ConwayLife.com - A community for Conway's Game of Life and related cellular automata
Home  •  LifeWiki  •  Forums  •  Download Golly

The Omniperiodicity Problem

For discussion of specific patterns or specific families of patterns, both newly-discovered and well-known.

The Omniperiodicity Problem

Postby gmc_nxtman » September 27th, 2015, 4:51 pm

I looked through 11 pages of this forum, and was surprised to see no thread devoted to finding all oscillator periods, so I thought I'd make one.

What oscillator periods are known:

•All periods under 43, except for 19, 23, 34, 38, and 41

•All periods above 43, using snark loops

What oscillator types are known:

Billiard table configurations, oscillators in which the rotor is encased in the stator. The definition of "encased" is not always clear.

Examples of billiard table oscillators:

x = 58, y = 13, rule = B3/S23
4bob2o42b2o$4b2obo26b2o14b2o$34b2o$4b3o12b2ob2o26b4o$2obobobo10bobobob
o9b4o11bo4bob2o$2obo3bo10bo3bobo6bobo4bo10bobo2bob2o$3bo3bob2o4b2obo3b
ob2o5b2obo3bo7b2obobo2bo$3bo3bob2o5bobo3bo11bo3bob2o4b2obo2bobo$4b3o9b
obobobo11bo4bobo8b4o$17b2ob2o13b4o$3bob2o45b2o$3b2obo30b2o13b2o$37b2o!


Hassler oscillators, oscillators in which an unstable object is perturbed by other oscillator(s) (or still life[s]). The perturbed object need not necessarily be unstable, but the perturbers have to affect it in some way.

Examples of hassler oscillators:

x = 69, y = 15, rule = B3/S23
38b2o$2o12b2o23bo21bo4bo$bo12bo24bobo9b2o8b6o$bobo8bobo5b3o10b3o4bobo
8bo9bo4bo$2b2o8b2o7b3o8b3o14bobo$49b2o$5b6o14b6o12b2o$43b2ob2o13b6o$2b
2o8b2o7b3o8b3o11b2o12bo6bo$bobo8bobo5b3o10b3o4b2o17bo8bo$bo12bo24bobo
18bo6bo$2o12b2o23bo8bobo10b6o$38b2o9bobo$51bo$51b2o!


Shuttle oscillators, oscillators usually of an even period in which an active object goes back and forth, typically reflecting off some sort of stabilization at the ends. Some oscillators are a trivial type of shuttle without stabilizations, like the Tumbler.

Examples of shuttle oscillators:

x = 180, y = 39, rule = B3/S23
79b2o2bo38bo2b2o$79bo2bobo36bobo2bo$80bobobo36bobobo$79b2obob2o34b2obo
b2o$78bo3bo2bo11bo22bo2bo3bo$77bob2o2b3o10bobo21b3o2b2obo$77bo4b4o10bo
bo14b2o5b4o4bo$78b3o2b3o11bo2b2o2b2o4b2obo6b3o2b3o$52b2o28bo2bo14bo3b
2o3bo10bo2bo$22bo28bobo22b5obob2o14b2o7bo10b2obob5o$20bobo28bo24bo2bo
2bobo24b3o9bobo2bo2bo$2o3bo7b2o4bobo23bo5b3o13bo13bo2bo36bo2bo$2ob2o8b
2o3bo2bo21b3o21b3o12b2o38b2o$4bobo12bobo20bo27bo$5bo14bobo9b2o9b3o21b
3o12b2o$5b3o2b2o10bo9bobo10bo5b3o13bo13bo2bo62bo4bo$10b2o22bo16bo24bo
2bo2bobo24b3o33b2ob4ob2o$34b2o15bobo22b5obob2o14b2o7bo11bo25bo4bo$52b
2o28bo2bo14bo3b2o3bo9b3o39bo$78b3o2b3o11bo2b2o2b2o4b2obo4bo43b2o8bo4bo
$77bo4b4o10bobo14b2o2bobo41b2o7b2ob4ob2o$77bob2o2b3o10bobo19bo53bo4bo$
78bo3bo2bo11bo$79b2obob2o34bo$80bobobo32bobobo$79bo2bobo32b2o2bo$79b2o
2bo37b2o$111b3o$110b5o$109bob3obo$107b3o5b3o$106bo3bo3bo3bo$107b4obob
3obo$111b3o3bo$106bob2obobobo$106b2obo2bo2b2o$110bo2bobo$111b2o2bo$
115b2o!


Other oscillators, any kind of oscillator that does not meet any of the above definitions.

What do you think the missing oscillator periods could be?

Examples:

I think that a p19 oscillator could be....

•A pre-pulsar shuttle with pulsars bouncing back and forth
•A loop of gliders reflecting between p19 dependent reflectors of some sort
•An extensible pattern based on a stabilization of the several known p19 wicks
•An unstable object being perturbed at p19 by a group of still lifes
•A billiard table configuration

I think that a p34 oscillator could be....

•Two p17s interacting to hassle something like a toad
•A pre-pulsar shuttle based on (for example) clocks or killer toads

What to post in this thread:

•Challening the definitions posted above
•Ideas for how to make an oscillator of period n
•Oscillators that break a record of being the smallest of that period
•Solutions to the problem
User avatar
gmc_nxtman
 
Posts: 1097
Joined: May 26th, 2015, 7:20 pm

Re: The Omniperiodicity Problem

Postby codeholic » September 27th, 2015, 5:15 pm

There is also somewhat high probability that the first known oscillators of yet unknown periods are going to be loops of signals, when any 90-degree signal turner is found.
Ivan Fomichev
User avatar
codeholic
Moderator
 
Posts: 1140
Joined: September 13th, 2011, 8:23 am
Location: Hamburg, Germany

Re: The Omniperiodicity Problem

Postby dvgrn » September 28th, 2015, 1:43 pm

codeholic wrote:There is also somewhat high probability that the first known oscillators of yet unknown periods are going to be loops of signals, when any 90-degree signal turner is found.

Agreed. Specifically, I'd bet a nickel that the omniperiodicity problem will eventually be solved by someone finding a 90-degree 2c/3 signal elbow, with a recovery time under 19 ticks.

The 2c/3 signal itself is so simple that it can be repeated every three ticks (?!) so there's some room to maneuver here.

I don't think that a SAT-solver approach is likely to produce any definitive results yet -- too many cells will be involved to do an exhaustive search. Maybe things will be different if Moore's Law has a few more decades to work its magic.

If I'm understanding the Garden of Eden #6 paper correctly, the search limit for current technology seems to be around 30-40 cells. That doesn't sound too different from the size of search that lifesrc/WLS/JLS might be able to manage, counting all unknown cells in all phases, and efficiently discarding unworkable combinations.

If someone can figure out how to state the elbow problem in a way that requires less than 40 unknown cells to be tested in combination, then there might be a chance of running a distributed search to test every possible workable elbow in that search space.

Another idea: we already have a perfectly good 2c/3 elbow, but it doubles the signal so the elbow can only be used once:

#C another piece of circuitry overdue for replacement
#C   (with Snarks and syringes, mostly)
x = 316, y = 425, rule = B3/S23
140b2o$140b3o$139bob2o$139b3o$140bo11$136bobo5bobo2bo$136b2o6b2o3bobo$
137bo7bo3b2o5$133bo2bo$131bobo2bobo$132b2o2b2o3$129b2o$129bo2bob2o$
130b3ob2o2$130b6o$129bo6bo$129b5o2bo$126bo7bobobo$126b6o2bo2b2o$132bob
o$124b6o2bob2o14bo4b2o$123bo6bobo16b2o3b2o$123b5o2bobo16bobo4bo$120bo
7bob2o$120b6o2bo$126bobo$118b6o2bob2o22b2o$117bo6bobo25bobo$117b5o2bob
o25bo$114bo7bob2o$114b6o2bo$120bobo$112b6o2bob2o$111bo6bobo$111b5o2bob
o$108bo7bob2o$108b6o2bo$114bobo$106b6o2bob2o$105bo6bobo$105b5o2bobo$
102bo7bob2o$102b6o2bo$108bobo$100b6o2bob2o$99bo6bobo$99b5o2bobo$96bo7b
ob2o$96b6o2bo$102bobo$94b6o2bob2o$93bo6bobo$93b5o2bobo$90bo7bob2o$90b
6o2bo$96bobo$88b6o2bob2o$87bo6bobo$87b5o2bobo$84bo7bob2o$84b6o2bo$90bo
bo$82b6o2bob2o$81bo6bobo$81b5o2bobo$78bo7bob2o$78b6o2bo$84bobo$76b6o2b
ob2o$75bo6bobo$75b5o2bobo$72bo7bob2o$72b6o2bo$78bobo$70b6o2bob2o$69bo
6bobo$69b5o2bobo$66bo7bob2o$66b6o2bo$72bobo$64b6o2bob2o$63bo6bobo$63b
5o2bobo$60bo7bob2o$60b6o2bo$66bobo$58b6o2bob2o$57bo6bobo$57b5o2bobo$
54bo7bob2o$54b6o2bo$60bobo$52b6o2bob2o$51bo6bobo$51b5o2bobo$48bo7bob2o
$48b6o2bo$54bobo$46b6o2bob2o$45bo6bobo$45b5o2bobo$42bo7bob2o$42b6o2bo$
48bobo$40b6o2bob2o$39bo6bobo$39b5o2bobo$36bo7bob2o$36b6o2bo$42bobo$34b
6o2bob2o$33bo6bobo$33b5o2bobo$30bo7bob2o$30b6o2bo$36bobo$28b6o2bob2o$
27bo6bobo$27b5o2bobo$24bo7bob2o$24b6o2bo$30bobo$22b6o2bob2o$21bo6bobo$
21b5o2bobo$18bo7bob2o$18b6o2bo$24bobo$16b6o2bob2o280bob2o$10b2o3bo6bob
o283b2obo$9bo2bo2b5o2bobo$8bob3o7bob2o282b5o$4b2obobo3b5o2bo263b2o20bo
4bo2b2o$5bobo3bo6bobo264bo23bo2bo2bo$5bobo2b6o2bob2o263bobo21b2obobo$
3bobobobo6bobo267b2o18bo5bob2o$2bob2o2bob4o2bobo286bobo4bo$2bo3bobobo
3bob2o287bo2bo2b2o$2ob2obobo3bobo291b2o$bobo2bob4obob3o257b2o$o2bobo7b
o3bo258bo$b3o2b8o262bobo$4bobo270b2o$3b2obo2b7o$2bo2b2obo7bo278b2o$2b
2o4bo2b6o278b2o$8bobo$7b2obo2b6o$10bobo6bo$10bobo2b5o$11b2obo7bo$14bo
2b6o$14bobo$13b2obo2b6o$16bobo6bo$16bobo2b5o$17b2obo7bo$20bo2b6o257bo$
20bobo262bobo$19b2obo2b6o254bobo$22bobo6bo254bo$22bobo2b5o$23b2obo7bo$
26bo2b6o$26bobo$25b2obo2b6o114b2o$28bobo6bo106b2o5b2o$28bobo2b5o106b2o
$29b2obo7bo140bo$32bo2b6o138b3o$32bobo111b2o17b2o11bo$31b2obo2b6o103b
2o17bo12b2o$34bobo6bo96b2o21bobo20bo$34bobo2b5o96b2o21b2o19b3o$35b2obo
7bo136bo$38bo2b6o136b2o$38bobo$37b2obo2b6o$40bobo6bo136b2o9b2o26b2o7b
2o$40bobo2b5o136bo11bo25bobo7b2o$41b2obo7bo131bobo11bobo21b3obobo$44bo
2b6o131b2o13b2o20bo5b2o39bo$44bobo174b2o45b3o$43b2obo2b6o216bo14bo$46b
obo6bo135bo78b2o12b3o$46bobo2b5o81bo51b3o91bo$47b2obo7bo78b3o48bo94b2o
$50bo2b6o81bo47b2o$50bobo86b2o$49b2obo2b6o221b2o$52bobo6bo201b2o17b2o$
52bobo2b5o201b2o$53b2obo7bo$56bo2b6o$56bobo98b2o$55b2obo2b6o90bo$58bob
o6bo87bobo42b2o$58bobo2b5o87b2o43b2o11b2o51b2o$59b2obo7bo142bo53bo$62b
o2b6o109b2o32b3o48bo$62bobo73b2o40bo35bo48b2o$61b2obo2b6o65b2o15b2o24b
3o85b2o$64bobo6bo81bobo25bo12b2o38b2o32bo$64bobo2b5o83bo38bo39bo30b3o$
65b2obo7bo80b2o38b3o37b3o27bo$68bo2b6o122bo39bo$68bobo$67b2obo2b6o$70b
obo6bo58b2o$70bobo2b5o58bo$71b2obo7bo53bobo$74bo2b6o53b2o$74bobo$73b2o
bo2b6o$76bobo6bo$76bobo2b5o67b2obo$77b2obo7bo64bob2o$80bo2b6o$80bobo
63b2o$79b2obo2b6o55b2o122b2o$82bobo6bo177bobo$82bobo2b5o177bo29b2o$83b
2obo7bo173b2o29bobo$86bo2b6o206bo$86bobo185b2o25b2o$85b2obo2b6o176bobo
4b2o$88bobo6bo38b2o135bo7bo$88bobo2b5o39bo134b2o4b3o$89b2obo7bo36bobo
138bo$92bo2b6o37b2o146b2o$92bobo190bobo$91b2obo2b6o182bo$94bobo6bo180b
2o$94bobo2b5o53bo$95b2obo7bo48b3o$98bo2b6o47bo$98bobo53b2o$97b2obo2b6o
193b2o$100bobo6bo168bo23bobo$100bobo2b5o49b2o117b3o23bo$101b2obo7bo46b
2o120bo22b2o$104bo2b6o167b2o14b2o$104bobo189b2o$103b2obo2b6o54b2o$106b
obo6bo53bo$106bobo2b5o51bobo$107b2obo7bo48b2o$110bo2b6o11b2o$110bobo
16bobo$109b2obo2b6o8bo$112bobo6bo6b2o$112bobo2b5o$113b2obo7bo15b2o$
116bo2b6o15b2o$116bobo137b2o7b2o30b2o$115b2obo2b6o61b2o66b2o7bobo29bob
o$118bobo6bo60b2o45bo27bobob3o29bo$118bobo2b5o107b3o25b2o5bo28b2o$119b
2obo7bo107bo30b2o$122bo2b6o55b2o49b2o$122bobo20b2o39b2o$121b2obo2b6o
11bobo26b2o43bo$124bobo6bo10bo29bo43b3o$124bobo2b5o9b2o29bobo44bo$125b
2obo7bo38b2o43b2o11b2o$128bo2b6o34b2o60b2o$128bobo40b2o$127b2obo2b6o$
130bobo6bob2o$130bobo2b5ob2o$131b2obo138b2o21b2o$134bo2b6o19b2o108bobo
21b2o$134bobo5bo19bo109bo17b2o$133b2obo2b3o18bobo21b2o85b2o17b2o$136bo
bo5b2o14b2o22b2o$136bo2bo4b2o93b2o$137b2o9bo46b2o39b2o2bo2b2o47b2o$
144b2o2b3o43bo2bo38b2obo3bo41b2o5b2o$144b2o5bo42bobo23b2o17bobobo10b2o
29b2o$150b2o43bo20bo3b2o14b2obob2o12bo$215bobo12b2o4bo2bo12b3o$214bobo
13b2o6b2o12bo$163b2o49bo$162bobo48b2o$163bo$177b2o$177b2o4$143b2o$143b
2o5$145bo$115b2o28b3o$108b2o5b2o31bo92bo$108b2o37b2o90b3o11bo$143b2o
93bo14b3o$143bo94b2o16bo14bo$110b2o33bo109b2o12b3o$110b2o32b2o69bo52bo
$104b2o109b3o50b2o$104b2o112bo$217b2o$267b2o$248b2o17b2o$141b2o105b2o$
141b2o17b2o$160b2o3$161b2o$161bo89b2o$148b2o12b3o41bo23b2o20bo$131b2o
16bo14bo40bobo22bo19bo$131bo14b3o57bo14b2o8b3o16b2o$124bo7b3o11bo74bo
11bo20b2o$122b3o9bo87b3o30bo$121bo102bo27b3o$121b2o129bo$113b2o93b2o$
113b2o92bobo$207bo$206b2o62b2o$222b2obo44bobo$222bob2o46bo$272b2o$130b
2o83b2o$130bo84b2o$128bobo$128b2o3$262b2o$262b2o2$253b2obo$253bob2o2$
132b2o$132bobo$134bo91b2o44b2o$134b2o90bo44bobo$124b2o98bobo44bo$124b
2o98b2o44b2o$105bob2o$105b2obo2$114b2o135b2o$114b2o136bo$252bobo$253b
2o15b2o$210b2o58b2o$210bo2b2o$211b2obo$212bo40b2o$124b2o86bobo4b2o31bo
bo$124bo88b2o5bo31bo$122bobo94bo31b2o$122b2o68b2o25b2o$166bo25bo$154bo
11b3o21bobo$152b3o14bo20b2o$136bo14bo16b2o$136b3o12b2o70b2o44b2o$139bo
58b2o24bo44bo$100b2o36b2o59bo13b2o6b3o46b3o$101bo97bobo11b2o6bo50bo$
101bobo96b2o$99b2ob2o35b2o$98bobo38b2o17b2o$98bobo57b2o$97b2ob2o20b2o$
97bo24bo$98bob2o18bobo108b2o$96bobob2o18b2o109b2o$96b2o$155b2o32b2o$
155bo20b2o11b2o$120b2o35bo19bo67b2o21b2o$120bobo33b2o16b3o67bobo21b2o$
122bo29b2o20bo53b2o14bo17b2o$122b3o27bo75bo14b2o17b2o$125bo27b3o37b2o
14b2o3b2o13bo$124b2o29bo38bo15bo3bo13b2o$191b3o13b3o5b3o46b2o$191bo15b
o9bo39b2o5b2o$257b2o4$114b2o$114b2o2$105b2o$106bo$103b3o45b2o$103bo47b
2o$111b2o32b2o$111bo33b2o$112bo$111b2o$147b2o$140b2o5b2o$140b2o!

The elbow's repeat rate is 15 ticks, plenty good enough to solve the remaining oscillator periods, so maybe it's worth looking for a component that reduces the double signal back to a singleton signal. I don't know whether thorough searches have already been done on this. (?)
User avatar
dvgrn
Moderator
 
Posts: 4570
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI

Re: The Omniperiodicity Problem

Postby codeholic » October 19th, 2015, 3:52 pm

What are the known signals besides the two 2c/3 (single and double ones)?
Ivan Fomichev
User avatar
codeholic
Moderator
 
Posts: 1140
Joined: September 13th, 2011, 8:23 am
Location: Hamburg, Germany

Re: The Omniperiodicity Problem

Postby dvgrn » October 19th, 2015, 8:31 pm

codeholic wrote:What are the known signals besides the two 2c/3 (single and double ones)?

There's Dean Hickerson's 5c/9 signal wire from 11 April 1997, which can be connected to a 2c/3 wire. I've quoted a sample signal injector at the end. The link will take you to a larger collection, but that's maybe not a likely line of research to get even as high as p19.

Dean Hickerson wrote:
#C A signal moves along a diagonal wire with speed 5c/9 and period 18/2.
#C When it reaches the end it fizzles out.  Successive signals must be
#C at least 8 gens apart.
#C Dean Hickerson, 4/11/97
x = 52, y = 50
4bob2o$4b2o2bo$7bo2bo$2b5ob2obo2bo$bo2bo3bo2b4o$bob2obobobo6bo$2obob4o
bo2b5o$3bo6bobo5b2o$2obob4obo2bob2obobo$o2bobo2bob2obobobo2bo$2b2o2bo
2bo3bobo4bob2o$4b2o4b4ob2o2b2o2bo$4bo3bobo6bo3bo$5b4obob5ob3o3bo$9bobo
4bobo2b4o$7bo3bo2bo3bobo6bo$7b2o2bobob4obo2b5o$10b2obo6bobo5b2o$13bob
4obo2bob2obobo$13bobo2bob2obobobo2bo$12b2o2bo2bo3bobo4bob2o$14b2o4b4ob
2o2b2o2bo$14bo3bobo6bo3bo$15b4obob5ob3o3bo$19bobo4bobo2b4o$17bo3bo2bo
3bobo6bo$17b2o2bobob4obo2b5o$20b2obo6bobo5b2o$23bob4obo2bob2obobo$23bo
bo2bob2obobobo2bo$22b2o2bo2bo3bobo4bob2o$24b2o4b4ob2o2b2o2bo$24bo3bobo
6bo3bo$25b4obob5ob3o3bo$29bobo4bobo2b4o$27bo3bo2bo3bobo6bo$27b2o2bobob
4obo2b5o$30b2obo6bobo5b2o$33bob4obo2bob2obo2bo$33bobo2bob2obobobo2b2o$
32b2o2bo2bo3bobo$34b2o4b4ob2o$34bo3bobo6bo$35b4obob5obo$39bobo4bobo$
37bo3bo2bo3b2o$37b2o2bobob3o2bo$40b2obo5bo$44bob3o$45b2o!

This showed up starting with a 2c/3 wire. Here's the connection,
also showing a different way for the 5c/9 to fizzle out:

#C Two 2c/3 signals are converted to 5c/9 signals and then fizzle out.
#C The original signals are 6 gens apart, but after conversion they're
#C 9 gens apart.  (Don't try this twice in a row.)
#C Dean Hickerson, 4/11/97
x = 61, y = 56
6bo2bo$4b6o$3bo$3bobob5o$2obobo6bo$2obobo2b5o$4b2obo7bo$7bobob5o$7bobo
$6b2obo2b6o$9bobo6bo$9bobo2b5o$10b2obo7b2o$13bo2b6o$13bobo7bo$12b2obo
2b5obo2b2o$15bobo6bo3bo$15bobo2b4ob3o3bo$14b2obobo3bobo2b4o$18bo2bo3bo
bo6bo$19b2ob4obo2b5o$20bo6bobo5b2o$20bob4obo2bob2obobo$17b2obobo2bob2o
bobobo2bo$17bob2o2bo2bo3bobo4bob2o$21b2o4b4ob2o2b2o2bo$21bo3bobo6bo3bo
$22b4obob5ob3o3bo$26bobo4bobo2b4o$24bo3bo2bo3bobo6bo$24b2o2bobob4obo2b
5o$27b2obo6bobo5b2o$30bob4obo2bob2obobo$30bobo2bob2obobobo2bo$29b2o2bo
2bo3bobo4bob2o$31b2o4b4ob2o2b2o2bo$31bo3bobo6bo3bo$32b4obob5ob3o3bo$
36bobo4bobo2b4o$34bo3bo2bo3bobo6bo$34b2o2bobob4obo2b5o$37b2obo6bobo5b
2o$40bob4obo2bob2obobob2o$40bobo2bob2obobobo2bob2o$39b2o2bo2bo3bobo4bo
$41b2o4b4ob2o2b2o$41bo3bobo6bo$42b4obob5ob3o$46bobo4bobo2bo$44bo3bo2bo
3bobobo$44b2o2bobob4obobo$47b2obo6bob2o$50bob4obo$47b2obobo2bobob2o$
47b2obobo2bobob2o$51bo4bo!

For a short time I thought I'd found a way for the 5c/9 signal to
turn a corner and become a 2c/3. But, like the glider-activated
2c/3 that I sent earlier, it only works once:

#C 5c/9 signal turns a corner, becoming 2c/3.  But the corner is
#C damaged.
#C Dean Hickerson, 4/11/97
x = 63, y = 40
4bob2o$4b2o2bo$7bo2bo$2b5ob2obo2bo$bo2bo3bo2b4o$bob2obobobo6bo$2obob4o
bo2b5o38b2o$3bo6bobo5b2o35bo2bo2b2o$2obob4obo2bob2obobo34bobobo2bo$o2b
obo2bob2obobobo2bo31b2obobo2b2o$2b2o2bo2bo3bobo4bob2o29bobobo$4b2o4b4o
b2o2b2o2bo29bobo2b4o$4bo3bobo6bo3bo28b2obobo5bo$5b4obob5ob3o3bo24bobob
o2b3o$9bobo4bobo2b4o24bobobo4bo$7bo3bo2bo3bobo6bo18b2obobo2b4o$7b2o2bo
bob4obo2b5o19bobobo$10b2obo6bobo5b2o3b2o12bobo2b4o$13bob4obo2bob2obobo
bobo9b2obobo5bo$13bobo2bob2obobobo2bobo10bobobo2b3o$12b2o2bo2bo3bobo4b
ob2o9bobobo4bo$14b2o4b4ob2o2b2o4bo4b2obobo2b4o$14bo3bobo6bo3b4obo4bobo
bo$15b4obob5ob3o5bo4bobo2b4o$19bobo4bobo2b6ob2obobo5bo$17bo3bo2bo3bobo
6bobobo2b3o$17b2o2bobob4obo2b2o2bobobo4bo$20b2obo6bobo2bobobo2b4o$23bo
b4obo2bobobobo$23bobo2bob2obobobo2b4o$20bob2o2bo2bo3bobobo5bo$20b2o2b
2o4b4obo2b3o$23bo2b4o5bo4bo$24bo5b4ob5o$25b3o2bo2bo$27bo3bo2b6o$28b3o
5bo2bo2$28b2obo$28bob2o!

[2 June 1997:]
A p11 can inject signals into a 5c/9 track. Here are 2 versions, showing
both fizzle reactions that I know about:

#C p11 oscillators inject signals into p18/2, speed 5c/9 diagonal tracks
x = 116, y = 55
7b2o57b2o$3b2obo2bo52b2obo2bo$2bobob2obo51bobob2obo$bo2bo4bob2o47bo2bo
4bob2o$bobob5o2b3obob2o40bobob5o2b3obob2o$2obo6bo4b2ob2o39b2obo6bo4b2o
b2o$3bob2ob2ob3o48bob2ob2ob3o$3bo3bobobo2b6o42bo3bobobo2b6o$4b2obo3bob
o6bo42b2obo3bobo6bo$6bob4obo2b5o44bob4obo2b5o$6bo6bobo5b2o42bo6bobo5b
2o$5b2ob4obo2bob2obobo40b2ob4obo2bob2obobo$6bobo2bob2obobobo2bo41bobo
2bob2obobobo2bo$6bo2bo2bo3bobo4bob2o38bo2bo2bo3bobo4bob2o$7b2o4b4ob2o
2b2o2bo39b2o4b4ob2o2b2o2bo$11bobo6bo3bo45bobo6bo3bo$7b5obob5ob3o3bo38b
5obob5ob3o3bo$7bo4bobo4bobo2b4o38bo4bobo4bobo2b4o$10bo3bo2bo3bobo6bo
38bo3bo2bo3bobo6bo$10b2o2bobob4obo2b5o38b2o2bobob4obo2b5o$13b2obo6bobo
5b2o39b2obo6bobo5b2o$16bob4obo2bob2obobo41bob4obo2bob2obobo$16bobo2bob
2obobobo2bo41bobo2bob2obobobo2bo$15b2o2bo2bo3bobo4bob2o37b2o2bo2bo3bob
o4bob2o$17b2o4b4ob2o2b2o2bo39b2o4b4ob2o2b2o2bo$17bo3bobo6bo3bo41bo3bob
o6bo3bo$18b4obob5ob3o3bo39b4obob5ob3o3bo$22bobo4bobo2b4o43bobo4bobo2b
4o$20bo3bo2bo3bobo6bo38bo3bo2bo3bobo6bo$20b2o2bobob4obo2b5o38b2o2bobob
4obo2b5o$23b2obo6bobo5b2o39b2obo6bobo5b2o$26bob4obo2bob2obobo41bob4obo
2bob2obobo$26bobo2bob2obobobo2bo41bobo2bob2obobobo2bo$25b2o2bo2bo3bobo
4bob2o37b2o2bo2bo3bobo4bob2o$27b2o4b4ob2o2b2o2bo39b2o4b4ob2o2b2o2bo$
27bo3bobo6bo3bo41bo3bobo6bo3bo$28b4obob5ob3o3bo39b4obob5ob3o3bo$32bobo
4bobo2b4o43bobo4bobo2b4o$30bo3bo2bo3bobo6bo38bo3bo2bo3bobo6bo$30b2o2bo
bob4obo2b5o38b2o2bobob4obo2b5o$33b2obo6bobo5b2o39b2obo6bobo5b2o$36bob
4obo2bob2obo2bo40bob4obo2bob2obobob2o$36bobo2bob2obobobo2b2o40bobo2bob
2obobobo2bob2o$35b2o2bo2bo3bobo45b2o2bo2bo3bobo4bo$37b2o4b4ob2o46b2o4b
4ob2o2b2o$37bo3bobo6bo45bo3bobo6bo$38b4obob5obo45b4obob5ob3o$42bobo4bo
bo49bobo4bobo2bo$40bo3bo2bo3b2o46bo3bo2bo3bobobo$40b2o2bobob3o2bo45b2o
2bobob4obobo$43b2obo5bo49b2obo6bob2o$47bob3o53bob4obo$48b2o52b2obobo2b
obob2o$102b2obobo2bobob2o$106bo4bo!

Then there's a c/2 by Hartmut Holzwart, I believe:

x = 60, y = 60, rule = B3/S23
bo$obo$bobo$2bobo$3bobo$5bo$7bo$3bo3bo$7b2o$4b3o3bo$8b3o$9b2o$12bo$12b
obo2$13bobo$14bobo$15bobo2bo$16bo5bo$17b2o3bo$22bo$19b3o$21bobo$23b2o$
22b3o2$25b2o$28bo$26bo2bo$28bobo$29bobo$30bobo$31bobo$32bobo$33bobo$
34bobo$35bobo$36bobo$37bobo$38bobo$39bobo$40bobo$41bobo$42bobo$43bobo$
44bobo$45bobo$46bobo$47bobo$48bobo$49bobo$50bobo$51bobo$52bobo$53bobo$
54bobo$55bobo$56bobo$57bobo$58bo!

-- and Jason Summers' orthogonal lightspeed beehive wire, I suppose, though the wire drifts and has to be moved back into place with a series of later signals... and Gabriel Nivasch's diagonal lightspeed signals, which are complicated enough that it seems unlikely that there will ever be a recipe to generate them.

(Might be missing a few. Anyone have other contributions?)
User avatar
dvgrn
Moderator
 
Posts: 4570
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI

Re: The Omniperiodicity Problem

Postby Alexey_Nigin » October 20th, 2015, 4:52 pm

Dave Greene wrote:(Might be missing a few. Anyone have other contributions?)


There exists this simple orthogonal lightspeed signal:

x = 26, y = 11, rule = B3/S23
bo2bo2bo2bo2bo2bo2bo2bo$b24o$25bo$b24o$o4bo$b4o4b16o$6bo18bo$b24o$o$b
24o$3bo2bo2bo2bo2bo2bo2bo2bo!


Now that we have collected so many of them, it seems a good idea to invent some sort of nomenclature for signal turners, like the one for Hershel conduits (or does there already exist one?).
There are 10 types of people in the world: those who understand binary and those who don't.
User avatar
Alexey_Nigin
 
Posts: 322
Joined: August 4th, 2014, 12:33 pm
Location: Ann Arbor, MI

Re: The Omniperiodicity Problem

Postby Scorbie » October 20th, 2015, 6:36 pm

Alexey_Nigin wrote: it seems a good idea to invent some sort of nomenclature for signal turners
Wish if we had any to name them... There are currently only three converters that I know of.
Best wishes to you, Scorbie
User avatar
Scorbie
 
Posts: 1348
Joined: December 7th, 2013, 1:05 am

Re: The Omniperiodicity Problem

Postby Sphenocorona » October 21st, 2015, 12:19 am

I was wondering if it would actually be possible to convert an orthogonal lightspeed signal to another signal (or even try turning it around a corner)... the simple one Alexey Nigin mentioned below looks the easiest to form (due to size, though some larger signals have the symmetry advantage), but I'm not aware of any way to even absorb it. I went and played around with some of the ones that can be absorbed, though, and a couple of them are kinda interesting:

#C Maybe a drifter search could do something with it (not necessarily this form exactly)
x = 45, y = 21, rule = B3/S23
32b2o$32bobo$34bo$34b2o$36bo$bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2b2obo$b
34o2bo$36b2ob2o$b35o2bobo2b2o$o5bo29bobobobobo$b4o4b28o2bo2bo$4bo2bo
29b2o3b2o$b4o4b28o$o4bo31b2o$b36o2bo$38bobo$b38obo$bo2bo2bo2bo2bo2bo2b
o2bo2bo2bo2bo2bo5b2o$37b3o$36bo2bo$36b2o!

#C Orthogonal lightspeed signal turns 135 degrees to become a 2c/3 signal - but one important cell is turned OFF in the process.
x = 42, y = 26, rule = B3/S23
3bo2bo2bo2bo2bo2bo2bo2bo2bo2bo2bo$3b33o$36bo$b4ob2obo2b25o3b2o$o4bo3b
2o26b2o2bo$b6o2bo2b25o2b2o$37bobo$b35obobo$bo2bo2bo2bo2bo2bo2bo2bo12bo
b2o$25b8o2bo$25bo7bobob2o$26b5o2bobobo$20b2obo7bobobobo$21bob6o2bobob
2o$20bo8bobobo$21b6o2bobobo$27bobob2o$19b6o2bobo$18bo6bobobo$18b5o2bob
ob2o$23bobobo$18b3o2bobobo$18bo2bobob2o$19bobobo$20bo2bob2o$23b2obo!
Sphenocorona
 
Posts: 470
Joined: April 9th, 2013, 11:03 pm

Re: The Omniperiodicity Problem

Postby dvgrn » October 21st, 2015, 11:49 am

Sphenocorona wrote:I was wondering if it would actually be possible to convert an orthogonal lightspeed signal to another signal (or even try turning it around a corner)...

To prove omniperiodicity it probably makes the most sense to look for a fast direct 90-degree turn where the input and output signals are the same -- or a 180-degree signal turner would be fine if it leaves space for two wires next to each other, but that seems a little less likely.

Otherwise the problem doesn't really get any smaller. If we could fix the ortho-lightspeed to 2c/3 signal turner, then to complete a loop we'd still need a 2c/3 to ortho-lightspeed turner, at minimum, or a chain of other signal converters that could be combined to produce that.

I'll be really enormously impressed and surprised if anyone comes up with a working X-to-ortho-lightspeed signal converter any time soon -- let's say, before someone figures out how to program the question into an actual working 1000-qubit quantum computer. Actually I doubt that even quantum computing will be enough of a boost. This is just a ridiculously big search space.

How Big Is Ridiculously Big?

I did some very rough estimating. If anyone wants some slightly more specific crackpot calculations, I can post the rest of my notes...

Bigger Signal Object Means Bigger Search Space

Creating an ortho-lightspeed signal involves figuring out how to adjust the states of at least 18 bits. Really it might be more like 9 bits for the leading p1 part and another 18 for the trailing p2 part, but let's just call it 18 cells.

We have to start from some kind of large still life, give it an input signal, get that specific ortho-lightspeed output signal going, and then have the still life return to its exact original state. Even if the input is just a one-bit spark somewhere nearby, quite a few more cells will be needed to guide that signal into the stabilized end of an ortho-lightspeed wire.

Longer Recovery Also Means Bigger Search Space

And even if that one-bit spark propagates at lightspeed, it's going to take at least 6 ticks for that signal to build a full-length ortho-lightspeed signal in the beginning of the wire. After that we can stop worrying -- the wire will recover naturally.

-- But we already have on the order of 6*18 = 108 cell states to worry about, not counting stator cells around the edges. (Yes, it's possible to do a lot of hand-waving and imagine vaguely that we could get away with less, but really it will probably be more.)

Summary: Brute Force Has Its Limits

Anywhere above 2^40 cases is pretty difficult -- that's a trillion, which is apgnano/Catagolue territory -- and we're talking a lot more than 2^40 sets of 2^40 cases. So we need an algorithm clever enough to remove a factor of something like 2^70 from this search problem...!

My theory is that if we had the tools needed to solve the ortho-lightspeed elbow problem, we would also be able to solve the 2c/3 elbow problem, about a million times more easily. A 2c/3 elbow -- or even just a double-signal to single-signal converter -- is all we really need to prove omniperiodicity.
User avatar
dvgrn
Moderator
 
Posts: 4570
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI

Re: The Omniperiodicity Problem

Postby gmc_nxtman » October 22nd, 2015, 10:18 am

This is close to a 180 signal turner:

x = 26, y = 11, rule = B3/S23
bo2bo2bo2bo2bo2bo2bo2bo$b24o$25bo$b25o$o4bo$5o4b17o$6bo18bo$25o$o$b24o
$3bo2bo2bo2bo2bo2bo2bo2bo!


Keep in mind, that there are an infinity of lightspeed signals, very few of which could be potentially be turned.

These even simpler lightspeed signal might also be turnable:

x = 29, y = 29, rule = B3/S23
3bo2bo2bo2bo2bo2bo2bo2bo2bo$b27o$o$b27o$28bo$b2o2b24o$ob2o$b2o2b24o$
28bo$b27o$o$b27o$3bo2bo2bo2bo2bo2bo2bo2bo2bo4$3bo2bo2bo2bo2bo2bo2bo2bo
2bo$b27o$o$b27o$28bo$b3o2b23o$o2bo$b3o2b23o$28bo$b27o$o$b27o$3bo2bo2bo
2bo2bo2bo2bo2bo2bo!
User avatar
gmc_nxtman
 
Posts: 1097
Joined: May 26th, 2015, 7:20 pm

Re: The Omniperiodicity Problem

Postby Freywa » October 22nd, 2015, 10:33 am

dvgrn wrote:Anywhere above 2^40 cases is pretty difficult -- that's a trillion, which is apgnano/Catagolue territory -- and we're talking a lot more than 2^40 sets of 2^40 cases. So we need an algorithm clever enough to remove a factor of something like 2^70 from this search problem...!

No such algorithm can exist. We are better off getting supercomputer time and running dr or Bellman to find stabilisers for what partial results we have. It may be easier to disprove omniperiodicity.
User avatar
Freywa
 
Posts: 274
Joined: June 23rd, 2011, 3:20 am
Location: Singapore

Re: The Omniperiodicity Problem

Postby dvgrn » October 22nd, 2015, 4:49 pm

gmc_nxtman wrote:This is close to a 180 signal turner:

x = 26, y = 11, rule = B3/S23
bo2bo2bo2bo2bo2bo2bo2bo$b24o$25bo$b25o$o4bo$5o4b17o$6bo18bo$25o$o$b24o
$3bo2bo2bo2bo2bo2bo2bo2bo!

Not a potentially useful 180-degree turner for the omniperiodicity problem, though, is it? To make a p[19|23|34|38|41] oscillator we'd have to be able to add several signals to one conduit, so a 180-degree turner would have to be offset so the return signal could travel back on a separate wire.

Freywa wrote:We are better off getting supercomputer time and running dr or Bellman to find stabilisers for what partial results we have.

Yeah, I've been really curious to see what a high-performing computing cluster could do with this problem. I had access to a system with several thousand cores for a while last year, but sadly had too many other things to do to spend much time playing around with it.

I did figure out that the cluster was just about powerful enough to brute-force the discovery of the loafer, and every other spaceship of any speed that's the same size or smaller: there are "only" 2^53 arrangements of cells in a triangle 10 cells on a side, with the corner cells removed... a few thousand modified copies of apgnano running on a cluster could classify every possible object inside that snub triangle, so we'd get a lot of loafers and who knows what else besides.

At 2000 soups per minute per core a completely brute-force method would take about ten years on 2000 cores (!) That could be cut down a lot more with symmetries and so on... and most of those bit patterns aren't very challenging soups, after all. If we were really just looking for spaceships instead of cataloguing everything, we could cut off several orders of magnitude I would think.

But I don't know whether a modified dr or a special-purpose Bellman would be a better choice for hunting for a 2c/3 elbow, or a 2c/3 period doubler/pulse divider.

Freywa wrote:It may be easier to disprove omniperiodicity.

I hope not, since it's clearly impossible to prove the non-existence of those five oscillators. As far as I can see, any or all of those periods could perfectly well pop out of a RandomAgar or symmetrical Catagolue soup search tomorrow.

You'd have to show mathematically that at least one of those periods could not possibly be attained by a B3/S23 pattern of any size. A proof of non-omniperiodicity would have to give good reasons why no 2c/3 elbow can possibly exist -- along with an infinite number of other possible objects, like 5c/9 elbows.

Even if you somehow knew that no 2c/3 elbow exists inside a 20x20 bounding box, the jury would still be out on 20x21...!
User avatar
dvgrn
Moderator
 
Posts: 4570
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI

Re: The Omniperiodicity Problem

Postby Scorbie » October 22nd, 2015, 6:35 pm

dvgrn wrote:At 2000 soups per minute per core a completely brute-force method would take about ten years on 2000 cores (!) That could be cut down a lot more with symmetries and so on... and most of those bit patterns aren't very challenging soups, after all. If we were really just looking for spaceships instead of cataloguing everything, we could cut off several orders of magnitude I would think.
Yep... I think modifying code to something similar to gsearch would make it faster, I think. (Or maybe one can use unmodified gsearch... Depending on whether gsearch's life iterator is as fast as apgnano's.)
Best wishes to you, Scorbie
User avatar
Scorbie
 
Posts: 1348
Joined: December 7th, 2013, 1:05 am

Re: The Omniperiodicity Problem

Postby Sphenocorona » October 22nd, 2015, 7:47 pm

Freywa wrote:It may be easier to disprove omniperiodicity.

Finding a proof of lack of omniperiodicity isn't completely impossible, but going this route would be incredibly, incredibly hard, far harder than searching for the existence of them in the first place, as well as still requiring a ton of this sort of searching anyway - by which point you may have already found what you've been attempting to prove impossible.
Sphenocorona
 
Posts: 470
Joined: April 9th, 2013, 11:03 pm

Re: The Omniperiodicity Problem

Postby calcyman » October 23rd, 2015, 9:23 am

Sphenocorona wrote:Finding a proof of lack of omniperiodicity isn't completely impossible


There is a systematic way to convert any Pi^0_2 statement to a cellular automaton, such that the cellular automaton is omniperiodic if and only if the Pi^0_2 statement is true. Also, asking whether a cellular automaton is omniperiodic is easily expressed as a Pi^0_2 statement, so these two problems are genuinely equivalent.

Fortunately, for B3/S23 there are only finitely many periods for which the existence of oscillators is unknown, so this is only a Sigma^0_1 statement. Resolving an arbitrary Sigma^0_1 statement is equivalent to solving the halting problem for an arbitrary Turing machine. This is still undecidable, but not nearly as undecidable as resolving an arbitrary Pi^0_2 statement -- so you have more of a chance of victory.

The notation is explained in https://en.wikipedia.org/wiki/Arithmetical_hierarchy if you haven't encountered it before.
What do you do with ill crystallographers? Take them to the mono-clinic!
User avatar
calcyman
 
Posts: 1567
Joined: June 1st, 2009, 4:32 pm

Re: The Omniperiodicity Problem

Postby dvgrn » October 23rd, 2015, 1:34 pm

calcyman wrote:Fortunately, for B3/S23 there are only finitely many periods for which the existence of oscillators is unknown, so this is only a Sigma^0_1 statement. Resolving an arbitrary Sigma^0_1 statement is equivalent to solving the halting problem for an arbitrary Turing machine. This is still undecidable, but not nearly as undecidable as resolving an arbitrary Pi^0_2 statement -- so you have more of a chance of victory.

I'm not so theoretically-minded, I guess. One infinitesimal chance of victory looks about the same as another. @Calcyman, it sounds like you're comparing "one chance in aleph_0" to "one chance in aleph_1" -- metaphorically speaking, of course.

In other words, B3/S23 is so clearly omniperiodic that it seems like a waste of time to even think about trying to prove the opposite. Any such proof would exceedingly complex, headache-inducingly subtle, and ridiculously long... and it's a safe bet it would contain at least one error somewhere.

Let's have another look at how close we are to proving omniperiodicity right now:

x = 137, y = 84, rule = LifeHistory
15.2A$14.3A$14.2A.A$15.3A$16.A11$7.A2.A.A5.A.A$5.A.A3.2A6.2A84.2A$6.
2A3.A7.A85.2A$109.A$105.5A$104.A$104.6A$20.A2.A84.A.A.2A$18.A.A2.A.A
76.7A.A.2A$19.2A2.2A71.2A3.A6.A.A$95.A2.A2.5A2.A.A$94.A.3A7.A.2A$26.
2A62.2A.A.A3.5A2.A$21.2A.A2.A63.A.A3.A6.A.A$21.2A.3A64.A.A2.6A2.A.2A$
89.A.A.A.A6.A.A$21.6A61.A.2A2.A.4A2.A.A$20.A6.A60.A3.A.A.A3.A.2A$20.A
2.5A58.2A.2A.A.A3.A.A$18.A.A.A7.A56.A.A2.A.4A.A.3A$18.2A2.A2.6A55.A2.
A.A7.A3.A$22.A.A62.3A2.8A$2A4.A14.2A.A2.6A57.A.A$.2A3.2A16.A.A6.A55.
2A.A2.7A$A4.A.A16.A.A2.5A54.A2.2A.A7.A$25.2A.A7.A51.2A4.A2.6A$28.A2.
6A57.A.A$28.A.A62.2A.A2.6A$3.2A22.2A.A2.6A57.A.A6.A$2.A.A25.A.A6.A56.
A.A2.5A$4.A25.A.A2.5A57.2A.A7.A$31.2A.A7.A57.A2.6A$34.A2.6A57.A.A$34.
A.A62.2A.A2.6A$33.2A.A2.6A57.A.A6.A.2A$36.A.A6.A.2A53.A.A2.5A.2A$36.A
.A2.5A.2A54.2A.A$37.2A.A65.A2.6A19.2A$40.A2.6A19.2A36.A.A5.A19.A$40.A
.A5.A19.A36.2A.A2.3A18.A.A$39.2A.A2.3A18.A.A39.A.A5.2A14.2A$42.A.A5.
2A14.2A40.A2.A4.2A$42.A2.A4.2A57.2A9.A$43.2A9.A61.2A2.3A$50.2A2.3A59.
2A5.A$50.2A5.A64.2A$56.2A2$135.2A$69.2A63.A.A$68.A.A64.A$69.A5$115.2A
10.D2.2D$49.2A10.D2.2D49.2A11.3D$49.2A11.3D64.D$63.D2$124.C.2C$58.C.
2C60.3C.2C$56.3C.2C59.C$55.C66.3C.2C$56.3C.2C62.C.C$58.C.C63.C.C$58.C
.C64.C$59.C!

We already have ways to get a 2c/3 signal started (though it's certainly not repeatable at p19 yet).

The same wire can carry either a single or a double signal, and we already have a way to turn the corner while doubling the signal.

We even already have a mechanism that can accept either a single or a double signal, and turn either one into the exact same spark.

It's clearly not easy to turn that spark, or one of its variants, back into a single 2c/3 signal again. But 'dr' searches have only been done up to a certain number of active cells, with recovery within a certain number of ticks.

Every time Moore's Law lets us bump up the search limits a little bit, we find more interesting stuff. Here's a stamp collection of fizzles from 2008, to give a sense of the range of possibilities. (@Calcyman, this should look suspiciously familiar to you, but I'm not sure everyone else here has a copy handy...!)

#C Top-left section: 2c/3 and corresponding TL fizzles.
#C Bottom-left section: various reactions that can't be categorised
#C anywhere else.
#C Right section: Up to 7 different variations for each reaction.
#C Each row
#C corresponds to a different key reaction, and each column
#C corresponds to a different input.
#C
#C A) Small TL eater
#C B) Medium TL eater
#C C) Large TL eater
#C D) Perpendicular TL eater (with split)
#C E) Small 2c/3 fizzle
#C F) Medium 2c/3 fizzle
#C G) Large 2c/3 fizzle
#C
#C Adam P. Goucher, 23 November 2008
x = 578, y = 336, rule = B3/S23
173bo51bo$172bobo24boo22b3o$173bo25boo21bo31boo$196bo25b6o22boobobbo$
149bo21b5o20b5o27bo22boboobbo18boo$148bobo19bo4bo24bo21b3obboo14boo5bo
4boo15boobobo$148bobo18bobbo23boo20boobobbo18bobo3bob3o5bo11bobobo$
123bo22b3oboo14bobboboo19boobobo21boboo22bo3bo4b6o11bo3boo67bo21bobo
25bobo$100bo22b3o19bo19bobobo5bo15bobobo4bo18bo5bo19booboob3o15boobbo
96bo21bo35bobo27bobobo$75boo3booboo14bobo3booboo16bo3booboo11b3oboo3b
ooboo6bobbo4bobo3booboo6boboboobbobo3booboo6boobo4bobo3booboo14bobo3b
ooboo8bobb6o3booboo57bo3bo19bo65bo65bobobo44bobo$75boo4boboo13bobbo4bo
boo15boo4boboo13boboo4boboo9boobbobbo4boboo5boobo4bobbo4boboo6booboobb
obbo4boboo10b3obobo4boboo8boo6bo4boboo85bo20bo36bo29bo86bo$81bo17boo5b
o24bo24bo17boo5bo11bo5boo5bo17boo5bo12bobboboo5bo16b3o5bo60bobobo19bob
o64bo64bo$78boobo5boo14boobo5boo14boobo5boo14boobo5boo14boobo5boo4bobo
7boobo5boo14boobo5boo5boo7boobo5boo9bo4boobo5boo81bo20bo36bo29bobobo
81bo$4booboo69boboboo4bo14boboboo4bo14boboboo4bo14boboboo4bo14boboboo
4bo5boo7boboboo4bo14boboboo4bo14boboboo4bo6boobo4boboboo4bo53bo3bo19bo
66bo64bobobo44bobo$3boboboboboo12boo53bobobobo18bobobobo18bobobobo18bo
bobobo18bobobobo18bobobobo18bobobobo18bobobobo8booboo5bobobobo83bo21bo
35bo29bo85bo$3bobobobobo14bobboo4bo20boo19bobboboboboo14bobboboboboo
14bobboboboboo14bobboboboboo14bobboboboboo14bobboboboboo14bobboboboboo
14bobboboboboo14bobboboboboo54bo3bo19bobo25bobo35bo65bo50bo$oobobobboo
bbo13bobobbobbobo20bo4boo12bobobobboo3bo12bobobobboo3bo12bobobobboo3bo
12bobobobboo3bo12bobobobboo3bo12bobobobboo3bo12bobobobboo3bo12bobobobb
oo3bo12bobobobboo3bo139bobo27bobobo82bo$oobobo4boobbo10boob4obbobobbo
16bo4bobbo9b3obobo4boobbo9b3obobo4boobbo9b3obobo4boobbo9b3obobo4boobbo
9b3obobo4boobbo9b3obobo4boobbo9b3obobo4boobbo9b3obobo4boobbo9b3obobo4b
oobbo208bo48bobo$3bobb4obb3o17b3obb3o16b4obb3o8bo4bobb4obb3o8bo4bobb4o
bb3o8bo4bobb4obb3o8bo4bobb4obb3o8bo4bobb4obb3o8bo4bobb4obb3o8bo4bobb4o
bb3o8bo4bobb4obb3o8bo4bobb4obb3o$3bo6boo13b6o4boo23boo11boo3bo6boo11b
oo3bo6boo11boo3bo6boo11boo3bo6boo11boo3bo6boo11boo3bo6boo11boo3bo6boo
11boo3bo6boo11boo3bo6boo$4b6obbo12bo5b4obbo18b4obbo16b6obbo16b6obbo16b
6obbo16b6obbo16b6obbo16b6obbo16b6obbo16b6obbo16b6obbo$9boboboboo9b5o3b
oboboboo14bobboboboboo17boboboboo17boboboboo17boboboboo17boboboboo17bo
boboboo17boboboboo17boboboboo17boboboboo17boboboboo$bb7obbobobo15b3obb
obobo16boobbobobo11b7obbobobo11b7obbobobo11b7obbobobo11b7obbobobo11b7o
bbobobo11b7obbobobo11b7obbobobo11b7obbobobo11b7obbobobo$bbo6boobbobo
12boobobboobbobo13b3obboobbobo11bo6boobbobo11bo6boobbobo11bo6boobbobo
11bo6boobbobo11bo6boobbobo11bo6boobbobo11bo6boobbobo11bo6boobbobo11bo
6boobbobo$3b3obbo4boboboo10bobobo4boboboo9bobbobo4boboboo9b3obbo4bobob
oo9b3obbo4boboboo9b3obbo4boboboo9b3obbo4boboboo9b3obbo4boboboo9b3obbo
4boboboo9b3obbo4boboboo9b3obbo4boboboo9b3obbo4boboboo105boo23boo6b3o
119boo$5bobb5obbobobo9bobob5obbobobo9bobob5obbobobo10bobb5obbobobo10bo
bb5obbobobo10bobb5obbobobo10bobb5obbobobo10bobb5obbobobo10bobb5obbobob
o10bobb5obbobobo10bobb5obbobobo10bobb5obbobobo96boobboo3bo23bobo5b3o
27boo39boo48bobo$15bobobo10bo9bobobo10bo9bobobo20bobobo20bobobo20bobob
o20bobobo20bobobo20bobobo20bobobo20bobobo20bobobo96bobobbobbo26b3o4bo
4boo14booboobbobbo30booboobbobbobboo43bo$10b5obboboboo12b5obboboboo12b
5obboboboo12b5obboboboo12b5obboboboo12b5obboboboo12b5obboboboo12b5obbo
boboo12b5obboboboo12b5obboboboo12b5obboboboo12b5obboboboo44boo3boo23b
oo19b3o3boo24bo3bo7bobbo14bobo4boobbo29bobo4b3obbo36boo5boo$10bo6bobob
o13bo6bobobo13bo6bobobo13bo6bobobo13bo6bobobo13bo6bobobo13bo6bobobo13b
o6bobobo13bo6bobobo13bo6bobobo13bo6bobobo13bo6bobobo44bobobbobbo14boob
o3bobbo17bo3b3o22bobbob3obbooboobboo15bobb4obb3o29bobb4o3boobbo34bobo
3bo$13b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4o
bbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo43bo5b3o14boboo
bobb3o14bobbob3o3boo20b4o4booboboboo16boo6boo31boo6b3obb3o36b3obboo$
13bo5boboboo13bo5boboboo13bo5boboboo13bo5boboboo13bo5boboboo13bo5bobob
oo13bo5boboboo13bo5boboboo13bo5boboboo13bo5boboboo13bo5boboboo13bo5bob
oboo40b6o5boo9bo5b3o5boo10b4o4b3obbo16bo6b3o4bobo4b3o11bobb6obbo29bobb
5o4boo38bo3boobbo$16b3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo15b
3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo
15b3obbobobo15b3obbobobo37boo6bo3b3o9b6o3bo3b3o7bo6b3o4boo17b7o3b4obob
4obbo11boo6boboboboo25boo6b4obbo33bobbob3obbobobbo$16bo4bobobo15bo4bob
obo15bo4bobobo15bo4bobobo15bo4bobobo15bo4bobobo15bo4bobobo15bo4bobobo
15bo4bobobo15bo4bobobo15bo4bobobo15bo4bobobo37bobb6o4boo7boo6b4o4boo7b
7o3b4o4boo20b3o3bo3bobbo16b6obbobobo28b6o3boboboboo29b4o4b3obb3o$17b4o
bboboboo13b4obboboboo13b4obboboboo13b4obboboboo13b4obboboboo13b4obbobo
boo13b4obboboboo13b4obboboboo13b4obboboboo13b4obboboboo13b4obboboboo
13b4obboboboo35boo6boo11bobb5o4boo18b3o3bo3b3o15b5o3b3obbobboo17bo5boo
bbobo28bo5b3obbobobo27bo6b3o4boo$23bobobo20bobobo20bobobo20bobobo20bob
obo20bobobo20bobobo20bobobo20bobobo20bobobo20bobobo20bobobo37bobb4obbo
11boo6b3obbo12b5o3b3o5boo14bo6boo3booboobb3o15bobobo4boboboo26boboobo
bboobbobo27b7o3b4obbo$19b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbo
bo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbo
bo37bobo4boo13bobb4o3boo12bo6boo3b3o18bobb4obboobbobobbobbo16boob5obbo
bobo26boobobbo4boboboo31b3o3boboboboo$19bo5boboboo13bo5boboboo13bo5bob
oboo13bo5boboboo13bo5boboboo13bo5boboboo13bo5boboboo13bo5boboboo13bo5b
oboboo13bo5boboboo13bo5boboboo13bo5boboboo33booboobbo15bobo4b3o14bobb
4obbobbobbo16boobo4boboboo4boo29bobobo33b4obbobobo25b5o3b3obbobobo$22b
3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo
15b3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo15b3obbobobo15b3obbobo
bo40bobo11booboobbobbo13boobo4bobo3boo17bobboobbobo34b5obboboboo36bobo
bo24bo6boo3boobbobo$22bo4bobobo15bo4bobobo15bo4bobobo15bo4bobobo15bo4b
obobo15bo4bobobo15bo4bobobo15bo4bobobo15bo4bobobo15bo4bobobo15bo4bobob
o15bo4bobobo41boo19boo14bobboobbobo25bo5bo35bo6bobobo31b6obboboboo21bo
bb4obbobo4boboboo$23b4obboboboo13b4obboboboo13b4obboboboo13b4obboboboo
13b4obboboboo13b4obboboboo13b4obboboboo13b4obboboboo13b4obboboboo13b4o
bboboboo13b4obboboboo13b4obboboboo77bo5bo25boo44b4obbobo31bo7bobobo21b
oobo4bobob5obbobobo$29bobobo20bobobo20bobobo20bobobo20bobobo20bobobo
20bobobo20bobobo20bobobo20bobobo20bobobo20bobobo77boo77bo5boboboo31b5o
bbobo21bobboobbobo9bobobo$25b4obbobo16b4obbobo16b4obbobo16b4obbobo16b
4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b4obbobo16b
4obbobo159b3obbobobo30bo6boboboo20bo5bo5b5obboboboo$25bo7boo15bo7boo
15bo7boo15bo7boo15bo7boo15bo7boo15bo7boo15bo7boo15bo7boo15bo7boo15bo7b
oo15bo7boo158bo4bobobo33b4obbobobo18boo11bo6bobobo$28b4obo19b4obo19b4o
bo19b4obo19b4obo19b4obo19b4obo19b4obo19b4obo19b4obo19b4obo19b4obo160b
4obboboboo30bo5bobobo34b4obbobo$28bo4bo19bo4bo19bo4bo19bo4bo19bo4bo19b
o4bo19bo4bo19bo4bo19bo4bo19bo4bo19bo4bo19bo4bo166bobobo32b5obboboboo
31bo5boboboo$29b4o21b4o21b4o21b4o21b4o21b4o21b4o21b4o21b4o21b4o21b4o
21b4o163b4obbobo39bobobo35b3obbobobo$471bo5boboboo31b5obbobo35bo4bobob
o$31boo23boo23boo23boo23boo23boo23boo23boo23boo23boo23boo23boo166b3obb
obobo31bo5boboboo33b4obboboboo$31boo23boo23boo23boo23boo23boo23boo23b
oo23boo23boo23boo23boo166bo4bobobo21boo6bo4b3obbobobo38bobobo$475b4obb
oboboo18boo6boo3bo4bobobo34b4obbobo$465boo14bobobo14bo18b4obboboboo31b
o5boboboo$459bo4bobo10b4obbobo13bobo3b4o16bobobo35b3obbobobo$458bobo3b
o12bo7boo12bobobbo4bo11b4obbobo35bo4bobobo$199bo51bo206bobobboo15b4obo
12booboobb5o11bo7boo35b4obboboboo$198bobo24boo22b3o147boo23boo6b3o22b
ooboo18bo4bo11bobbobboo19b4obo24boo16bobobo$199bo25boo21bo31boo109boo
bboo3bo23bobo5b3o21bobbobb3o16b4o10bobboobbobbobboo14bo4bo23bobo12b4o
bbobo$222bo25b6o22boobobbo108bobobbobbo26b3o4bo4boo14bobboobbo3bo29boo
bo4b3obbo16b4o24bo14bo7boo$175bo21b5o20b5o27bo22boboobbo18boo38boo3boo
23boo19b3o3boo24bo3bo7bobbo13boobo4boobbo16boo11bobb4o3boobbo34boo5boo
17b4obo$174bobo19bo4bo24bo21b3obboo14boo5bo4boo15boobobo38boobbobbo14b
oobo3bobbo17bo3b3o22bobbob3obbooboobboo15bobb4obb3o16boo11bo6b3obb3o
16boo16bobo3bo19bo4bo$174bobo18bobbo23boo20boobobbo18bobo3bob3o5bo11bo
bobo37bo6b3o14boboobobb3o14bobbob3o3boo20b4o4booboboboo17bo6boo33b5o4b
oo19boo18b3obboo18b4o$51boo96bo22b3oboo14bobboboo19boobobo21boboo22bo
3bo4b6o11bo3boo36b7o5boo9bo5b3o5boo10b4o4b3obbo16bo6b3o4bobo4b3o14b6o
bbo37b4obbo37bo3boobbo$23booboo4boo18bobboo69bo22b3o19bo19bobobo5bo15b
obobo4bo18bo5bo19booboob3o15boobbo46bo3b3o9b6o3bo3b3o7bo6b3o4boo17b7o
3b4obob4obbo19boboboboo26b7o3boboboboo29bobbob3obbobobbo16boo$22bobobo
bobobbo18bobobbo3bo39boo3booboo14bobo3booboo16bo3booboo11b3oboo3booboo
6bobbo4bobo3booboo6boboboobbobo3booboo6boobo4bobo3booboo14bobo3booboo
8bobb6o3booboo30b6o4boo7boo6b4o4boo7b7o3b4o4boo20b3o3bo3bobbo15b7obbob
obo27bo6b3obbobobo30b4o4b3obb3o16boo$22bobobobob3o18boob4obbobo22bo15b
oo4boboo13bobbo4boboo15boo4boboo13boboo4boboo9boobbobbo4boboo5boobo4bo
bbo4boboo6booboobbobbo4boboo10b3obobo4boboo8boo6bo4boboo29bo6boo11bobb
5o4boo18b3o3bo3b3o15b5o3b3obbobboo16bo6boobbobo28booboobobboobbobo27bo
6b3o4boo$19boobobobboo29b3obbo17bobbobo20bo17boo5bo24bo24bo17boo5bo11b
o5boo5bo17boo5bo12bobboboo5bo16b3o5bo32bobb4obbo11boo6b3obbo12b5o3b3o
5boo14bo6boo3booboobb3o14b3obbo4boboboo27boobobbo4boboboo24b7o3b4obbo$
19boobobo4boo20b6o4boo18b4obbo16boobo21boobo21boobo21boobo21boobo11bob
o7boobo21boobo12boo7boobo16bo4boobo31boobo4boo13bobb4o3boo12bo6boo3b3o
18bobb4obboobbobobbobbo16bobb5obbobobo22b3o8b4obbobobo30b3o3boboboboo$
22bobb4obbo19bo5b4o4boo18boo17boboboo4boo13boboboo4boo13boboboo4boo13b
oboboo4boo13boboboo4boo4boo7boboboo4boo13boboboo4boo13boboboo4boo5boob
o4boboboo4boo23bobboobbo15bobo4b3o14bobb4obbobbobbo16boobo4boboboo4boo
29bobobo20bobbo15bobobo25b5o3b3obbobobo$22bo6boo21b5o3bo3b3o14b4o4boo
15bobobobobbo15bobobobobbo15bobobobobbo15bobobobobbo15bobobobobbo15bob
obobobbo15bobobobobbo15bobobobobbo5booboo5bobobobobbo25bobbobboo12boob
oobbobbo13boobo4boo4boo17bobboobbobo34b5obboboboo17boo3bo7b6obboboboo
21bo6boo3boobbobo$23b6o4boo22b3o5boo14bobbo3b3o12bobbobobob3o13bobbobo
bob3o13bobbobobob3o13bobbobobob3o13bobbobobob3o13bobbobobob3o13bobbobo
bob3o13bobbobobob3o13bobbobobob3o27booboobbo13bobbobboo14bobboobbo27bo
bbobbo35bo6bobobo22boo7bo7bobobo22bobb4obbobo4boboboo$28bo3b3o19boobo
bb3o19boo5boo11bobobobboo16bobobobboo16bobobobboo16bobobobboo16bobobo
bboo16bobobobboo16bobobobboo16bobobobboo16bobobobboo32bobobbo10b3obboo
boobb3o13bobbobboo26booboo39b4obbobo34b5obbobo21boobo4bobob5obbobobo$
21b7o5boo20bo4bobbo20b3o13b3obobo4boo12b3obobo4boo12b3obobo4boo12b3obo
bo4boo12b3obobo4boo12b3obobo4boo12b3obobo4boo12b3obobo4boo12b3obobo4b
oo30bobo3b3o5bobbo4bobobbobbo14booboobbo26bobo40bo5boboboo31bo6boboboo
18bobboobbobo9bobobo$21bo6b3o22bobo5boo21bobbo11bo4bobb4obbo10bo4bobb
4obbo10bo4bobb4obbo10bo4bobb4obbo10bo4bobb4obbo10bo4bobb4obbo10bo4bobb
4obbo10bo4bobb4obbo10bo4bobb4obbo30bo6bo5boo3bobbobo3boo16bobobbo27bob
o43b3obbobobo33b4obbobobo19bobbobbo5b5obboboboo$24boobbobbo21boo30boo
12boo3bo6boo11boo3bo6boo11boo3bo6boo11boo3bo6boo11boo3bo6boo11boo3bo6b
oo11boo3bo6boo11boo3bo6boo11boo3bo6boo48boo3bo22bobo3b3o25bo44bo4bobob
o33bo5bobobo20booboo6bo6bobobo$24boo3boo74b6o4boo13b6o4boo13b6o4boo13b
6o4boo13b6o4boo13b6o4boo13b6o4boo13b6o4boo13b6o4boo73bo6bo71b4obbobob
oo31b5obboboboo18bobo10b4obbobo$110bo3b3o18bo3b3o18bo3b3o18bo3b3o18bo
3b3o18bo3b3o18bo3b3o18bo3b3o18bo3b3o158bobobo39bobobo19bobo10bo5bobob
oo$103b7o5boo11b7o5boo11b7o5boo11b7o5boo11b7o5boo11b7o5boo11b7o5boo11b
7o5boo11b7o5boo154b4obbobo34b5obbobo20bo14b3obbobobo$103bo6b3o15bo6b3o
15bo6b3o15bo6b3o15bo6b3o15bo6b3o15bo6b3o15bo6b3o15bo6b3o158bo5boboboo
32bo5boboboo32bo4bobobo$106boobbobbo17boobbobbo17boobbobbo17boobbobbo
17boobbobbo17boobbobbo17boobbobbo17boobbobbo17boobbobbo160b3obbobobo
29bo4b3obbobobo32b4obboboboo$106boo3boo18boo3boo18boo3boo18boo3boo18b
oo3boo18boo3boo18boo3boo18boo3boo18boo3boo161bo4bobobo29boo3bo4bobobo
38bobobo$475b4obboboboo32b4obboboboo31b4obbobo$481bobobo39bobobo32bo5b
oboboo$477b4obbobo35b4obbobo35b3obbobobo$477bo7boo34bo7boo34bo4bobobo$
480b4obo38b4obo36b4obboboboo$424boo6b3o45bo4bo38bo4bo24boo16bobobo$
391boo31bobo5b3o46b4o40b4o24bobo12b4obbobo$372bo18bobo32b3o4bo4boo24b
oo34boo8bo42bo14bo7boo$342boo3boo19boobobo19b3o29bo3bo7bobbo16boo4bobo
17boo9boo5bo6b3o16boo16boo5boo17b4obo$342boobbobbo14bobbobobobbo17bo3b
obbo21bobbob3obbooboobboo11boo5bo4bo19boo9bobo4boboobbo19boo16bobo3bo
19bo4bo$339bo6b3o15b4o3b3o14bobbob3obbobo20b4o4booboboboo13bobo4boboob
oo31bobboobobo3b3o36b3obboo18b4o$339b7o5boo8bo6b3o5boo10b4o4b3obbo16bo
6b3o4bobo4b3o11bobboobobo3bo29boobobo4b3o3bo34bo3boobbo$346bo3b3o8b7o
3bo3b3o7bo6b3o4boo17b7o3b4obob4obbo10boobobo4boobbo31bobb4o3boobbo29bo
bbob3obbobobbo16boo$341b6o4boo15b4o4boo7b7o3b4o4boo20b3o3bo3bobbo16bo
bb4obb3o31bo6b3obb3o29b4o4b3obb3o16boo$340bo6boo14b5o4boo18b3o3bo3b3o
15b5o3b3obbobboo17bo6boo35b5o4boo29bo6b3o4boo$340bobb4obbo12bo6b3obbo
12b5o3b3o5boo14bo6boo3booboobb3o15b6obbo39b4obbo28b7o3b4obbo$337boobob
o4boo13bobb4o3boo12bo6boo3b3o18bobb4obboobbobobbobbo20boboboboo28b7o3b
oboboboo31b3o3boboboboo$338bobboobobo12boobobo4b3o14bobb4obbobbobbo14b
oobobo4boobboo4boo16b7obbobobo29bo6b3obbobobo27b5o3b3obbobobo$336bobo
4boboo13bobboobobo3boo9boobobo4boo4boo16bobboobobo3bo24bo6boobbobo32b
4o3boobbobo26bo6boo3boobbobo$53boo281boo5bo14bobo4boboobbobbo9bobboobo
bo22bobo4boboo4boo25boobo4boboboo29bobbobbo4boboboo23bobb4obbobo4bobob
oo$52bob3o285boo14boo5bo6boo8bobo4boboo22boo5bo8bo25boob5obbobobo34b5o
bbobobo19boobobo4b3ob5obbobobo$48boobo5bo306boo16boo5bo31boo8bobo33bob
obo41bobobo20bobboobobo11bobobo$45boobbobob3obbo329boo42boo28b5obbobob
oo33b5obboboboo15bobo4boboob4ob5obboboboo$45bo3bobbo3boo404bo6bobobo
34bo6bobobo16boo5bo4bobbobo6bobobo$47bobo4bobo408b4obbobo37b4obbobo22b
oo4bobo5b4obbobo$43b4obob5obo211bo196bo5boboboo34bo5boboboo26bo6bo5bob
oboo$42bo3bobo6bo211bobo198b3obbobobo36b3obbobobo35b3obbobobo$42boo4b
4oboo210b3obo198bo4bobobo36bo4bobobo35bo4bobobo$40boobbobbo3bobo210bo
4boo198b4obboboboo34b4obboboboo33b4obboboboo$41bobobboboobobobobboo35b
oo166bobboo3bo203bobobo41bobobo40bobobo$41bob4obobboboobobbo35bo167bob
oob3o200b4obbobo37b4obbobo36b4obbobo$38boobo6bobo5boo38bo5boo137boobbo
3bo14bo206bo5boboboo34bo5boboboo33bo5boboboo$35boobbobob4obobb5o39boo
5bo88boo4boo43b4obbobo14boboo205b3obbobobo36b3obbobobo35b3obbobobo$35b
o3bobbo3bobo6bo41boo4bo87bobbobobbobbo18bo18bo5b3obbo11boboboo205bo4bo
bobo36bo4bobobo35bo4bobobo$37bobo4bobobb4o35boo3b4obbobboo88b3ob3obbob
o17b3o16b5o4boo12boo9boobo197b4obboboboo34b4obboboboo33b4obboboboo$33b
4obob5ob3o3bo35boobbo3bobobo98b3obbo14boo3bo20b4o4boo16bobboboo12bobbo
105boo23boo6b3o27boo17bobobo41bobobo22boo16bobobo$32bo3bobo6bo3bo42bob
oobboboboo90b4o4boo14bobb3o16boobbo3bo3b3o15bobobo15b4o45boo54boo3bo
23bobo5b3o27bobo12b4obbobo37b4obbobo21bobo12b4obbobo$32boo4b4oboobboo
bbo41bobboobbobobo88bo4b4o4boo11bobo4boo12bobbob3o5boo11boo3bobbo19boo
42bobbo3boo49bobbo26b3o4bo4boo24bo12bo7boo36bo7boo16boobbo14bo7boo$30b
oobbobbo3bobo4boboo43bo4bobobo87bobobbo3bo3b3o10boo5b3o8boo3boboo3b3o
13b3obo5boo12b6obbo42b3obbobbo23bo21b3o3boo17boo5bo3bo7bobbo23boo14b4o
bo19boobboo15b4obo17bobboo17b4obo$31bobobboboobobobobbo47b4obboboboo
85bobbob3o5boo11bobo4boo8bobboboo3bobobbo11bo4b6obbo10bo6boo48b3o16boo
bboobobo15boobbo3b3o19bobbobbob3obbooboobboo22boo16bo4bo15bobbobbobbo
15bo4bo18boo19bo4bo$31bob4obobboboobobo44b3o6bobobo87b3obobb3o14bobb3o
14boo4b3obobo13b4o6boo11b7o4boo39b5o5boo12bobbobobobbo14bobbob3o3boo
19b5o4booboboboo20bobbobboo15b4o12bobbobobboboobboo14b4o16b3obboo18b4o
$29b3o6bobo5boo45bobb6obbobo90bo3bobbo14boo3bo14bob3o5bo12bobo3b6o4boo
13bo3b3o38bo5bo3b3o13b3o3b3o16b3o4b3obbo23b3o4bobo4b3o12bobbobobbobobb
o29b4ob4o3boobbo28boobbo3boobbo$28bo4b4obobb5o49bo8boboboo86bo5boo17b
3o13bobobobbo17boo3bo5bo3b3o7boob3o5boo38b7o4boo16b3o5boo15b3o4boo19b
5o3b4obob4obbo12b4ob4obb3o15boo21b3obb3o15boo11bobbob3obbobobbo16boo$
24boobbobboo3bobo6bo50b3obb3obbobobo85boo23bo15boo4boo22boob3o5boo8bob
obb3o49boo15b5o3bo3b3o10b5o3b4o4boo14bo5b3o3bo3bobbo24boo18boo14b6o4b
oo18boo12b3o4b3obb3o16boo$24bobobobobobobobb4o55bobbo4bobobo157bobobb
3o12bobbobobbo39b4ob4obbo13bo5b4o4boo9bo5b3o3bo3b3o14b6o3b3obbobboo18b
7obbo33bo5b4obbo34b3o4boo$26bobobobboob3o3bo59b7oboboo152bobobbobobbo
12bobobboo40bobbobobbobo14b6o4boo13b6o3b3o5boo21boo3booboobb3o15bo5bob
oboboo30b5o3boboboboo25b5o3b4obbo$25boobobo4bo3bo68bobobo153boobboobob
o14boo48bobbobboo20b3obbo19boo3b3o16b4ob4obboobbobobbobbo16b5obbobobo
36b3obbobobo25bo5b3o3boboboboo$28bobb4obb3o64b4obbobo161bo69boo13b4ob
4o3boo11b4ob4obbobbobbo15bobbobobbobobboo4boo24boobbobo33b3o3boobbobo
25b6o3b3obbobobo$28bo6boo67bo5boboboo230boo11bobbobobboboo13bobbobobbo
bo4boo20bobbobbobo27b3obo4boboboo29bo3bobo4boboboo29boo3boobbobo$29b6o
bbo69b3obbobobo229bo16bobbobbobb3o13bobbobbobo28booboo26bobbob5obbobob
o28booboob5obbobobo19b4ob4obb3o4boboboo$34boboboboo65bo4bobobo227bobo
20boobboobbo16booboo29bo30boo9bobobo41bobobo19bobbobobbobo3b4obbobobo$
27b7obbobobo67b4obboboboo224boo28boo17bo30bobo36b5obboboboo33b5obbobob
oo20bobbobbobo7bobobo$27bo6boobbobo73bobobo272bobo30boo37bo6bobobo34bo
6bobobo25booboob6obboboboo$28b3obbo4boboboo44bobo19b4obbobo272boo73b4o
bbobo37b4obbobo26bo3bobo6bobobo$30bobb5obbobobo44boo19bo5boboboo344bo
5boboboo34bo5boboboo21boboboo5b4obbobo$40bobobo44bo23b3obbobobo346b3o
bbobobo36b3obbobobo20boobbo6bo5boboboo$35b5obboboboo65bo4bobobo346bo4b
obobo36bo4bobobo24bobo7b3obbobobo$35bo6bobobo50booboobo10b4obboboboo
344b4obboboboo34b4obboboboo22boo7bo4bobobo$38b4obbobo51boboboo16bobobo
351bobobo41bobobo33b4obboboboo$38bo5boboboo48bo17b4obbobo347b4obbobo
37b4obbobo39bobobo$41b3obbobobo44boobo17bo5boboboo3boo339bo5boboboo34b
o5boboboo32b4obbobo$41bo4bobobo45boboo19b3obbobobobobo342b3obbobobo36b
3obbobobo31bo5boboboo$42b4obboboboo34boo3b3o23bo4bobobobo344bo4bobobo
36bo4bobobo34b3obbobobo$48bobobo35boobbo27b4obboboboo344b4obboboboo34b
4obboboboo31bo4bobobo$44b4obbobo39boboo30bobo353bobobo41bobobo33b4obbo
boboo$44bo5boboboo37bobbo25b4obboboo346b4obbobo37b4obbobo39bobobo$47b
3obbobobo38boo25bo5boboo346bo7boo36bo7boo34b4obbobo$47bo4bobobo68b3o
353b4obo40b4obo35bo7boo$48b4obboboboo64bo356bo4bo40bo4bo38b4obo$54bobo
bo65boo106boo4boo242b4o42b4o39bo4bo$50b4obbobo174bobb3o333b4o$50bo7boo
173bobo4bo243boo44boo$53b4obo171boobobobboobo242boo44boo42boo$53bo4bo
140boo8boo20bobobobobob3o330boo$54b4o142bo8boo20bobobobobbo3bo$200bobo
23boobboobobobo3b3o118boo95boo36boo$56boo143boo23bobbobbobobbobobo98b
oo20bobo5boo88bo4boo31bobboo4bo$56boo169boobobobobobobo98bobbo21bo4bo
bbo86bo4bobbo30bobobbobbobo$221boo5bobobobobobbo99b3o21boobobb3o87b4o
bb3o29boob4obbobobbo$195boo24boo5bobobboboboo98boo5boo20b3o5boo87boo
39b3obb3o$196bo32bobobobobo98bobbo3b3o15b5o3bo3b3o83b4obbo31b6o4boo$
196bobo32bobobobbo97b4o4boo14bo5b4o4boo83bobboboboboo27bo5b4obbo$197b
oo31bobboobobo101boo18b6o4boo88boobbobobo29b5o3boboboboo$226boobb3o3b
oo98b4obbo24b3obbo84b3obboobbobo34b3obbobobo$226b3o4bobo100bobbobo18b
oob4obbobo84bobbobo4boboboo28boobobboobbobo$226boobb3o3boo102bo20bobo
bbo3bo86bobob5obbobobo28bobobo4boboboo$230bobboobobo122bobboo92bo9bobo
bo28bobob5obbobobo$231bobobobbo121boo101b5obboboboo26bo9bobobo$229bobo
bobobo225bo6bobobo32b5obboboboo$197bo23boo5bobobboboboo227b4obbobo32bo
6bobobo$197bobo21boo5bobobobobobbo226bo5boboboo32b4obbobo$197b3o27boob
obobobobobo228b3obbobobo31bo5boboboo$199bo26bobbobbobobbobo229bo4bobob
o34b3obbobobo$226boobboobobobo232b4obboboboo31bo4bobobo$209boo20bobobo
bo238bobobo33b4obboboboo$209boobboo16bobobobobo232b4obbobo39bobobo$
213bobo14boobobobboo232bo5boboboo32b4obbobo$190boo23bo17bobo4boo233b3o
bbobobo31bo5boboboo$191bo23boo16bobb3obbo233bo4bobobo34b3obbobobo$188b
3o41boo4boo236b4obboboboo31bo4bobobo$188bo293bobobo33b4obboboboo$478b
4obbobo39bobobo$478bo7boo34b4obbobo$481b4obo35bo7boo$481bo4bo38b4obo$
482b4o39bo4bo$526b4o$484boo$484boo42boo$528boo9$111bobbo26bobbo26bobbo
26bobbo$109b6o24b6o24b6o24b6o$105boobo26boobo26boobo26boobo$105boobobo
b5o18boobobob5o18boobobob5o18boobobob5o$108bobo6bo20bobo6bo20bobo6bo
20bobo6bo$108bobobb5o20bobobb5o20bobobb5o20bobobb5o$109boobo7bo18boobo
7bo18boobo7bo18boobo7bo$112bobb6o21bobb6o21bobb6o21bobb6o$112bobo27bob
o27bobo27bobo$111boobobb6o18boobobb6o18boobobb6o18boobobb6o$114bobo6bo
20bobo6bo20bobo6bo20bobo6bo$114bobobb5o20bobobb5o20bobobb5o20bobobb5o$
115boobo7bo18boobo7bo18boobo7bo18boobo7bo$118bobb6o21bobb6o21bobb6o21b
obb6o$118bobo27bobo27bobo27bobo$117boobobb6o18boobobb6o18boobobb6o18b
oobobb6o$120bobo6bo20bobo6bo20bobo6bo20bobo6bo$120bobobb5o20bobobb5o
20bobobb5o20bobobb5o$121boobo7bo18boobo7bo18boobo7bo18boobo7bo$124bobb
6o21bobb6o21bobb6o21bobb6o$124bobo27bobo27bobo27bobo$123boobobb6o18boo
bobb6o18boobobb6o18boobobb6o$126bobo6bo20bobo6bo20bobo6bo20bobo6bo$
126bobobb5o20bobobb5o20bobobb5o20bobobb5o$127boobo7bo18boobo7bo18boobo
7bo18boobo7bo$130bobb6o21bobb6o21bobb6o21bobb6o$130bobo27bobo27bobo27b
obo$129boobobb6o18boobobb6o18boobobb6o18boobobb6o$132bobo6bo3boo15bobo
6bo3boo15bobo6bo20bobo6bo$132bobobb5obbobbo14bobobb5obbobbo14bobobb5o
20bobobb5o$133boobo7b3obo14boobo7b3obo14boobo7bo18boobo7bo$136bobb5o3b
oboboo13bobb5o3boboboo13bobb6o21bobb6o$136bobo6bo3bobo14bobo6bo3bobo
14bobo27bobo$135boobobb6obbobo13boobobb6obbobo13boobobb6o18boobobb6o$
138bobo6bobobobo14bobo6bobobobo14bobo6bo20bobo6bo$138bobobb4obobboobo
13bobobb4obobboobo13bobobb5o20bobobb5o$139boobo3bobobo3bo14boobo3bobob
o3bo14boobo7bo18boobo7bo$142bobo3bobobooboo15bobo3bobobooboo15bobb6o8b
o12bobb6o$139b3obob4obobbobo13b3obob4obobbobo16bobo14b3obo8bobo15boo$
139bo3bo7bobobbo12bo3bo7bobobbo14boobobb6o9boo7boobobb6o8bo$143b8obb3o
17b8obb3o18bobo6boboobboo13bobo6boboo3bo$150bobo27bobo21bobobb5oboobbo
b3o10bobobb5oboo3boo$141b7obboboo17b7obboboo21boobo15bo10boobo$140bo7b
oboobbo15bo7boboobbo23bobb6o6boo13bobb6o10boo$140b6obbo4boo6bo8b6obbo
4boo23bobo5bo5bo15bobo5bo5boobobbo$129boo15bobo8bob3o14bobo28boobobb3o
5boboo12boobobb3o5bobob3o$129bo8b6obboboo7boo9b6obboboo30bobo5bobbobo
16bobo5bobbobo$130bo3boobo6bobo13boobboobo6bobo33bobbo4b4obo16bobbo4b
4ob3o$129boo3boob5obbobo10b3obobboob5obbobo34boo9bo18boo9bo3bo$142bob
oo10bo15boboo42b4o26b3obobboo$122boo10b6obbo13boo6b6obbo46bo29bobo$
122bobboboo5bo5bobo15bo5bo5bobo44bo29bo$123b3obobo5b3obboboo12boobo5b
3obboboo43boo28boo$127bobobbo5bobo16bobobbo5bobo$125b3ob4o4bobbo16bob
4o4bobbo$124bo3bo9boo18bo9boo$124boobbob3o26b4o$129bobo29bo$133bo29bo$
132boo28boo9$252bobbo$250b6o$171bobbo71boobo$169b6o40bobbo27boobobob5o
$130bobbo31boobo44b6o30bobo6bo$128b6o31boobobob5o32boobo36bobobb5o$
124boobo40bobo6bo31boobobob5o29boobo7bo$89bobbo31boobobob5o32bobobb5o
34bobo6bo31bobb6o$87b6o34bobo6bo32boobo7bo31bobobb5o31bobo$83boobo40bo
bobb5o35bobb6o32boobo7bo27boobobb6o$51bobbo28boobobob5o33boobo7bo32bob
o41bobb6o30bobo6bo$49b6o31bobo6bo35bobb6o31boobobb6o33bobo36bobobb5o$
45boobo37bobobb5o35bobo40bobo6bo31boobobb6o29boobo7bo$9bobbo32boobobob
5o30boobo7bo31boobobb6o32bobobb5o34bobo6bo31bobb6o$7b6o35bobo6bo32bobb
6o34bobo6bo32boobo7bo31bobobb5o31bobo$3boobo41bobobb5o32bobo40bobobb5o
35bobb6o32boobo7bo27boobobb6o$3boobobob5o34boobo7bo28boobobb6o33boobo
7bo32bobo41bobb6o30bobo6bo$6bobo6bo36bobb6o31bobo6bo35bobb6o31boobobb
6o33bobo36bobobb5o$6bobobb5o36bobo37bobobb5o35bobo40bobo6bo31boobobb6o
29boobo7bo$7boobo7bo32boobobb6o30boobo7bo31boobobb6o32bobobb5o34bobo6b
o31bobb6o$10bobb6o35bobo6bo32bobb6o34bobo6bo32boobo7bo31bobobb5o31bobo
$10bobo41bobobb5o32bobo40bobobb5o35bobb6o32boobo7bo27boobobb6o$9boobo
bb6o34boobo7bo28boobobb6o33boobo7bo32bobo41bobb8o28bobo6boboo$12bobo6b
o36bobb6o31bobo6bo35bobb6o31boobobb4o35bobo8bo27bobobboobbobo$12bobobb
5o36bobo37bobobb5o35bobo40bobo4bo33boobobb4obobo27boobobbobobbo$13boob
o7bo32boobobb6o30boobo7bo31boobobb4o34bobobboobo3boo27bobbobo3bobobo
30bobobboboo$16bobb6o35bobo6bo32bobb6o34bobo4bo34boobo3bo3bo30boobobob
oboboo29bob4o$16bobo41bobobb5o32bobo40bobobboobo36bobooboobo31bobboobo
bbo32bo4boobo$15boobobb6o8bo25boobo7bo28boobobb4o35boobo3bo32boboobobo
bobo32bo4bobobo33b3oboboo$18bobo6boboobobobo27bobb6o31bobo4bo37boboob
oo30boobobobobo35b4obboboo34boo$18bobobb5oboboobobo27bobo37bobobboobo
33b3obobobobo34bobobbo36bo3bobo$19boobo10bobo27boobobb4o30boobobo3bo
32bo4bobobobbo32bobbo3bo37bobobo$22bobb4o4bo29bobbobo4bo32bobobooboo
31b3oboboboboobbo29boobobob3o34booboo$22bobo4bobboo30b3obobboobbo30bo
bbobobo34bobobobo3b3o31boboo4bo$21boobobboobo37boobboboo31bobobobobo
34bobobbobo34bo4b3o$22bobobo3b4o30boobo4bo33booboboboobo33bob4oboo34b
3obo$22boboboboo4bo30boboobobo36bobo4bo34bo43boo$23bobobobob3o31bobbob
oo37bobob3o37bo$25bobobobo34boo42boobo38boo$24boboobbo$24bo3boo$22boob
oo$23bobo$23bobo$24bo7$182boo$182bo$184bo$114boo67boo$114bo67bo6bo$
115b3o3boo24boo3boo29b4obobo$117bobbobbo23boobbobbo31bobobbo$120b3o28b
3o29boo3b3o$115b5o5boo18b6o5boo25bob3o5boo$114bo5bo3b3o17bo6bo3b3o23b
oo5bo3b3o$114b7o4boo15bobb7o4boo22bobb6o4boo$121boo19boobo6boo26bobo6b
oo$114boob4obbo21bob5obbo24booboobb3obbo$115bobobbobo22boo4bobo25bo5bo
bbobo$115bobbo3boo24bobobboo25booboobbobboo$116boo4bo22booboo3bo27bobo
4boo$124bo20bobo7bo25bobo6boo$123boo22bo6boo26bo7bo$147boo39bobo$188b
oo!

The LifeViewer is pretty good for looking at these -- space bar and 'b' keyboard shortcuts highly recommended. Look at how far some of those signals can travel before they fizzle out -- including across the 2c/3 wire-end boundary! And most of those paths, not counting initial traffic-light stages, could easily accept another signal within 19 ticks.

All someone has to do to prove omniperiodicity is find a way to adapt any one of these, or any similar traveling signal, to allow them to be chained linearly at any angle, and also to turn a 90-degree corner. That's an awful lot of possibilities that would all have to be ruled out somehow before a proof of the nonexistence of p19 oscillators would be within reach.
User avatar
dvgrn
Moderator
 
Posts: 4570
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI


Return to Patterns

Who is online

Users browsing this forum: Google [Bot] and 8 guests