Breeders

gmc_nxtman · Post by **gmc_nxtman** » June 17th, 2015, 11:08 pm

Kiran wrote:What is the precise definition of an "engine"?
My "slab" definition is, at least, well defined in the sense that there is no ambiguity.

In this case, the definition of "engine" is an object which produces other objects at any rate, as long as it's not zero. Breeders can be a lot of things, but my definition is simply

"Any two stage process of objects producing other objects unboundedly"

This makes linear-growth patterns NOT breeders, and, gott's dots a breeder.
Even logarithmically producing caber tossers is a breeder.

Also, has it been proven impossible for exponential growth to exist in Conway's game of life? Linear is steady growth, quadratic is the growth rate increases, and logarithmic is where it decreases. But is exponential possible?

Kiran · Post by **Kiran** » June 17th, 2015, 11:20 pm

But is exponential possible?

No, a finite pattern's building box cannot expand faster then the speed of light in any life-like cellular automaton.
Therefore no pattern can grow in population faster then the filled X*Y building box in the rule B12345678/S012345678.
Therefore the upper limit for the population of a pattern that fits in an X*Y building box is X*Y+t*2(X+Y)+t^2*4 a quadratic polynomial.
You can check the formula by running solid rectangles in that rule.
There is another explanation here:
http://www.conwaylife.com/wiki/Talk:Infinite_growth
EDIT:
Define "creating other objects."
You could say the LWSS creates a tail spark, so is a LWSS gun a breeder?
For the same reason you can say this puffer is a breeder:

Code: Select all

x = 139, y = 190, rule = B3/S23
41bo13bo$40b3o11b3o$40bob2o4bo5bob2o14bo13bo$41b3o3b3o5b3o13b3o11b3o$
41b2o3b2o2bo4b2o13b2obo5bo4b2obo$48b3o19b3o5b3o3b3o$71b2o4bo2b2o3b2o$
77b3o2$54b3o$53bo2bo$56bo14b3o$56bo14bo2bo$53bobo15bo$45b2o24bo$45b2o
25bobo$81b2o$39bo41b2o$38b3o7bo$37b2obo5b2ob2o37bo$37b3o6b2ob2o28bo7b
3o$38b2o6b2ob2o26b2ob2o5bob2o$48b2o4bo5bo16b2ob2o6b3o$53b3o3b3o15b2ob
2o6b2o$52b2obo3bob2o4bo5bo4b2o$52b3o5b3o3b3o3b3o$53b2o5b2o3b2obo3bob2o
$65b3o5b3o$46b2o9bo8b2o5b2o$47b2o7b3o$46bo8bo3bo10bo9b2o$55bo3bo9b3o7b
2o$22b3o11b3o29bo3bo8bo$21bo2bo10bo2bo12b2o2bo3bo8bo3bo$24bo4b3o6bo13b
2obo3bo29b3o11b3o$24bo4bo2bo5bo12bo3b2ob2o8bo3bo2b2o12bo2bo10bo2bo$21b
obo4bo3bo2bobo18bobo9bo3bob2o13bo6b3o4bo$28bobob2o23bo10b2ob2o3bo12bo
5bo2bo4bo$29b2ob2o35bobo18bobo2bo3bo4bobo$30b3o37bo23b2obobo$23bo70b2o
b2o$22b3o70b3o$21b2obo6bo72bo$21b3o6b2o26bo44b3o$22b2o7bo24b2ob2o35bo
6bob2o$28bo27b2ob2o8bo26b2o6b3o$26b4obo18bo6b3o7b2ob2o24bo7b2o$27bo2bo
18b3o15b2ob2o27bo$28b3o18bob2o15b3o6bo18bob4o$50b3o23b3o18bo2bo$38b3o
9b3o22b2obo18b3o$3bo13bo19bo2bo9b2o23b3o$2b3o11b3o21bo34b3o9b3o$b2obo
5bo4b2obo21bo35b2o9bo2bo$b3o5b3o3b3o19bobo47bo$2b2o5bob2o3b2o69bo$88bo
bo18b3o11b3o$35bo7b3o3b3o56bo2bo10bo2bo$36b2o5bo2bobo2bo59bo4b3o6bo$
12bo22b2o6bo7bo24b3o3b3o7bo18bo4bo2bo5bo$10b2o4b3o24bo7bo24bo2bobo2bo
5b2o16bobo4bo3bo2bobo$16bo2bo24bobobobo25bo7bo6b2o22bobob2o$13bo2bo59b
o7bo31b2ob2o$8bo7bo30bo29bobobobo33b3o$7bo3b2o4bobo26b3o8b2o51bo$7bo
37b2ob2o7bobo20bo28b3o$8bo36b2ob2o8b2o9b2o8b3o26b2obo6bo$18b3o3b3o19bo
bo19bobo7b2ob2o25b3o6b2o$bo9bo5bo2bo3bo2bo19bo20b2o8b2ob2o26b2o7bo$3o
6b3o8bo3bo54bobo33bo$ob2o5bo10bo3bo55bo32b4obo$b3o5bobo5bobo5bobo17b2o
b2o7b2o43bo5bo5bo2bo$b2o7b2o28bo3bo5bo5bo2bo41b3o3b3o5b3o$21b3o17b2o2b
o3bo7b2o10b2o7b2ob2o17b2obo3bob2o14b3o$21b3o16b2o4b3o19bo2bo5bo5bo3bo
12b3o5b3o14bo2bo$21bobo22bobo20b2o7bo3bo2b2o14b2o5b2o15bo$22bo24bo31b
3o4b2o37bo$22bo24bo31bobo23bo20bobo$8bo71bo23b3o$8b2o20bo49bo22bo3bo$
7bobo10bo3bo4b3o71bo3bo$20b2ob2o4bob2o64bo$21bobo6b3o63b3o4bo3bo11b3o$
13bo16b2o63b2obo4bo3bo11bo$13b2o7bo72b3o5b2ob2o12bo$12bobo7bo73b2o6bob
o$22bo82bo$45bo68b3o$18bo27b2o66bo$18b2o25b2o35bo32bo$17bobo60b2o$81b
2o$15bo93b3o$14b3o6bo85bo$14bob2o5b2o32b2o51bo$15b3o4bobo22b2o8bobo$
15b3o29b2o9b2o9b2o48b3o11b3o$15b2o51bobo8b2o23b3o11bo2bo10bo2bo$68b2o
9b2o23bo16bo4b3o6bo$105bo15bo4bo2bo5bo$118bobo4bo6bobo$47bobo7b2o71bo$
47bobo6bo2bo64bo4bo$46bo10b2o10b2o7bobo43bo$46b3o19bo2bo6bobo39bo5bo$
42b2o3b3obo17b2o10bo37b3o3b3o$42b2o6b2o27b3o36b2obo3b3o$50b2o24bob3o3b
2o20b2o10b3o4b2o$38bo37b2o6b2o20b2o11b2o5b5o$38b2o36b2o34bo15b3o$37bob
o71b3o$110b2obo4bo6bo$89b3o18b3o4b3o5bo$89bo20b3o4bob2o6bo7b3o$90bo19b
3o5b3o6b2o6bo2bo$111b2o5b2o6b2o7bo$126bo8bo$115bo20bobo$114b3o$113bo3b
o3$93bo13bo5b2ob2o$46b2o5bo38b3o11b3o6bo12b3o$46b2o5b2o2b2o33bob2o4bo
5bob2o18bo$52bobo2bobo20b2o11b3o3b3o5b3o19bo$58b2o9b2o9b2o11b2o3b2o2bo
4b2o4bo3bo$68bobo3b3o23b3o9bobobobo$68b2o4bo38b5o5b3o$75bo37b2ob2o5bo$
114bobo7bo$106b3o5b3o$105bo2bo$108bo9b3o$108bo9bo$105bobo11bo$97b2o$
97b2o23bo$121b3o$91bo24b2o3bob2o$90b3o7bo15b3o3b3o$89b2obo5b2ob2o19b3o
$89b3o6b2ob2o19b2o$90b2o6b2ob2o$100b2o4$69b2o32b3o$70b2o31bo$69bo28b2o
4bo4bo$58b2o39b2o7b3o$57bo3bo36bo9bob2o$46b2o8bo5bo7bo38b3o$46b2o8bo3b
ob2o4bobo38b2o$56bo5bo3b2o12b2o21b2o$57bo3bo4b2o12b2o22b2o$58b2o6b2o
35bo$68bobo$70bo4$84b2o$85b2o$84bo5$77bo$77b2o$76bobo6$69b2o$70b2o$69b
o$58b2o$57bo3bo$46b2o8bo5bo7bo$46b2o8bo3bob2o4bobo$56bo5bo3b2o12b2o$
57bo3bo4b2o12b2o$58b2o6b2o$68bobo$70bo!

Or even this oscillator:

Code: Select all

x = 402, y = 145, rule = B3/S23
349bo$347b3o$346bo$346b2o$151bobo30b2o$151bo3bo28bobo157bo$135bo19bo
23b2o4b3o12bo142bobo$100bo32b4o14bo4bo4b2o12b4o2bo4b3o8b4o4bo136bo3bo$
98b3o31bobob2o4bo12bo5b2o12b3ob2o4b3o8b4o5bo137b3o$97bo29b2o2bo2bob3o
5b2obo3bo3bo28bobo9bo2bo9b2o130b2o3b2o$97b2o9b2o17b2o3bobob2o5b4obo2bo
bo30b2o10b4o9b2o$74bo33b2o23b4o6b2o2b2o39bo8b4o122b2o$74b3o58bo27b2o8b
2o12bo12bo93bo29bo$77bo85b2o8b2o12b3o101b4o29bobo6bobo$65b2o9b2o95b2o
115b4o16bo14b2o5bo2bo4bo$65b2o106bo11b2o96b2o5bo2bo8b2o5bobo19b2o5b2o$
92b2o3b2o54b2o9b2o6bobo10bobo95b2o5b4o14bo3b2o15b2o3bo9bo$92bo5bo54b2o
7bob2o6bob2o9bo105b4o3b2obobo4bo3b2o17b2o11bo$162bo131bo2bo3b2o5bo3b2o
18bo2bo$93bo3bo51bo12bo17bo117bo10bobo3b2o16bobo18b2o$94b3o34b2o16b2o
11bo2bo7b2o5bobo98b2o15bo2bo8bo4bobo37bo$64b3o11b3o50b2o15bobo12b2o8b
2o5b2o98bobo34bo37bobo10bo$63b2ob2o9bo3bo25b5o92b2o73b3o35b2o37b2o9bob
o$63b2ob2o8bo5bo23bob3obo79b3o2bo5bo3bo71b2o86b2obo7b2o$63b5o8bo5bo24b
o3bo76bo3bo9bo5bo73b2o7bo61b2o12b2ob2o6bobo$62b2o3b2o10bo28b3o75bobo4b
o8bo3bob2o2b2o67b3o6bo61b2o13b2obob3o6bo$77bo3bo27bo21bo52b2o16bo5bo3b
2o76b3o2b3o56bo12bobo2bo2bo2bo2bo7b2o$78b3o17bo30b2ob2o7b2o41b2o17bo3b
o87bo72bo4b2o6bo7b2o$64b2o13bo16bobo43b2o40b2o18b2o75b2o13bo81bobo$63b
5o29b2o29bo5bo6bo37b2o5bobo92b2o95b2o$63bo47bo66bobo7bo120b2o$65b3o12b
3o10bo17bo16b2obob2o43bo99b2o24bo4b2o42b2o10bobo$67bo12b3o9b3o15bobo
63b3o99bo19b2ob5o6b2o39b2o10b2o$65b2o12bo3bo7b5o13b2ob2o46b2o13bo93bo
6bobo18bo4b2o2bo5b3o5b2o44bo$65b2o6bobo14b2o3b2o6b2o3bo5bo19b2o24bo4bo
9b2o92bobo4b2o18bo8b2o5b2o6b2o40b2o$73b2o3b2o3b2o6b5o7b2o6bo21bobo15bo
6bobo4bobo61b2o39bobo23bo7bo4b2o43b2o6b2o22b2o$74bo16bo3bo12b2o3b2o19b
o16bobo4b2o5b2o5b3o54b2o26b2o11bo2bo22bo12b2o41bob2o30bo$60b2o3b2o25bo
bo38b2o17bobo17b3o33b2o16b2o10bo4bo13b2o11bobo14b2o8bo54bo16bo7bo6bobo
$60bobobobo3b2o19b3o39b3o3b2o11bo2bo52b2o8b2o5b3o10bo4bobo23bobo14bobo
9b2o52bo15b4o5bobo4b2o$61b5o4b2o18b2o5bo14bo21b2o3b2o11bobo63b2o6b2o
10bo7b2o21bo16bo65bo2bo7b2o2bo2b2o8b2o$62b3o26bo4bo15bo19bo18bobo11bo
63b2o2b2o11b2o4b2o31b2o66b2o8b2o2b2o11b2o$63bo24b3o5b3o14bo37bo12bo5b
2o3b2o52b2o2bo2b2o8b2o4b2o106b2o10bo7b2o$88bo75b3o4b5o58b4o5bobo113b3o
10bo4bobo$83b2o87b3o41b2o17bo7bo116b2o10bo4bo$83bo26b2o21bo17bobo19bo
42b2o136b2o7b2o$84b3o14bo8b2o20b3o16b2o200bobo7b2o$60b2o17bo6bo10b2o2b
2ob3o24b5o16bo63bo60b2o74bo$61bo18b2o15b2o4b4o23b2o3b2o78bobo8bobo3bo
44bobo72b2o$58b3o18b2o20b2o28b5o71b3o5bo2bo7b2o3b2o45b3o4b3o75b2o$58bo
72bo3bo70bo3bo5bo2bo7bo3bobo21b2o22b2o4bo78bo$34bo30bo66bobo32b3o5b2o
28bo5bo43bo20b2o8bo77bobo5b2o$32b3o30b3o65bo28b2o3bo7bo30bo3bo5bo2bo
24b2o7bobo20b3o86b2o5bobo$31bo36bo93bobo3bo7b3o28b3o6b2o26bobo6b2o118b
3o12bo$31b2o9b2o12b2o9b2o5bo88b3o12bo28b3o24bo4b2o6bo37b3o86b3o8b4o$
42b2o12b2o16b3o56b2o29b2o67bobo2bo2bo2bo2bo13bo15b2o8bo85b3o8b4o9b2o$
71bo5bo55b2o26b2o10b2o58b2obob3o6bo12bo16b2o7bo6b2o69b2o6bobo9bo2bo9b
2o$71bo4b2o83b3o9b2o58b2ob2o6bobo13b3o30bobo67bobo6b2o10b4o5bo$233b2ob
o7b2o8bobo36bo68b3o11bo8b4o4bo$80bo124b3o15b2o8bobo18b2o106b2o11bo12bo
$43bo23b2o3b2o6bo81b2o40b2ob2o3bo9bobo9bo20bo109b2o8b3o$29bo13bo24b5o
6bo82b2o40b2ob2o4b2o7bo55b2o84b3o$28b3o11bobo24b3o22bo109b5o3b2o7b2o
54bobo$27b5o9b2ob2o24bo24b2o69b2o35b2o3b2o69bo$26b2o3b2o7bo5bo6b2obob
2o16b2o3b2o11b2o69bobo100bo20b2o9b3o9b2o31b2o17b2o$27b5o11bo32bo5bo52b
o31bo99b4o18bo10bo9bo2bo30b3o17b2o$27bo3bo8b2o3b2o6bo5bo75bobo44b3o81b
2obobo6bobo6bobo5bo5bo7bo34b2obo$28bobo46bo3bo53b2o45bo82b3obo2bo3bo3b
o6b2o4b4o12bo35b3o44b2o$29bo24b2ob2o19b3o102bo82b2obobo4bo15bobob2o11b
o36bo33b3o2bo5bo3bo$42b4o10bo92bo8b2o14b2o19bo61b2o8b4o4bo4bo10bo2bob
3o11bo2bo5b2o55bo3bo9bo5bo3b2o$41b2o2bo23bo77bobo6bo2bo14bo20b2o11b2o
46bobo9bo7bo14b2obob2o14b2o5bobo52bobo4bo8bo3bob2o2b2o$27bo13b2obo22b
2o66bo10bobo7bo3bo2b2o7bobo9bo5b2o4b2o10bo47bo19bo3bo8b3ob4o24bo50b2o
16bo5bo$25b2ob2o38b2o24bobo37bo10bo2bo7bo2b2o2b3o6b2o10bobo3b2o4b3o10b
3o43b2o21bobo7bobo4bo25b2o49b2o17bo3bo$34bo7b2o37b2o12b2o37b3o9bobo16b
2obo16bobo2b2o4b2o13bo76bo83b2o18b2o$24bo5bo4bo6bo38bo13bo46b2o3bobo4b
3o8bo2bo16bo2bo6b2o7b2o81b2o78b2o5bobo$33b3o46b3o36bobo17bobo5bo15b2ob
o16bobo7bo8bobo159bobo7bo$24b2obob2o22b3o28bo36b2o18bo21b3o8b2o8bobo
19bo159bo$42b2obob2o3bo3bo12b2o3b2o33bo12bo17b2o21b2o8bobo8bo21b2o157b
2o$36bo5bo5bo12b2o8b3o36b2o61bo$35bo7bo3bo3bo5bo3b2o7bo3bo34b2o61b2o$
35b3o6b3o4b2o3b2o13bobo169b2o58b2o$72bo169b4o56b4o$242b2ob2o55b2ob2o$
24b2o28bo11bobo5b2o65b2o58b2o41b2o15b2o41b2o15b2o$25bo27bobo11b2o5bo
64b2ob2o55b2ob2o55b2ob2o55b2ob2o$22b3o28bobo11bo7b3o61b4o56b4o56b4o56b
4o$22bo30bo23bo62b2o58b2o58b2o58b2o$53bo67bo59bo59bo59bo$44b2o7bo2bo
63b3o3b2o52b3o3b2o52b3o3b2o52b3o3b2o38bo$35bo8b2o8b2o4bo2b2o54b5ob3o9b
3o39b5ob3o9b3o39b5ob3o9b3o39b5ob3o9b3o25bo$31b2o2b2ob3o18bo3b2o2b2o54b
3o9bo3b2o42b3o9bo3b2o31bobo8b3o9bo3b2o31bobo8b3o9bo3b2o24b3o$31b2o4b4o
18bo7b2o54b3o9bobo2b2o41b3o9bobo2b2o31b2o8b3o9bobo2b2o31b2o8b3o9bobo2b
2o$35b2o23b4o60bo9bo5bo43bo9bo5bo32bo10bo9bo5bo32bo10bo9bo5bo21b2o$21b
o101bo11bo2b2o43bo11bo2b2o43bo11bo2b2o43bo11bo2b2o22b2o$20bo114bo59bo
59bo59bo$20b3o114bo59bo59bo29bobo27bo$70bo2bo56bo2bo56bo2bo56bo2bo34b
2o20bo2bo$12bo61bo6b2o51bo6b2o51bo6b2o51bo6b2o25bo25bo6b2o$12bobo55bo
3bo4b2ob2o46bo3bo4b2ob2o46bo3bo4b2ob2o46bo3bo4b2ob2o46bo3bo4b2ob2o$2o
11bobo55b4o4b4o48b4o4b4o48b4o4b4o48b4o4b4o48b4o4b4o$2o11bo2bo63b2o58b
2o58b2o58b2o20bobo35b2o$13bobo267b2o$12bobo268bo$12bo50b2o181b2o8b2o8b
2o8b2o8b2o8b2o8b2o8b2o8b2o8b2o$63bo28bo153bobo7bobo7bobo7bobo7bobo7bob
o7bobo7bobo7bobo7bobo$55bo5bobo23b3ob2o2b2o151bo9bo9bo9bo9bo9bo9bo9bo
9bo9bo$16bo37b2o5b2o24b4o4b2o151b2o8b2o8b2o8b2o8b2o8b2o8b2o8b2o8b2o8b
2o$16bo21b2o13b2o36b2o7b2o$15bobo20b3o11b3o45b2o$14b2ob2o21b2obo9b2o3b
2o$13bo5bo20bo2bo10b2o22bo$16bo23b2obo11bo6b3o13b3o$13b2o3b2o11b2o5b3o
10bo12bo3b2o11bo$30bobo5b2o9bobo11bo3bobo10b2o$30bo19b2o14b3o31b3o$15b
o13b2o35b2o22b3o$15bo53b2o19bo9bobo$14bo53b3o20bo7b5o$56b2o40b2o3b2o$
56b2o40b2o3b2o$16b2o50b2o20b3o$16b2o27bo22b2o12b3o7bo$44b4o33bo3bo5bo
6b2o$32b2o9b2obobo6bobo22bo5bo9b2obo$32b2o8b3obo2bo3bo3bo23bo3bo10bo$
43b2obobo4bo12b2o14b3o$44b4o4bo4bo8b2o14bo2bo11b3obo$45bo7bo29b3o15bo$
53bo3bo25b2obo14bo$55bobo26bobo$85bo3$82b2o3b2o7b2o3b2o$82bobobobo9b3o
$83b5o9bo3bo$84b3o11bobo$85bo13bo4$98b2o$87b2o9b2o$87bo$88b3o$90bo!

Kiran · Post by **Kiran** » June 23rd, 2015, 10:28 pm

Another problem with your definition:
You say an engine is a "pattern producing other patterns", but how can you define which is the producer and which the produced?

gmc_nxtman · Post by **gmc_nxtman** » June 23rd, 2015, 10:35 pm

Kiran wrote:Another problem with your definition

My definition has many flaws.

By the way, I had an interesting idea. Has anyone created a superstring-based breeder? By that, I mean a wickstretcher which stretches a string extender; however, I don't think any non-lightspeed superstrings exist with a string extender, let alone a non-agar debris-laying one....

Kiran · Post by **Kiran** » June 24th, 2015, 12:15 pm

By the way, I had an interesting idea. Has anyone created a superstring-based breeder? By that, I mean a wickstretcher which stretches a string extender; however, I don't think any non-lightspeed superstrings exist with a string extender, let alone a debris-laying one....

Like this?

Code: Select all

x = 27, y = 27, rule = s23/b3
18bo8b$17b3o7b$12b3o4b2o6b$11bo2b3o2bob2o4b$10bo3bobo2bobo5b$10bo4bobo
bobob2o2b$12bo4bobo3b2o2b$4o5bobo4bo3bob3o2b$o3b2obob3ob2o9b2ob$o5b2o
5bo13b$bo2b2obo2bo2bob2o10b$7bobobobobobo5b4o$bo2b2obo2bo2bo2b2obob2o
3bo$o5b2o3bobobo3b2o5bo$o3b2obob2o2bo2bo2bob2o2bob$4o5bobobobobobo7b$
10b2obo2bo2bob2o2bob$13bo5b2o5bo$b2o9b2ob3obob2o3bo$2b3obo3bo4bobo5b4o
$2b2o3bobo4bo12b$2b2obobobobo4bo10b$5bobo2bobo3bo10b$4b2obo2b3o2bo11b$
6b2o4b3o12b$7b3o17b$8bo!

EDIT:
And this:

Code: Select all

x = 1487, y = 159, rule = B3/S23
166b6o$165bo5bo95bobo$171bo95bo2bo$165bo4bo42b2o34b4o17b2o51b2o$167b2o
43bo2bo32b6o14bo3bo48bo4bo62bo$212bobo33b4ob2o14bob3o53bo61b4o$211bo
40b2o4bo15bo46bo5bo62b3o$242b2o14b3obo6b4o2bo46b6o42b6o14bo3bo78b2o$
185b3o54bobo14b2o2bo8bo2bo93bo5bo16b3o76bo4bo$259bo3b2o4b3ob3o99bo3bo
10bo4bo81bo$179b2o24b2o7bobo5bo34b3o5bo6b2o86b3o6bo4bo4bob2o8b6o74bo5b
o95bobo119bo$138b2o39b2o24b2o6bo2bo4bobo19b3o9bo2bo2b2ob3o3bo88bo3bo7b
2o9b4o10b2o74b6o95bo2bo51b6o43b2o16b4o$136bo4bo49bobo19bo3bo3bobo19bo
5b4o4bo9bo3bobo61b3o20bo5bo19b2o5b2o3bo179b2o49bo5bo41bo4bo15b3o117bob
o$113b4o25bo19b2o23b2obo3bo18b2o3bo3bo26b9o3bo4b3o4b2o48b2o15bo17bo3b
2o14bob2obo3bo2bo2b2o163b2o14bo3bo54bo47bo14bo3bo52b2o61bo2bo$112bo3bo
19bo5bo19b2o22bo2bo4bo18bo28bobo3b2o2bo7bo11b2o49b2o14bobo17bo2b2o13bo
2bo4b3o4bobo159b4ob2o14bob3o47bo4bo42bo5bo16b3o48b4ob2o42b4o17b2o$116b
o20b6o19b2o23b2ob3o22b2o2bo23b2o3b3o3b4o3b2o5b5o66bobo19b3o9b2o3b2o2b
2o5bo3b3o158b6o4bo15bo33b3o12b2o14bo30b6o3bo10bo4bo47b6o42b6o14bo3bo$
107bobo2bo2bo47bo50bo4bo28b3o2bo3b5o7bobo68bo20b3o9bob4o8b3o5bo128b2o
29b4o5b3obo6b4o2bo32bobo6bo23bo37bob2o8b6o47b4o43b4ob2o14bob3o$164b2o
23b2ob2o21b3o38bo12bo31bo28bo3bo34b2ob2o7bobo7bo3bo102b3o22b6o36b2o2bo
8bo2bo32bobo5bobo19bo3bo25b2o12b4o10b2o97b2o4bo15bo$107bo2bo52b4o20b4o
b2o51bo2b3o2bo3b5o7bobo28b3o27b2ob2o34b2ob2ob4o4bo5b2o3bo72b3o51b5o3bo
35bo3b2o4b3ob3o29b5o30b4o23bo2bo13b2o5b2o3bo36b2o28b2o36b3obo6b4o2bo$
107bo3bo52b4o18bob2o56b5o3b4o3b2o5b5o56bobobobo37bo5b5o8b2o74bo23b2o
15bo10b2o4b3o6bo27b3o5bo6b2o32b3o5bo4bo10b2o6bo3bo4bo19bo2bo7bob2obo3b
o2bo2b2o37bobo28b2o37b2o2bo8bo2bo112bobo$108bobo27b2ob2o24bo17bo2bo8b
2o45bob4o2bo7bo11b2o26b2o26b3o2bo41bobo11b2o77bo23b2o14bobo8bo2b2obobo
6bobo19b3o2bo2bo2b2ob3o3bo34b2o23b2o9bo4bobo19bo8bo2bo4b3o4bobo37bo7bo
bo13b2o7bo26b2o8bo3b2o4b3ob3o49b2o61bo2bo$108bobo22b2o2bobo24bo21b2o4b
2o50bobo2b9o3bo4b3o4b2o31b2o19bo3bo3bo37bo5b5o8b2o24bo2bo45b3o38bobo9b
ob2o10bobo19b3o4bo9bo3bobo65bo6bobo25bo3b2o2b2o5bo3b3o32b2o2bo2bo3bo2b
o12b3o5bobo5bo21bo6b3o5bo6b2o47b4ob2o42b4o17b2o$108bo24b3obobobob2o17b
obo3b2o21b3o4bo45b2o3b4o4bo9bo3bobo6bo47bo4bo35b2ob2ob4o4bo5b2o3bo27bo
20b6o27bo32bo10bob2o4b3o4bo21b7o3bo4b3o4b2o39bo3bo28bo20bo6b4o8b3o5bo
32b2o10bobo12bo7b2o5bobo20bo4bo2bo2b2ob3o3bo50b6o27b3o12b6o14bo3bo50b
2o$104bob2o29b3o5bo8b2o4b2ob2o3b2o85bo2bo2b2ob3o3bo7bo3bo20b2o24b2o2bo
35b2ob2o7bobo7bo3bo22bo3bo19bo5bo26bobo49b2ob2o27bo6bo11b2o42bo51bo12b
obo7bo3bo40bo4bo13b2obobo9bobo20bo6bo9bo3bobo48b4o28b2ob2o10b4ob2o14bo
b3o47bo4bo62bo$102b2obo2bo24b7o3bobo7bo2bo3b3o20b2ob2o6bobo34b5o21b3o
5bo6b2o9bo18b2o2b2o21bob5o37bob4o8b3o5bo23b4o25bo26b2o3bo19b2o25b2ob2o
20bo6b5o3b2o5b5o71b2o23bo7b2o4bo5b2o3bo37bo4b2o32bo25b4o3bo4b3o4b2o98b
2o4bo15bo52bo61b4o$105bo2bo22bo3bo4bo2b3o5bo34b2o7bo34bo4bo23bo3b2o4b
3ob3o2bo4bo20bo24b6obo36b2o3b2o2b2o5bo3b3o46bo4bo31bobo18bobo26bo22bo
10b5o7bobo34b2o36b2o24b5obo2b5o8b2o39bob4o56bo7bo11b2o80b2ob2o20b3obo
6b4o2bo45bo5bo62b3o$103bob3o26b2o16b2ob2o9bo16b3o49bo23b2o2bo8bo2bo3b
5o13b2o30b2o2bo3bo39bo2bo4b3o4bobo49b2o33bobo19b2ob4o52bobo12bo38bo30b
o34bo2bo11b2o37b3o3b3o31b2o25b6o3b2o5b5o58bo21b2o2b2o21b2o2bo8bo2bo46b
6o42b6o14bo3bo49b4o62bobo53b2o$90b5o8b4o24bo3b2o3bo12b2o10bobo62bo3bo
23b3obo6b4o2bo20bo2bo36bo41bob2obo3bo2bo2b2o85bo18b2obobob2o44bo10b5o
7bobo33bo2bo8bobo18b5o27b5obo2b5o8b2o74b2o25bo4b5o7bobo59bo21b5o22bo3b
2o4b3ob3o93bo5bo16b3o48b6o61bo2bo50bo4bo62bo$89bo4bo10b2o25b3o4bo24bo
2bo64bo25bo15bo21bo34bo52b2o5b2o3bo77b2o23b2obobo3bo43bo6b5o3b2o5b5o
31bobobo12bo15bo5bo25bo7b2o4bo5b2o3bo34b2o10b4o54bo12bo48b2o11bo3bo17b
o2bo13bo7b3o5bo6b2o101bo3bo10bo4bo47b4ob2o63b2o55bo61b4o$94bo70b2o102b
ob3o21bo35bo18b6o26b4o10b2o56bo18bo2bo23bo2b2o2bo51bo6bo11b2o33b3o5b2o
2bo2b2o12b3o2bob2o24bo11bobo7bo3bo33b3o8b7o16b2o30bo4b5o7bobo46b2o14bo
bo16b3o14bo5bo2bo2b2ob3o3bo38b3o27b3o27bo4bo4bob2o8b6o50b2o46b2o14bo3b
o33bo14bo5bo62b3o$89bo3bo7bo3bo162bo3bo9b2o12bobo9b2o21bo17bo5bo23bob
2o8b6o24b3o30bo15bo2bo2bo4bo17b2o2b3obo21b2o26b7o3bo4b3o4b2o4b5o23b3o
6bo2bo2bo14b2o2bobo25b2o4b4o8b3o5bo31b2o3bo7bob2ob2o15bo2b2o28b6o3b2o
5b5o53bo7bobo17b6o10bo7bo9bo3bobo34bo2bo6bo23b2o25bo12b4o10b2o32bo60b
4ob2o14bob3o32b2o14b6o42b6o14bo3bo$91bo10bo2bobo20b2o22b3o115b2o10b2o
13bo10b2o45bo23bo10bo4bo23b2ob2o29b2o10bo5b4obo3bobo14b2o4bo34bo17b3o
4bo9bo3bobo4bo4bo23b3o5b2obobo17bo31bobobo3b2o2b2o5bo3b3o31bo12bobo2b
2o15bo4bo27bo7bo11b2o51b3o7bo20b2ob2o14b4o3bo4b3o4b2o31bo3bo5b2ob2o17b
o3b2o26bo13b2o5b2o3bo32bobo59b6o4bo15bo31b3o60bo5bo16b3o$104bo2b3o18b
2o24bo112bo2bo78bo4bo37b3o23bo3bo26bo13b2ob2obob3o7b2o16b3obo32bo3bo
15b3o2bo2bo2b2ob3o3bo12bo20b2ob2o7b2ob3o2bo14b3o35bo2bo4b3o4bobo8b2o
28bo6b4o19bo3bo29b4o3bo4b3o4b2o49b2o29bobob2o10b4o6bo11b2o31bob3o21b2o
9b2o4bo20bobo6bob2obo3bo2bo2b2o33b2ob2o5bo51b6o5b3obo6b4o2bo32b2o65bo
3bo10bo4bo68b6o$104bobobo43bo114bobo81b2o37bo3bo22bo3bo26b3o5b3o4b2o6b
o2b2o21b3o41bo21b3o5bo6b2o4bo3bo22bobo15bo16bo10b2o25bob2obo3bo2bo2b2o
5b3ob2o34bobobo19b3o26bo6bo9bo3bobo52bobo26bo3bo11b2ob5o3b2o5b5o32bobo
8bo3bo10bo7b3o5bobo20bo6bo2bo4b3o4bobo32bo3bo6bo22bo2bo24b2o11b2o2bo8b
o2bo32b3o53b2o3bo4bo4bob2o8b6o66bo5bo95bobo$105b3o45bo157bo78b3o4bo19b
obobo25bo4bo10b2o3bo3b2o3bo58bo4bo23bo3b2o4b3ob3o4bo25bo11b5o26bo4bo
29b2o5b2o3bo3b5o23b2ob2o11bobo46bo4bo2bo2b2ob3o3bo51bo5bo27bo2bo11b2ob
o3b5o7bobo43b5o27bobo20bo4bo2b2o2b2o5bo3b3o30bo33bo3bo23bo2bo10bo3b2o
4b3ob3o28bo4bo54bobo4b2o9b4o10b2o71bo95bo2bo$106bo202bob3o75b4o2bo3bo
19bob2o25b2o2b3o17bo5bo58b5o23b2o2bo8bo2bo40b3o2bo33bo26b4o10b2o3b3o
26bo14b2o46bo6b3o5bo6b2o23b2o29bo27b3o16bobo12bo78bo22b8o8b3o5bo34bo
20b2o8bo2bo3bo19b3o9b3o5bo6b2o34bo30b3o23b2o17b2o5b2o3bo66bo4bo78b4o
17b2o$103bo25bo180b5o74bo10bo17bob3o26b6o9bobo6b2o3bo85b3obo6b4o2bo13b
2o24bob2o30bo5bo23bob2o8b6o43b2o30b4o26bo3b2o4b3ob3o20bo2bo22bobo2b2o
42b2obo3b5o7bobo99bo9bobo7bo3bo30b2o23b2o14bobo28bo2bo2b2ob3o3bo56b2o
15bo21b2o10bob2obo3bo2bo2b2o70b2o79b6o14bo3bo$102bobo23bobo176b3o2b3o
80bo4bo16bob2obo31bo10bo8bo2bo86bo15bo14b2o23bo2bo32b6o23bo10bo4bo5bo
37bo2bo28b6o25b2o2bo8bo2bo19b3ob2o20bo4bob2o41b2ob5o3b2o5b5o32bobo37b
2o23bo5b3o4bo5b2o3bo72bobo26bo3bo9bo3bobo53b2o14bobo18bob2o9bo2bo4b3o
4bobo151b4ob2o14bob3o$102bo2bo22bo2bo175b6o83b5o16bobo155bob3o118b3o5b
o12b2o24bo2bo28b4ob2o23b3obo6b4o2bo19bo3bo21b6o3b2o39b4o6bo11b2o31bobo
37b2o24b2obo4b5o8b2o74bo27b5o3bo4b3o4b2o68bobo25b2obo3b2o2b2o5bo3b3o
154b2o4bo15bo$19bo7b5o71b2o24b2o176b3o108bo58bo96bo3bo46b3o68bo3bo6bo
11b2o24bo2bo32b2o24bo15bo6b2o13bo2bo21bo2bo5b3o42b4o3bo4b3o4b2o32bo9b
2o55b2obo11b2o98b2o12bo11b2o70bo19bo6bo2b5o8b3o5bo122b2o36b3obo6b4o2bo
$18bobo4b2o5b2o274b2o165bo2bo97b2o118b3o47b2o70bob3o4b3ob2o12bobo23bob
ob4o2b2o36bo7bo9bo3bobo32bo2bo7b3o20bobo28b2obo4b5o8b2o71b2o23b5o3b3o
3b2o5b5o91bo2bo13bobo7bo3bo62b3o28b3o26b2o37b2o2bo8bo2bo$20bo4bo7bo
407b2o29b2obo2bo94bo2bo46bo71b4o118bo3bo5b5o39bobob3o3b3o35bo5bo2bo2b
2ob3o3bo9bo24b2o2bo8b2ob2o16b2o2b2o25bo5b3o4bo5b2o3bo33bo36b2o24b7o2b
5o7bobo32bo35bo24b2o3b2o3b4o4bo5b2o3bo63b3o50b2o7bo26b2o8bo3b2o4b3ob3o
$17bo2bo3bo7b2o406bo25b2o2bob3o2bo95bobo48bo70bo50bo72b2o7b3o41b3o3b2o
b2o36bo7b3o5bo6b2o4bo3bo22b4o10bob3o14b3o2b2o26bo9bobo7bo3bo31bobo29b
2obo34b3o12bo33b2o30bo3b2o30b2o6b5o8b2o65bo23b2o15bo9b3o5bobo5bo21bo6b
3o5bo6b2o$17b3o3bo6b2o71b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
127bob2o22bo3bo2bo2bo145b3o119b2ob2o67bo2bo59b3o17b4o26bo3b2o4b3ob3o7b
o23b2o7bob2obobo15b2o30b8o8b3o5bo30bo2b2o28b5o27b7o2b5o7bobo30bo2bo28b
3ob2o35bobo11b2o45b2o21bo23b2o14bobo8bo14bobo20bo4bo2bo2b2ob3o3bo$28bo
74b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24bo132bo22bo2bo150b2o193bobo
61bo17b6o25b2o2bo8bo2bo2bo4bo23b2o8bobo20b4o31bo2b2o2b2o5bo3b3o30b2o9b
obobobo13bo6b2o24bob3o3b3o3b2o5b5o28b2o12bo17b2obo3b3o23bo4b2o6b5o8b2o
40bo4bo18b3o38bobo8b2obobo5bo3bobo20bo6bo9bo3bobo$8b2o13bo4bo387bo22bo
4bo175bo2bo123bobo90b2o10b2o44b4ob2o23b3obo6b4o2bo3b5o22bo2bo7bo3bob2o
16b2o26b3o5bo2bo4b3o4bobo6b3o23bo8b2o3b4o12bo6b2o23b2o12bo11b2o6bo20bo
bo7bo3b2o23bo26b2ob2o3b4o4bo5b2o3bo16b4o25bo17b3o6bo32bo17b2obo4bo25b
4o3bo4b3o4b2o$8b2o14bob2o387bobo21b2ob2o301bo3bo90b2o10b2o48b2o24bo15b
o31bobo12b3o18bo35bob2obo3bo2bo2b2o5b5o22bo3bo7b6o12bo2bo3b2o24b3o3b5o
3bo4b3o4b2o3bo3bo18bobo6b2o4b2o23b3o22b3o12bobo7bo3bo14bo3bo19bo5bo25b
obo49b2ob2o27bo7bo11b2o$107bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo
25bo21bo2bo22bo182b2o119bo2b2o189bob3o33bo13b2o61b2o5b2o3bo3b3ob2o30bo
2bo18b2o2b2o26b3o3bo3bo9bo3bobo9bo17bo7b2obo28bo24bo4bo2b5o8b3o5bo18bo
20b6o25b2o3bo19b3o25b3o28b6o3b2o5b5o$106bobo23bobo23bobo23bobo23bobo
23bobo23bobo23bobo23bobo23bobo23bobo23bobo21b2o204bo2bo120b3o189bo3bo
43bo2bo30b6o26b4o10b2o5b2o20b2o3bo5b7o17b3o35bo2bo2b2ob3o3bo7bo4bo17bo
b2o5b4o20bo2bob2o26bo3b2obo3b2o2b2o5bo3b3o14bo2bo56bobo17b2ob5o21b3o
28bo4b5o7bobo$105bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2b
o22bo2bo22bo2bo22bo2bo22bo2bo227b3o122bo192b2o18b2o22b3o2b3o27bo5bo23b
ob2o8b6o28b2ob2o8bo5bo15b3o37b3o5bo6b2o5b5o27b5o19bobob2o26bo9bo2bo4b
3o4bobo75bobo17b2obobob2o21bo33bo12bo$106b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o543bo2bo19b2o23b2o2b2o34bo23bo10bo4bo31b
o11b5o58bo3b2o4b3ob3o27bo2bo7b3o18b4o39bob2obo3bo2bo2b2o76bo20b2obo3bo
50bo4b5o7bobo$618b2o24b2o24b2o24b2o24b2o213bobo49b2o28bo4bo37b3o42bo4b
2o31b2o25b2o2bo8bo2bo24b2o4bo7b2obo66b2o5b2o3bo68b2o25b2o3bob2o49b6o3b
2o5b5o$416b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o18b2o24b2o24b2o24b2o
24b2o264b3o30b2o37bo3bo15b2o24bo34b4ob2o23b3obo6b4o2bo26b4o8b2o34b6o
26b4o10b2o14b2o51bob2o21bo2bo5bo50bo7bo11b2o$2o6bo3bo403b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o388b3o69b3o17b2o25b3o31b6o24bo15bo11b2o15bo
11bo33bo5bo23bob2o8b6o13b4o30b3o9bo8bob2o3bo16b3ob2ob2o54b4o3bo4b3o4b
2o$2o5b3obo2bo973b3o68b4o48bo30b4o36bob3o12b2o24b2ob2o38bo23bo10bo4bo
13bo4bo25bo3b3o8b3o8b2o3bobo15bo4bo53bo6bo9bo3bobo$7bobobo977bo69bo51b
o69bo3bo39b2ob2o32bo4bo37b3o14bo3b2o24bobo12b2obo7b2o5b2o15bob3o54bo4b
o2bo2b2ob3o3bo$13bo7bobo967bo119bo71b2o41b2ob2o33b2o37bo3bo14bo4bo23bo
8b3o3bobo2bo2b2o2bo23b2o38b5o14bo6b3o5bo6b2o$13b2ob2obob3o4b2o956b2o3b
o188bo2bo42b3obo72b3o19b2o24bo3bo10bo2bo4bo3b2o22b2o37bo4bo23bo3b2o4b
3ob3o$7bo3bobo3b2o2bo6b2o71b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o129b2o119bobo70bobo42bo2bo73b4o45bo
4bo10bobo5bo3bob2o63bo23b2o2bo8bo2bo$11bobo87b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o133bo115b2obo
114bo3bo73bo19bo29bo3b4o9b2o5bo2bob2o58bo3bo23b3obo6b4o2bo$24b2o70b2o
888bo2bo116bo2bo114bo4bo91bobo32bo20b2o64bo25bo15bo$23b4o8b3o56b2o2bo
888b3o116b2o115bo2bo2bo91bobo31bo124bob3o$23bo2b2o5b7o54b2ob2o6b2o24b
2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b
2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b
2o24b2o24b2o24b2o24b2o144bo114bo97b2o57bo97bo3bo$24b3obo3b9o53b2ob2o6b
2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b
2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b
2o24b2o24b2o24b2o24b2o24b2o144b2o113b2obobo152bo98b2o$24bob2o3b3o7bo
54bo1010bo2bo116b2o149bo99bo2bo$31bo6b3o80b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o22b3o115b3o61b2o53b2o24b2o2bo7bo95bobo$38b2o70b2o
9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b
2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o
13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b
2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o
9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o9b2o13b2o8bo15b
2o24b2o24b2o24b2o21b2o62b2o6b2o47b3o21b2o5b4o$36bo73b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o22b3o68bobo44bo
2b2o28bo$35bo1190b3o69bo21bo22bo2bobo$34bo1192bo91bobo25bo$97b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24b2o24b2o23b2o25bo2bo20bo3bo$50bo45bo2bo22bo2bo22bo2bo22bo2bo22bo2bo
22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo
22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo
22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo
22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo
22bo2bo20bob2obo24b2o23bo$49bobo45b2o24b2o24b2o24b2o24b2o24b2o24b2o24b
2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b
2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b
2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o21b4o2b2o$48b2ob2o
1238b2obob2o$48b2ob2o1239b2o2b5o$51b3o1241b2o2b2o$47bo3b3o1239bobo$46b
o4b3o1240bo$36b2o7bo5b2o$36b2o12b3o$50b2o$49b2o3$50bo$49b3o49bo$48bo3b
o49b2o$47b2o19b3o30b2o$48b2o$67bo$66b2o4b2o$67bo5bo9b2o21bo25bo25bo25b
o25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo
25bo25bo25bo25bo25bo31b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24bo$67b2o
8bo5bobo19bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo
23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bo
bo23bobo23bobo23bobo23bobo30b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o23bo
bo$70b3o4b2o4bo2bo18bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bo
bo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo
23bobo23bobo23bobo23bobo23bobo23bobo26bo213bo$72bo11bo2bo18bo25bo25bo
25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo
25bo25bo25bo25bo25bo25bo26b2o$72b2o10bobo695b2o2bo22b2o182b2o$73bo2bo
36b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o24b2o
24bo25bo25bo252b4o22bo2bo181bo2bo$42bo6b2o22b2obo35bo2bo22bo2bo22bo2bo
22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo
22bo2bo22bobo23bobo23bobo27b2o131b3o88b4o22bobo181bo5bo$41bobo6bo23bo
38bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bobo23bo
bo23bobo23bobo22bo2bo22bo2bo22bo2bo22b2o227b2o2bo21bo183bob2o2bo$40bo
3bo3bo65bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo25bo24b2o24b
2o24b2o22bo3bobo29b2o24b2o24b2o24b2o24bo92bobo204bo4bo74bobo$41bobo5b
4o26bo198bo275bo4bo29b2o24b2o24b2o24b2o24bo84b3o4bo3bo43b3o159bo2bo74b
o2bo$42bo7bo27b2o197bobo274bo4bo130bo2bo85b3o3bo2bo43bo3bo115bo42bobo
78b2o$47bo3bo2bo3bo19b2o196b2o186bo90bo2bo130bo90bobo3bobo21bo21bo4bo
114b4o118bo3bo$46bobo8bobo216b2o2bo181bo3bo89bo132b4o88bo5bo20bo3bo18b
o5bo115b3o79b4o36bob3o$46b2o7b3obo33b3o181bobob2o184bo5b2o213b2obobo
115bo2bob2o15bo3bo117bo3bo9b2o24b2o39b6o24bo15bo$52bo2b2obo36bo181bo4b
o179bo4bo3bo4bo17bo69b2o122b3o86b2o32bo2bo16bo3bo9b2o108b3o9b2o12b3o9b
2o39b4ob2o23b3obo6b4o2bo$54b2o35bob2o182bo4bo180b5o9bo14bo3bo66b2obo
79bo41b3o2b2o79b3ob2o32bobo9b2o17b2obo16b2o77bo10bo4bo22b2o32b3o20b2o
25b2o2bo8bo2bo$88b2o30b2o24b2o24b2o24b2o24b2o24b2o24b4o191bo5bo19bo67b
3o78b4o39b2o2b2o72bobo4b5o33b2o12bo5b3o7bo3bo14b2obo77bob2o8b6o21bo2bo
34bo45bo3b2o4b3ob3o$89bobo27bo2bo22bo2bo22bo2bo22bo2bo22bo2bo22bo2bo
22b3obo23b2o167b6o14bo4bo26bo40b2o80b3o40b2obo73bo2bo4b3o34b2o9bobo15b
o3b2o14b2o55b2o25b4o10b2o21b2o37bo41b3o5bo6b2o$90bo29b2o24b2o24b2o24b
2o24b2o24b2o24b2o2bo21bo2bo25bo129b2o30b5o26b4o35b3o81bo3bo38bo4bo74b
2o39b2o9b2obo2bo11b2obobo15b3o51b3ob2o26b2o5b2o3bo6b2o14b2o30bo5b2o39b
o2bo2b2ob3o3bo$277b3obo20bobo21b2o3b2o127bo64b3o35b2o84b3o43bo72bo3bo
38bo2b4o2b2ob2obo2bo9bob2obo70b5o21bob2obo3bo2bo2b2o5b3ob2o19bobo23b2o
b5o37bo3bo9bo3bobo$104b3o172b3o21bo22b2o3b3o128b2obo39b6o14bo3bo16b2o
37b3o19b6o23bo10bo4bo116bob3o37bo2b4o2b2o3b2o2bo10bo24bo7b3o40b3o17bo
3bo2bo4b3o4bobo5b5o20bo2bo24bo41b6o3bo4b3o4b2o$104bobo173bo49b3obo21bo
103bob4o25bo12bo5bo16b3o16b2o37bo2bo17bo5bo23bob2o8b6o10b2o24b2o40b2o
24bo15bo38b2ob2o4b3o3bo12bo5b2o16b2o2bo66b2o3b2o2b2o5bo3b3o5b3o17bo3b
2ob2o4bo17bo38b2o2bo7bo11b2o$104bob2o164b2o55b2o24b2o103bobobo26bo18bo
3bo10bo4bo53bo3bo23bo26b4o10b2o9b2o14b2o9bo36b4ob2o23b3obo6b4o2bo39b3o
5b2obo14b3o2b3o15bo2b2o68b5o8b3o5bo24bo7b2o4bo15b2ob4o34b2o2bob4o3b2o
5b5o$104b3o165b2o57bo2b2o18bo99bobo5b2o24bo2bo12bo4bo4bob2o8b6o53b2o2b
2o15bo4bo29b2o5b2o3bo25b5o44b6o25b2o2bo8bo2bo39b3o7bo15b2o3b3o16bob2o
3bo58b2o4bo6bobo7bo3bo32bo21b2ob2o3bo36b2o3b5o7bobo$108b3o218b3o3bo17b
2obo97bo2bo32bo15b2o9b4o10b2o3b3o43bo3b3o2bo16b2o25bob2obo3bo2bo2b2o
27bo4bo6bo22b2o13b4o26bo3b2o4b3ob3o67bobob2o16bo8bo19bo36b3o4b4o4bo5b
2o3bo27b2o26bo7bo38bobo12bo$104bo229b2o16b2o3bo2b2o95b2o27bo4bo27b2o5b
2o3bo3b5o36b2o7b3o3bo34bo7bo2bo4b3o4bobo6b3o29bo22bob3o39b3o5bo6b2o71b
o2bo17b2obo23bo2bo38b4o3b5o8b2o29bo29b3o39b2o3b5o7bobo$51b2o24b2o24bob
o231b2o12bo8bobo92bo3bo29b2o22bob2obo3bo2bo2b2o5b3ob2o9b2o23b7o3b2o4bo
33bobo3bo3b2o2b2o5bo3b3o4b5o18b3o5bo5bo17b2o4bo36bo2bo2b2ob3o3bo71bo2b
o2bo16bo3bo22bobo42bo2bo11b2o31b3o26b3ob3o33b2o2bob4o3b2o5b5o$50bo2bo
22bo2bo22bo3b2o22b2o25bobo177bobo16bo5bo93bob3o24b2o5bo19bo2bo4b3o4bob
o8b2o10b2o25bob3o3b2o40bo3b5o8b3o5bo4b3ob2o17bo6bo26bo2bobo24b2o3b4o4b
o9bo3bobo48b4o16b3o20bo3b3o20b2o41b4o3b5o8b2o97b2o2bo7bo11b2o$51b2o24b
2o24bo25bo2bo204bo2bo9bo3b2o4bobo82bo15bo11b5o27bobobo3b2o2b2o5bo3b3o
21bo21b2obo7bo26bo11bo13bobo7bo3bo7b2o16b2ob2o4bo3bo27bo24bobo2b9o3bo
4b3o4b2o24b2o20b6o13bo5b2o16bo3bo2b2o58b3o4b4o4bo5b2o3bo67bo21b2o9b6o
3bo4b3o4b2o$104b3o2bo19bobo21b2o2bo144b2o27bo8bo10bob2o6bo83b3obo6b4o
2bo9bo4bo11bo14b2o4b4o8b3o5bo19bo23b2ob3o3b3o25b3o10b2o5b5o4bo5b2o3bo
26b2o7bo2bo2b3o22b3o23bob4o2bo7bo11b2o23b2ob2o19b4ob2o12b2o29b2o51bo6b
2o4bo6bobo7bo3bo65bobo18bobo11bo3bo9bo3bobo$106b3o21bo22b2o3b3o140bo2b
o3bo20b2o6b2o2bo10bo5b2o58b5o23b2o2bo8bo2bo14bo10bobo13bo11bobo7bo3bo
19bo3bo21b2o2bo4bo24b2obo12bo3b2o4b5o8b2o27b2obo6b2o3bobo22bo2bo24b5o
3b4o3b2o5b5o24b4o24b2o40b2ob2ob3o26bo20b3o11b5o8b3o5bo49b2o14bobo17bo
16bo2bo2b2ob3o3bo$161bo21b2o117bobo2bob2o19b2o4bobo2bo10b3o4bo57bo4bo
23bo3b2o4b3ob3o9bo3bo27bo7b2o4bo5b2o3bo12b4o6bobo26b2o3bo23bo3bo19bo
11b2o31bobo11b2o16b2ob2o2b2o25bo2b3o2bo3b5o7bobo26b2o40b2ob2o23bobo30b
obo21b2o9b2o3b2o2b2o5bo3b3o49b2o11bo3bo18b2ob2o14b3o5bo6b2o$99b2o55b2o
3bo20bo120bo2b2ob3o20b2obo4b2o11b4o2b4o5bo53bo21b3o5bo6b2o13bo14b2ob2o
11b5obo2b5o8b2o12b6o36b3o23bo3bo12bo3b2o4b5o8b2o28b2obo28bo2b3o39bo12b
o89b2o24b2o14bobo17bo4b2o8bo3bo2bo4b3o4bobo63bo22b2ob2o16bo3b2o4b3ob3o
$99b2o55b2o23bo99bo25bob3o21b3o18b4o2bob2o4bo48bo3bo20bo2bo2b2ob3o3bo
31bo2b2o15bo2bo11b2o15b4ob2o62bo13b2o5b5o4bo5b2o3bo29b3o28b3obobo29b3o
2bo3b5o7bobo87b2o24b2o15bo17b2o4b2o13bob2obo3bo2bo2b2o33bo29bo22bo2b2o
16b2o2bo8bo2bo$157bo3b2o18bobo97b4o71b2ob2o4bo2bo16b4o42b2o3b4o4bo9bo
3bobo29b3o12b5obo2b5o8b2o16b2o31b3o30bo12bo13bobo7bo3bo28b2o34b2o24b2o
3b3o3b4o3b2o5b5o93bo25bo33bo20b2o5b2o3bo31bo53b2o8b4o5b3obo6b4o2bo$
158bo4bo16bob2o3b2o93b3o70b3ob2obo3bo18bobob2o25b2o13bobo2b9o3bo4b3o4b
2o29bo12bo7b2o4bo5b2o3bo48b2ob3o6bo36bo3b5o8b3o5bo89bobo3b2o2bo7bo11b
2o92bo25bo27bo24b4o10b2o93b6o4bo15bo$160bo18b2ob2o3b2o93bo3bo68b2o5bo
3bo17bo2bob2o24b2obo12bob4o2bo7bo11b2o43bo11bobo7bo3bo48b2o8bobo21b2o
10bobo3bo3b2o2b2o5bo3b3o69bo26b9o3bo4b3o4b2o91bo25bo28bo3bo16bob2o8b6o
94b4ob2o14bob3o$164b3o11b2ob2o6bo94b3o79bo24bo24bobo14b5o3b4o3b2o5b5o
44b2o4b4o8b3o5bo42b2o4b2ob2o5bobo15b2o6bo10bo7bo2bo4b3o4bobo69bobo19bo
5b4o4bo9bo3bobo148b3o8b6o3bo10bo4bo51b2o46b2o14bo3bo$160bo3b3o11b2obo
5b2o82bo10bo4bo23bo44b2o6b2o25b3o22bo15bo2b3o2bo3b5o7bobo46bobobo3b2o
2b2o5bo3b3o42b2o6bo8bo15b2o4bob2o19bob2obo3bo2bo2b2o53b2o14bobo19b3o9b
o2bo2b2ob3o3bo161bo5bo16b3o48b4ob2o63b2o$129b2o26bobo7bo10bo8b2o82bob
2o8b6o22bo13b2o40b2o23b2o22bo26bo12bo55bo2bo4b3o4bobo55b3o19bo4b3o26b
2o5b2o3bo51b2o11bo3bo34b3o5bo6b2o164bo14bo3bo48b6o61bo2bo$128bo2bo3bo
21bo6b2o2bo11bobo2b2o60b2o25b4o10b2o33bo4bo37b2o15b2ob3o43bo2b3o2bo3b
5o7bobo54bob2obo3bo2bo2b2o82b2o25b4o10b2o63bo40bo3b2o4b3ob3o116bo39bo
4bo15b3o51b4o62bobo$129bobo2bobo7bo12b2o6b4o10bo4bobo57b3ob2o26b2o5b2o
3bo11b4o25bo33bo4bo15b5o28b3o13b5o3b4o3b2o5b5o59b2o5b2o3bo80bo23bob2o
8b6o34bo29bo23bobo14b2o2bo8bo2bo78b2o36bo41b2o16b4o$130bo3bob2o6bo15bo
b2o3bo12bo6b2o55b5o21bob2obo3bo2bo2b2o12bo3bo19bo5bo32bobo20bo45bob4o
2bo7bo11b2o56b4o10b2o47b2o31bo13b6o3bo10bo4bo35bo53b2o14b3obo6b4o2bo
74b4ob2o95bo$134bo3b2o20b2o19b3o3bob2o3b3o48b3o11bo9bo2bo4b3o4bobo16bo
20b6o32bobo31bo34bobo2b9o3bo4b3o4b2o52bob2o8b6o46bo4bo25b4o13bo5bo16b
3o36bo63b2o4bo15bo75b6o$135bo47bobobo3bo21b2o44bo3b2obo3b2o2b2o5bo3b3o
11bo2bo60bo31bobo15bo17b2o3b4o4bo9bo3bobo53bo10bo4bo53bo24bo22bo14bo3b
o32bo2bo13b4o43b4ob2o14bob3o77b4o$188b5o18b2obob2o25b3o12bo4bo2b5o8b3o
5bo65b2o3b3o18b2o14bobo13b2ob2o6bo19bo2bo2b2ob3o3bo69b3o48bo5bo25b3o
13bo4bo15b3o33bo2bo13b6o42b6o14bo3bo$136bo3bo41b2o3bobob3o20bo2bo25bo
14b3o12bobo7bo3bo65b2o3bobo18b2o15bo14bo2bo3bobobo21b3o5bo6b2o64bo3bo
49b6o25b2o16b2o16b4o34b2o14b4ob2o42b4o17b2o$183bo7b2o17bo3bo28bobo13b
2ob2o3b4o4bo5b2o3bo71b3o51b2o3bobob2o23bo3b2o4b3ob3o62b3o118bo57b2o61b
o2bo$137bobo43b2o6bo25bob2o23b2o12bo4b2o6b5o8b2o99b3o29bo3bo24b2o2bo8b
o2bo61b4o239bobo$138bo55b2o15bo7bo24b3o20bobo11b2o153b4o5b3obo6b4o2bo
61bo$151b4o39bobo14bo2bo3bo25b3o11bo4b2o6b5o8b2o149b6o4bo15bo$128b2o
20b6o34bo3bo16b3ob2o28bo13b2ob2o3b4o4bo5b2o3bo128b2ob3o14b4ob2o14bob3o
$126b2ob2o19b4ob2o32bobo66b3o12bobo7bo3bo126bo24b2o14bo3bo$126b4o24b2o
33bobo31bo34bo4bo2b5o8b3o5bo126bo3bo38b2o$127b2o61bo31bobo14bobo17bo3b
2obo3b2o2b2o5bo3b3o65b6o59bo35bo2bo$180b2o4bo19b2o14bobo13bo2bo6bo10bo
9bo2bo4b3o4bobo65bo5bo58bo36bobo$180b2o2b2ob2o17b2o15bo14bo2bo4bo2b2o
19bob2obo3bo2bo2b2o71bo$186bo26bo25b2o3bobo29b2o5b2o3bo63bo4bo$213bo
30b2ob2o25b4o10b2o64b2o$213bo32b2o23bob2o8b6o$246b2o14b6o3bo10bo4bo$
244bobo14bo5bo16b3o$241b3o23bo14bo3bo$241b5o15bo4bo15b3o$183b2o58b3o
17b2o16b4o$179b4ob2o95bo$179b6o$180b4o!

gameoflifemaniac · Post by **gameoflifemaniac** » January 24th, 2017, 12:06 pm

Paul Tooke wrote:The "super breeders?" topic had the side effect of sparking off some discussion of what the definition of a breeder is. My dissatifaction with the published definitions led me to create a few breeders which challenge these definitions. I have started this topic to share these patterns and hopefully to encourage others to produce similar patterns and/or share their opinions of what a breeder is.

My understanding of the term "breeder" as a class of patterns is: "Any object which creates an unbounded sequence of objects which in turn create an unbounded sequence of objects." Such a pattern may grow quadratically, but I don't see why this is a necessity. Furthermore the classification of existing breeders into four types (MMM, MMS, MSM or SMM) as described in several definitions on the web can mislead the casual reader into thinking that these are the only types possible. They aren't.

As an example of sub-quadratic breeding, I have produced what I would call an SSS-type breeder. The pattern is too large to post, but can be downloaded from here.

The pattern consists of a "Pianola" Universal Constructor complete with an instruction tape for the repeated construction of a slide gun which extends a line of beehives.

The Pianola is a simplified Universal Constructor inspired by Andrew J. Wade's Gemini constructor. It comprises two mirror-image gangs of seven glider guns. Each gun consists of two stable reflectors, one by Paul Callahan, the other by Stephen Silver. These circulate a stream of gliders, duplicating them at the Northern end. The gliders are timed to create volleys of gliders matching those produced by the Chapman-Greene construction arm, but reflected, so that the guns are on the opposite side of the stream.

The slide gun is a Spartan (i.e. built from easily constructible still lifes) gun that fires a salvo of gliders at a block. The result of the collision is to extend a line of beehives, move the block 10 cells diagonally and to send back a glider which is then duplicated several times to produce the next glider salvo. The salvo+block reaction is a variation of that used in Dean Hickerson's "Sqrtgun 10.1". The line of beehives grows at a rate of O(sqrt(t)) and therefore the entire pattern grows at a subquadratic rate.

EDIT: to be ...

Is it possible to make a type SSS breeder that grows quadratically?

Mr. Missed Her · Post by **Mr. Missed Her** » February 9th, 2017, 4:11 pm

gameoflifemaniac wrote:Is it possible to make a type SSS breeder that grows quadratically?

I'd say yes. An example would be a universal constructor that repeatedly builds an SS pattern, like blockstackers.
If the blockstackers get in the way somehow, you could make a gemini which creates a blockstacker each time it dupliactes itself. You could also program it to leave the previous constructor section, for slightly more growth. (Extra linear growth, not extra quadratic growth.)
Also, does the blockstacker count as an SS object? Instead of an S(M/S) object? Because after infinity generations, only 41.2%/39.4% of the infinite population are blocks, the rest being the weightships that indirectly create the blocks. (There's two numbers because of the varying population of weightships.)

gameoflifemaniac · Post by **gameoflifemaniac** » February 12th, 2017, 3:32 am

1. Yes, Blockstacker is a type-SS linear growth pattern.
2. Instead of calling it 'weightship', call it *WSS.

Mr. Missed Her · Post by **Mr. Missed Her** » February 13th, 2017, 10:40 am

gameoflifemaniac wrote:Yes, Blockstacker is a type-SS linear growth pattern.

Yes, but why is the portion of the population that is the weightships (_WSS) not counted? Is it because they're not permanent?

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