n>=4:
`x = 9, y = 24, rule = B2ein3inr4ikw5jky6cei8/S012-an3ijkqy4acjkq5jkq6in72bo\$5bo\$6bobo\$5bo7\$bo\$5bo\$6bobo\$5bo7\$o\$5bo\$6bobo\$5bo!`

(1,0)c/(4n+1), p(4n+1), n>=2:
`x = 8, y = 34, rule = B2e3ijknq4aeijqr5ekqr6e78/S01c2cen3aenqr4anqrtw5enqr6aen7c3bo\$b4o\$2b2obobo\$3b2o7\$2bo\$4bo\$5bobo\$4bo7\$bo\$4bo\$5bobo\$4bo7\$o\$4bo\$5bobo\$4bo!`

(1,0)c/(4n+2), p(4n+2), n>=4:
`x = 8, y = 24, rule = B2en3ainq4aceiknr5-ceij6aik78/S01c2-ak3ijkry4ajnrty5ijq6ein7e82bo\$4bo\$5bobo\$4bo7\$bo\$4bo\$5bobo\$4bo7\$o\$4bo\$5bobo\$4bo!`

(1,0)c/(4n+3), p(4n+3), n>=3:
`x = 8, y = 24, rule = B2en3ainq4-ackt5acekn6-ac7c8/S01c2cen3-acek4cjt5-aeqr6-ce7e82bo\$4bo\$5bobo\$4bo7\$bo\$4bo\$5bobo\$4bo7\$o\$4bo\$5bobo\$4bo!`

There are many more similar ships, even with the same c/2, I've just chosen a set with minimum population of 5 covering all speeds c/n (for n>15).

----

(1,0)c/2n, p(2n), n>=7 (moon bouncer):
`x = 12, y = 35, rule = B2aen3aen4inqtwyz5-ekny6aen7e8/S1e2n3ejry4knqrtw5cnq6787bo\$8bo\$4bo3bo\$3bobobo2bo\$11bo6\$7bo\$8bo\$2bobo3bo\$3bo3bo2bo\$11bo6\$7bo\$8bo\$2bo5bo\$bobo3bo2bo\$11bo6\$7bo\$8bo\$obo5bo\$bo5bo2bo\$11bo!`

I tried to find stable reflectors but didn't have any luck completing a ship with that restriction, so I tried p2 instead.

Edit: Actually, just found one with stable reflectors and smaller minimum population:

(1,0)c/2n, p(2n), n>=6:
`x = 11, y = 36, rule = B2-ci3nqr4aceqry5ceqy6cn/S01e2i3-aein4jkqr5k6ek2bobo\$3b3o\$3b2obobobo\$3b2obo\$3b3o\$2bobo5\$2bo\$5bo\$6bobobo\$6bo\$5bo\$2bo5\$bo\$5bo\$6bobobo\$6bo\$5bo\$bo5\$o\$5bo\$6bobobo\$6bo\$5bo\$o!`
wildmyron

Posts: 1073
Joined: August 9th, 2013, 12:45 am

### Re: Rules with small adjustable spaceships

wildmyron wrote:
AforAmpere wrote:This means that all speeds C/40+4n for n>0 have max population of 29:

This limit can be reduced to 5 for all speeds c/n (given 3 cell ships for most speeds c/n where n<100)

This set of adjustable speed spaceships reduces the upper limit on population for all speeds c/n and also 2c/n to 4 cells (given 3 cell examples for all c/n and 2c/n with n < 25 [except 2c/3 which is 4 cells]). Several of them convert the ship to a different 3 cell ship traveling in the backwards direction, specifically, the c/n ships - there are similar ships at 2c/n not shown here.

(1,0)c/4n, p(4n), n>=5:
`x = 10, y = 33, rule = B2ek3aij4eir5a6ac/S01c2-an3eiqy4aenrt5aiq6ik7e3bo3bo\$9bo\$7bo8\$2bo4bo\$9bo\$7bo8\$bo5bo\$9bo\$7bo8\$o6bo\$9bo\$7bo!`

(1,0)c/(4n+1), p(4n+1), n>=5:
`x = 9, y = 33, rule = B2ek3aiq4ai5y6k/S01c2-an3acekr4ey5ijkqy6cik3bo2bo\$8bo\$6bo8\$2bo3bo\$8bo\$6bo8\$bo4bo\$8bo\$6bo8\$o5bo\$8bo\$6bo!`

(1,0)c/(4n+2), p(4n+2), n>=4:
`x = 10, y = 33, rule = B2ek3ai4eir6ac/S01c2-an3eiqy4anrt5aeiq6k3bo3bo\$9bo\$7bo8\$2bo4bo\$9bo\$7bo8\$bo5bo\$9bo\$7bo8\$o6bo\$9bo\$7bo!`

(1,0)c/(4n+3), p(4n+3), n>=5:
`x = 10, y = 33, rule = B2ek3aik4einrwyz5a6c/S01c2-an3eqy4ajkqrt5-cknr6k83bo3bo\$9bo\$7bo8\$2bo4bo\$9bo\$7bo8\$bo5bo\$9bo\$7bo8\$o6bo\$9bo\$7bo!`

(2,0)c/4n, p(4n), n>=4 (With n=4 the forward direction ship doesn't actually fully form):
`x = 9, y = 43, rule = B2ek3ij4ry5e/S012cei3ijqr4ar5aeky6i4bobo\$8bo\$6bo8\$3bo2bo\$8bo\$6bo8\$2bo3bo\$8bo\$6bo8\$bo4bo\$8bo\$6bo8\$o5bo\$8bo\$6bo!`

(2,0)c/(4n+1), p(4n+1), n>=5:
`x = 9, y = 33, rule = B2ek3ij4rwz5e7e/S012-an3ijqr4aer5ai6c7e3bo2bo\$8bo\$6bo8\$2bo3bo\$8bo\$6bo8\$bo4bo\$8bo\$6bo8\$o5bo\$8bo\$6bo!`

(2,0)c/(4n+2), p(4n+2), n>=5:
`x = 9, y = 33, rule = B2e3ijqry4nr5r6k7e/S012-an3eir4a5q6c3bo2bo\$8bo\$6bo8\$2bo3bo\$8bo\$6bo8\$bo4bo\$8bo\$6bo8\$o5bo\$8bo\$6bo!`

(2,0)c/(4n+3), p(4n+3), n>=4:
`x = 9, y = 33, rule = B2e3ijry4qr5ek7e/S012cei3eij4aj5e7e3bo2bo\$8bo\$6bo8\$2bo3bo\$8bo\$6bo8\$bo4bo\$8bo\$6bo8\$o5bo\$8bo\$6bo!`
wildmyron

Posts: 1073
Joined: August 9th, 2013, 12:45 am

### Re: Rules with small adjustable spaceships

Here's something a bit different: a family of c/2 ships with adjustable period and fixed minimum population. They consist of a c/2 diagonal ship bouncing between two c/2 orthogonal ships

(1,0)c/2, p(4n), n>3, 20 cells (except p12 which has min pop 21 in phase 1)
`x = 50, y = 17, rule = B2aei3cekq4jknrt5ejkry6-kn8/S1c2-ae3cinqr4aentw5-ackr6ace7c2bo14bo14bo14bo\$2o2bo10b2o2bo10b2o2bo10b2o2bo\$2bobo12bobo12bobo12bobo\$o3bo10bo3bo10bo3bo10bo3bo2\$obo12bobo12bobo12bobo\$3bo14bo14bo14bo\$46bo\$3bo14bo14bo14bo\$43bob2o\$45bo3bo\$30bo3bo12bobo\$15bo3bo12bobo10b2o2bo\$o3bo12bobo10b2o2bo12bo\$2bobo10b2o2bo12bo\$2o2bo12bo\$2bo!`

Finding similar constructions for other periods is more difficult because two different reactions are required. I haven't tried searching for such ships yet, though I can say there are a lot of similar ships to the one above (this is just the smallest I could find). Here's a reflection which restores the diagonal ship after an odd number of generations (again, just one of many):
`x = 6, y = 11, rule = B2aei3cekn4ciknqt5cjnqr6in78/S12ci3ceiqr4-acer5cjq6i78bo2bo\$3b3o\$obo2bo\$3b3o\$bo2bo3\$bobo\$4bo2\$4bo!`

It's probably feasible to search for compatible pairs of reflections which can bounce the ship with period of 4n+2 (the other reflection has to delay the diagonal ship by 2 gen overall so that it matches speed with the two escorts). I haven't tried this search yet.

Does anybody know what the smallest such ships in Life are? And are there examples with period not divisible by 4?

P.S. Thanks again to Macbi for lls, which is extremely useful in this kind of endeavour.
wildmyron

Posts: 1073
Joined: August 9th, 2013, 12:45 am

### Re: Rules with small adjustable spaceships

muzik wrote:All speeds of the form 2c/(4n+2), n>3:
`x = 98, y = 24, rule = 1e2-en3aejnr4ir5y7c/2c3a4ak6ack7e/3.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A4.A\$A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A2.3A\$ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA2.ABA\$2.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$7.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$12.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$17.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$22.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$27.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$32.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$37.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$42.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$47.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$52.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$57.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$62.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$67.A2.A.A2.A.A2.A.A2.A.A2.A.A2.A.A\$72.A2.A.A2.A.A2.A.A2.A.A2.A.A\$77.A2.A.A2.A.A2.A.A2.A.A\$82.A2.A.A2.A.A2.A.A\$87.A2.A.A2.A.A\$92.A2.A.A\$97.A!`

Generations feels like it has a lot more potential for adjustable-speed technology, with the whole one-way-signal mechanic.

I think it's actually 2c/(4n+6), n≥3; the fastest is actually a 2c/18.

Here's a slightly different set of ships that do travel at 2c/(4n+2), n≥3:
`x = 17, y = 63, rule = 12-en3ejnr4r5jry/2c3a4a6ack7e/312.4A\$14.B2A\$13.3A3\$11.5A\$14.B2A\$12.4A3\$10.6A\$14.B2A\$11.5A3\$9.7A\$14.B2A\$10.6A3\$8.8A\$14.B2A\$9.7A3\$7.9A\$14.B2A\$8.8A3\$6.10A\$14.B2A\$7.9A3\$5.11A\$14.B2A\$6.10A3\$4.12A\$14.B2A\$5.11A3\$3.13A\$14.B2A\$4.12A3\$2.14A\$14.B2A\$3.13A3\$.15A\$14.B2A\$2.14A3\$16A\$14.B2A\$.15A!`
`x = 4, y = 3, rule = B3-q4z5y/S234k5j2b2o\$b2o\$2o!`

LaundryPizza03 at Wikipedia

LaundryPizza03

Posts: 266
Joined: December 15th, 2017, 12:05 am
Location: Unidentified location "https://en.wikipedia.org/wiki/Texas"

### Re: Rules with small adjustable spaceships

There is a chance that adjustable ships may exist in B2a rules:
`x = 67, y = 4, rule = B2ae3q/S0o\$4bo9bo5bobobo3bobo3bo3bo3bo5bo3bobobo3bo3bobo\$4bo9bo5bobobo3bobo3bo3bo3bo5bo3bobobo3bo3bobo\$o!`

This reaction moves at C/7, with replicators and a backend. If someone can find something that can be pushed by the replicators, there might be a new class of adjustable ships.
Things to work on:
- Find a (7,1)c/8 ship in a Non-totalistic rule (someone please search the rules)
- Find a C/10 in JustFriends
- Find a C/10 in Day and Night
AforAmpere

Posts: 986
Joined: July 1st, 2016, 3:58 pm

### Re: Rules with small adjustable spaceships

AforAmpere wrote:There is a chance that adjustable ships may exist in B2a rules:
`x = 67, y = 4, rule = B2ae3q/S0o\$4bo9bo5bobobo3bobo3bo3bo3bo5bo3bobobo3bo3bobo\$4bo9bo5bobobo3bobo3bo3bo3bo5bo3bobobo3bo3bobo\$o!`

This reaction moves at C/7, with replicators and a backend. If someone can find something that can be pushed by the replicators, there might be a new class of adjustable ships.

(m/g,0)c/(n/g) , period n [g=gcd(m,n) , (m=1 AND n=5) OR (0<5m<n AND m==n(mod 2))]

m=1 and n={5,7,9,11} (reaction={(5),(7),(9),(11)}):
`x = 114, y = 36, rule = B2a3jkq/S01c3e103bo\$102bo10bo\$106bo5bo\$106bo5bo\$102bo10bo\$103bo5\$79bo\$78bo34bo\$85bobo5bo3bo7bobobo\$85bobo5bo3bo7bobobo\$78bo34bo\$79bo5\$bo\$o112bo\$3bobo5bo7bo3bobo7bobo3bobobo7bobo9bobo5bo3bobo3bo3bobo11bo5bobo\$3bobo5bo7bo3bobo7bobo3bobobo7bobo9bobo5bo3bobo3bo3bobo11bo5bobo\$o112bo\$bo5\$7bo\$6bo106bo\$13bobo9bo5bobo5bo3bo3bo5bo11bo3bobobo3bo3bo9bobo5bo3bobobobo\$13bobo9bo5bobo5bo3bo3bo5bo11bo3bobobo3bo3bo9bobo5bo3bobobobo\$6bo106bo\$7bo!`
m=3 and n=17 (reaction=(5,5,7)):
`x = 540, y = 6, rule = B2a3jkq/S01c3ebo\$o538bo\$7bobo3bo5bo9bobo11bo3bo5bobobobo3bo3bobobo9bo3bo3bo3bo3bo3bobobobo7bobo11bo3bo3bobo7bo3bo5bo3bo3bobo3bo5bo5bobo3bobo3bobobo17bobo5bo7bobo3bobobo5bobobo3bobo3bo9bobobo3bo5bo13bobo3bobo3bo5bobo5bo9bobo3bo7bobo3bobo3bobobo3bobo5bobobo5bo5bobo9bo3bobo7bo11bobobo5bobo11bobobo9bobo3bobo9bo3bo3bo5bobo11bobo7bo3bobo3bobo\$7bobo3bo5bo9bobo11bo3bo5bobobobo3bo3bobobo9bo3bo3bo3bo3bo3bobobobo7bobo11bo3bo3bobo7bo3bo5bo3bo3bobo3bo5bo5bobo3bobo3bobobo17bobo5bo7bobo3bobobo5bobobo3bobo3bo9bobobo3bo5bo13bobo3bobo3bo5bobo5bo9bobo3bo7bobo3bobo3bobobo3bobo5bobobo5bo5bobo9bo3bobo7bo11bobobo5bobo11bobobo9bobo3bobo9bo3bo3bo5bobo11bobo7bo3bobo3bobo\$o538bo\$bo!`
Iteration of sigma(n)+tau(n)-n [sigma(n)+tau(n)-n : OEIS A163163] (e.g. 16,20,28,34,24,44,46,30,50,49,11,3,3, ...) :
965808 is period 336 (max = 207085118608).
AbhpzTa

Posts: 460
Joined: April 13th, 2016, 9:40 am
Location: Ishikawa Prefecture, Japan

Next