EvinZL wrote: January 13th, 2024, 8:50 pm
WLOG we can take the white cell to be ON in generation 0 and OFF in generation 2. Then we search for such a patch and find that it does not exist. Thus a period p phoenix does not exist. (I'm not sure how apg chooses the patch but I assume it's along the lines of start with 25x25, if there is a solution expand?)
Just while we're waiting for a more official writeup on this: here's an example pattern that APG posted on Discord to explain why 25x25 was chosen. The patch has to be centered on the key ON at T=0, OFF at T=2 cell, so its dimensions will be odd numbers. At 23x23 a SAT solver can still find solutions for a partly-period-4 patch of possible phoenix:
x = 23, y = 23, rule = LifeHistory
4BCBC5BC9BC$C7BC3BCBCBCBCBCB$4BC3BCBC9BC$B2C7BC5BC3B$4BCBCBC5BC3BC3.A
$B2C7BCBC3BCB$6BCBC3BCBC2B3.2A$2BC5BCBC5B$4BC7BCBC5.A$4BC3BCBC3BA3.A$
2BC3BCBC4B3.A3.A$C5BC4BE6.A3.A$BCBC6B7.A3.A$5BCBCB.2A$2B2C4B5.A.A.A.A
.A$7BA.A.A.A$BC4B13.A.A$BCBCB4.A11.A$3BC$3B2.A3.A.A$BC.A.A$B6.A.A2.2A
$4.A!
But at 25x25 there apparently aren't any such patches that could theoretically be part of a larger p4 phoenix -- if you set up that SAT problem, you get UNSAT. It sounds like 25x25 was a big enough patch that all higher periods up to 69 have also returned zero solutions.
... Now I'm wondering whether there are any periods above 4 where 23x23 is a big enough patch after all -- i.e., maybe the odd periods actually get gradually easier to prove UNSAT for. Or if 25x25 is always both sufficient and necessary for periods 4 through 69, that would be interesting too!
I realized that the general-case phoenix problem proof (which is the non-existence of non-p2 finite phoenices in Life) has not made it to the forums yet (it's already been on the discord since yesterday), and so I'll summarize apgoucher's proof here:
calcyman wrote:Yes... at a high level the proof proceeds as follows:
1. prove that no 2x2 square of a finite phoenix (of any period) can contain more than 2 live cells;
2. deduce the correctness of wwei23's table at https://conwaylife.com/wiki/User_talk:P ... ransitions
3. suppose that we have a finite phoenix of some period other than 2, and consider the first (by diagonal reading order) cell that's not vacuum or p2;
4. there must be some time T such that the cell is off at generation T and on at generation T+2;
5. run a SAT solver on the 29x29x32 spacetime cuboid that's spatially centred on that 'first non-p2 cell' and consists of generations [T-17, T+14] to determine all possible 39-cell patches that are locally consistent with these constraints in this cuboid;
6. for each of the 20 different patches that arise in this manner, set up a new SAT problem to find all 39-cell patches that can be the grandparent of each of these patches;
7. keep repeating this until we obtain the set of all possible 39-cell patches that can occur at any time T-2*k (k >= 0) in the history;
8. note that all 197 of these patches have the central cell OFF, so can't match generation T+2 which has central cell ON;
9. as such, it can't be periodic at all (because even if the phoenix had odd period p, then it would also be periodic with period 2p, which is ruled out).
This also means that all finite phoenices in Life are of constant population:
calcyman wrote:As a corollary, we establish the conjecture on https://conwaylife.com/wiki/Phoenix that every (finite) phoenix has constant population: if you consider the arrangement of red and blue cells, corresponding to cells that are live in odd and even generations, then every red cell has 3 blue neighbours and every blue cell has 3 red neighbours, so there are equally many of each colour. (Every infinite phoenix technically has constant population too -- because its population is aleph_0 in every generation -- but there are examples of higher-period phoenix agars where the population density is non-constant.)
x = 42, y = 24, rule = B3/S23
23bo5bo$22bobo4bo$12bo12b2ob2obob2o$12bo6b2o9b2o3bo$7b2obob2o2bob2o2b
2obo4b2o$6bo3b2o9b2o6bo4bo2bo$10b2o3b2o12b2o4b2o$4bo2bo4bo4bobo9bo7b2o
$5b2o4b2o5bo19bo$3b2o7bo21b3ob2o$3bo30b2o4b2o$2b2ob3o25b2obobo$2o4b2o
28bobobo$3bobob2o$bobobo22bobo7bo$23bo4bo2bo6bo$3bo7bobo9bobo5bo3b3o$
3bo6bo2bo5bo3bo11bo$4b3o3bo7bobo2b4ob2ob3o$6bo9bo8bo2b3o2bo$8b3ob2ob4o
2bobo4b3o$8bo2b3o4bo3bo$11b3o2bobo$18bo!
The quoted post contains an almost-strictly-volatile p4, in which there are only four cells that are oscillating with p2 instead of p4.
I found all the statorless parts using Nicolay Beluchenko's WLS modification, downloadable at viewtopic.php?p=98433#p98433
#C A period 4284 oscillator based on kickbacks.
#C This uses reactions between two phase-changing period 56 glider
#C streams produced by four period 28 glider guns. Pairs of gliders
#C alternately perform kickbacks and annihilations. A wave of
#C reactions travels down the glider streams from one pair of glider
#C guns to the other, reversing direction whenever the guns are reached.
#C The period can be increased by separating the pair of guns further.
#C David I. Bell, 22 January 2024
x = 242, y = 245
6boo8boo33bobbo$4bobbo8boo9bo18boo3b6o$4booboboo15bobo18bo9bo$5bobobbo
11bob3obo18bob7obo$bb3obbo14boo4boo14boobobobbobboboboo$bo23boo3bo14b
oobo7boboo$boboobobo13boobob4o12bo3bo9bo3bo$oobo4boo12bobbobo15b4ob3o
3b3ob4o$bobo5bo15bobbo20booboboo$boboobobbo19bo15boobb7obboo$oobo5bo6b
3o5boboboo9boo4boo4b3o4boo$3bo4boo4booboo4boo8bo6bo$3boobobo5b3o6bo7b
3o6bobo$23bo6bo10boo7bo$b4obbo15bo5bobboo11boo3b3o$bo3bobobbo12boobbo
bboobo10boo4b3ob3o$4booboboo10b3oboo4bo28boo$4bobbo23bo12b3ob3o4boo3b
oo$6boo9bobo7b3oboo15b3o3boo$12boo4boo6bobbo20bo$13bo4bo7booboboo6bo$
10b3o16bobbo5bo$10bo18boo7b3o3boob3o4booboo$45bob5obboobo$48boboo6bo$
24bobo18boo3b3ob3obo$25boo14boob3o6bobboboo$25bo15boboo7boboo4bo$43boo
6bobobob3o$39b5o3bo3boobbobo$39bo6bobo6boo$41bob4obbo$31bobo6boobo3bob
o$32boo12booboo$32bo12$45bobo$46boo$46bo12$59bobo$60boo$60bo12$73bobo$
74boo$74bo12$87bobo$88boo$88bo12$101bobo$102boo$102bo12$115bobo$116boo
$116bo$$115boo$114boo$116bo12$129boo$128boo$130bo12$143boo$142boo$144b
o12$157boo$156boo$158bo12$171boo$170boo$172bo3$224boo8boo$214bo8bobo8b
obbo$213bobo15booboboo$213bob3obo11boobobo$212boo4boo15bob3o$211bo3boo
5boo10bobo3bo$211b4oboboo12b3obboobo$214boboboo12b3o3boboo$213bobo15b
oo5bobo$185boo25bobobo6bo3boobb3o3boobo$184boo26boobb3o3bobobobboboo5b
oboo$186bo21bo7b3obbo7bobb3o3bo$208b3o7boobobbobobo3b3obboo$211bo5b3o
bboo3bo6bobo$208boobbo5boo15bob4o$208boboo3boo14boobobo3bo$210bo4boob
oo4bo6booboboo$210bobb3o3bo4bobo7bobbo$209boob3o9boo8boo$212bobbo12boo
$209booboboo12bo$209bobbo16b3o$211boo18bo$199boo16bo$198boo17bobo$200b
o16boo5$206b3obo13booboo$205booboobbo5boobo3bobo$207bobobo7bob4obbo$
217bo6bobo6boo$217b5o3bo3boobbobo$221bo7bobobob3o$219bobobbo7boo4bo$
213boo4boobbo5bo4boboo$212boo9bo7b4obo$214bo9bo3boo6bo$223b6o4bobo$
223bo8booboo$223bo$$220boo$219boo7boo$221bo6boo8boo$227bobo8boo$227boo
$227boo$219boo$218bobo$218bo$217boo4boo11boo$223bo4b5o4bo$227boo3boo$
221b5o9b5o$221bo5bobbobbo5bo$223boo11boo$222boobobobbobboboboo$225bob
7obo$225bo9bo$224boo3b6o$229bobbo!
IMO categorifying these reactions as oscillators ("dependent reflector loops") does not tell the whole story. The most interesting part is the dependent reflector itself, and those components deserve to be a separate category of discoveries, not necessarily presented in a completed oscillator form.
127:1 B3/S234cUser:Confocal/R (isotropic CA, incomplete)
Unlikely events happen.
My silence does not imply agreement, nor indifference. If I disagreed with something in the past, then please do not construe my silence as something that could change that.
glider_rider wrote: January 23rd, 2024, 6:47 pmThat's fair, especially for the 0-degree ones or the 180-degree ones that can't be looped. I just like looping them when possible.
Actually, shifters can be "looped" too, if you are willing to loop the entire universe into a torus:
#C p77 dependent glider shifter by carsoncheng
#C https://conwaylife.com/forums/viewtopic.php?p=163501#p163501
x = 61, y = 60, rule = B3/S23:T76,76
14b2o$obo11b2o$b2o$bo8$3b2o2b2ob2o$3b2ob2o3b2o$7b2ob2o$8b3o$9bo2$6b2o
15bo$7bo13bobo$4b3o15b2o$4bo16$41bo$42b2o$41b2o$51b2o$51b2o6b2o$59bo$
52bo4bobo$51b3o3b2o$51bob2o$54b2o$51bob2o$51b3o$52bo2$40b2o$40b2o6$58b
o$59bo$57b3o!
Also, If I remember correctly, when this was discussed on the forums, there was no observable consensus to describe those oscillators "loops" without adding the word 'dependent'.
I agree the oscillators visually resemble loops, but I still believe they are different enough from the actual glider loops. (And hence deserve a different description.) The gliders belong to the rotor and cannot be removed without breaking the oscillator.
NO, they are just oscillators with a rotor composed partially of glider streams. If you touch the rotor, the oscillator is destroyed. If you want to call them "loops", the word "dependent" should be used.
5) is it okay to rename the p71 glider shuttle to "p71 dependent reflector loop" while we're at it, with a redirect?
YES, it fulfills the criteria used so far. A DR-LOOP, something that visually resembles a "true" glider loop.
dvgrn wrote: July 1st, 2023, 6:37 am[...]
Time for a minor nitpick on terminology, I think: it doesn't seem like any of these should be referred to as "glider loops", without the word "dependent" in there somewhere at least. "Dependent" would give a good hint that you can't remove one of the gliders from the "loop" and expect the thing to still work.
However, there isn't actually any kind of signal loop at all in these things -- it's two or four separate segments. So any use of "loop" seems like not such a good idea. You wouldn't call something like this a "glider loop", right? [...]
glider_rider wrote: July 1st, 2023, 8:02 am
You could argue that the signals are sub-lightspeed, it's just that the signal corresponding to each glider is actually the first one that wouldn't appear if it weren't there. I do think "dependent glider loop" is a good clarification to make, though: [...]
127:1 B3/S234cUser:Confocal/R (isotropic CA, incomplete)
Unlikely events happen.
My silence does not imply agreement, nor indifference. If I disagreed with something in the past, then please do not construe my silence as something that could change that.
glider_rider wrote: January 23rd, 2024, 6:47 pm
That's fair, especially for the 0-degree ones or the 180-degree ones that can't be looped. I just like looping them when possible.
Does there exist a toaster variant with this frontend? I have been using WLS on it for around two weeks, but so far nothing, even though WLS shows it's promising.
A slower gun at the lower right could fix this, but I fear that its period would be related to the glider stream period, so to make the whole thing pointless.
#C A period 880 oscillator based on kickback reactions.
#C This uses many streams produced by period 80 glider guns.
#C The base glider stream period here is multiplied by 11.
#C Adding or removing glider guns at the top left will change the
#C multiplication factor, which is one more than the number of guns.
#C If such a change is made, simply adding eaters if necessary and
#C running the pattern will eventually result in the new oscillator.
#C David I. Bell, 26 January 2024
x = 1052, y = 1024
58boobb3o$58bobo8boo$59bo10bo$60boo8bobo3booboo$61booboo5b3oboo3bo$61b
o4bo6boobb3o$60bo4bo7b4o$60boo3bo8bobb3o$64bo10boobbo$63boo13bo$60bo
18b3o$59bobo19bo$60bo$47boo18bobo$47bo18bobbo$48bo19boo$35bo11boo$bbo
32b3o11boo12boo11boo$bb3o10bo22bo8boobbo10bobo11bobo$5bo7b3o21boo9bobo
11bo15bo$4boo6bo19boo7bo4boboboo9boo15boo$12boo17bobo6bobo3boo$29b3obo
bo4bobbo$28bo5boo5boo$28boo35bo$63boo5bo$51bo12boo3bobo$51b3o16bo6b3o$
20bo24boo7bo14bo7b3o$19boboo9boo10boo7boo13bobo6b3o$18bo4bo8boo12bo20b
o6b3o$19bo48bobo3b3o$20b4o45bo4b3o$9boo$oo4boobo30bo$oo4bobbo29bobo26b
oo$33boo4bobo4boo15boo3boo$33bobbo3bo3bobbo8boo5bobo35boobb3o$4bobboo
25b3o7b3o9boo7bo35bobobo6boo$4bobo14boo42boo35bo10bo$5bo15bobo8b7o3b7o
54bobo7bobo3bobboo$23bo8bo6bobo6bo3bo55bo5boo3boobbo$6boo15boo8boboboo
3boobobo3boboboo46bob3obo7bobb3o$6bobo25boobo5boboo4bobobobo44bobboobb
o7boo$5booboboo27bobo8boobobobobbo42boobo10bobb3o$5boobobbo26booboo8bo
bboob4o44bo12boobbo$5boobobo40bo4bo49bobo12bo$8boo42b3obobboo42bo18b3o
$54boo3boo41bobo19bo$103bo$90boo19b3o$11bo78bo19bo3bo$12boo77bo17bo5bo
$11boo65bo11boo18bo3bo$45bo32b3o11boo12boo3b3o5boo$45b3o10bo22bo8boobb
o10bobo11bobo$48bo7b3o21boo9bobo11bo15bo$47boo6bo19boo7bo4boboboo9boo
15boo$55boo17bobo6bobo3boo$72b3obobo4bobbo$71bo5boo5boo$71boo$53bo59bo
$52b3o26bo12bo17bobo$51boobbo25bo12b3o10boo4bo7boo$51bo3bo20bo4bo15bo
8bo12boboo$51bobo21boboo8bo8boo13boo5bo$75boboo7bobo22b3o7bo$52booboo
19b4obo6bo9b3o10b3o3boobo$54bo25boo5bo9bo3bo15boo$51bo44bo5bo$31bo11b
oo4bo3bo29bo12bo5bo$32boo9boo9bo28bo12bo5bo8boo$31boo18boobo21boo5bo5b
oo6bo3bo4boo3boo$48bobobobbo20bobo4bo4bobo7b3o5bobo$49boobobbo21bo4b3o
4bo18bo$51bobbo9boo15bobobo22boo$51bobbo9bobo8b8ob8o55boobb3o$51b3o12b
o8bo6bobo6bo3bo51bobobo6boo$66boo8boboboo3boobobo3boboboo48bo10bo5bo$
77boobo5boboo4bobobobo48bobo7bobob3obboo$82bobo8boobobobobbo50bo5boob
oo4bo$81booboo8bobboob4o45bob3obo9b4o$94bo4bo48bobboobbo9bo$95b3obobb
oo45boobo11bob3o$97boo3boo47bo13bobbo$152bobo12bo$149bo18b3o$148bobo
19bo$149bo$136boo19boo$51bo84bo19bobbo$52boo83bo19b3o$51boo71bo11boo$
91bo32b3o11boo12boo11boo$91b3o10bo22bo8boobbo10bobo11bobo$94bo7b3o21b
oo9bobo11bo15bo$93boo6bo19boo7bo4boboboo9boo15boo$101boo17bobo6bobo3b
oo$118b3obobo4bobbo$67bo49bo5boo5boo$65bobo49boo$66boo84bobo4bo$140bo
11boo4bobo6bo$134bo5b3o10bo3bobbo5b3o$108bobbo21boo8bo21bob3o$107bo3bo
9boo10bobo6boo12bobbo4bo3bo$107bo4bo8boo33bo6bo3bo$107bo4bo43bobo3b3ob
o$98bo9bo3bo44boo4b3o$96b4o9bobo52bo$71bo17boo4bo3bo29bo$72boo15boo4bo
bo30bobo26boo$71boo23bo25boo4bobo4boo15boo3boo$122bobbo3bo3bobbo8boo5b
obo$94bobo26b3o7b3o9boo7bo36boobb3o$93bobbo13boo42boo35bobo8boo$94bobo
13bobo8b7o3b7o54bo10bo$94b3o15bo8bo6bobo6bo3bo51boo8bobo6boo$95b3o14b
oo8boboboo3boobobo3boboboo48booboo5boobboo3bo$87bo9b3o23boobo5boboo4bo
bobobo47bo4bo8bob3o$85bobo8booboo27bobo8boobobobobbo43bo4bo8boo$86boo
6bobbobbo26booboo8bobboob4o43boo3bo9bob3o$94boo3bo40bo4bo51bo11bobbo$
95b4o42b3obobboo46boo13bo$97bo45boo3boo43bo18b3o$192bobo19bo$193bo$
180boo19boo$100bobo77bo19bobbo$101boo78bo18booboo$91bo9bo66bo11boo19bo
bo$92boo41bo32b3o11boo12boo4bo6boo$91boo42b3o10bo22bo8boobbo10bobo11bo
bo$138bo7b3o21boo9bobo11bo15bo$137boo6bo19boo7bo4boboboo9boo15boo$145b
oo17bobo6bobo3boo$162b3obobo4bobbo$161bo5boo5boo$161boo$107bo88bo6bo7b
oo$105bobo35bo40bo10bo6bobo$106boo34b3o25boo12b3o8b3o5bo6bo3bo$141bobb
oo23boboo14bo9bo11bo4bo$141bo3bo19b3o3boo13boo13bo6bobobo$142boobo19b
3obobo5boo22b3o3bobobo$144boo24boo3bobo27bo4bo$170bo5bo11b3o14bo3bo$
140bobo44bo3bo$120bobo10boo4bobbo43bo5bo14boo$121boo10boo4bo3bo29bo13b
o3bo9boo$111bo9bo16boobo24boo3booboo3boo7b3o5boo3boo$112boo25bo3boo21b
obb4ob4obbo15bobo$111boo31bo22b4obbobb4o18bo$140bobbo10boo42boo$141b3o
10bobo8b7o3b7o55boobb3o$156bo8bo6bobo6bo3bo51bobo8boo$156boo8boboboo3b
oobobo3boboboo48bo10bo$167boobo5boboo4bobobobo48boo8bobo3booboo$172bob
o8boobobobobbo46booboo5b3oboo3bo$127bo43booboo8bobboob4o46bo4bo6boobb
3o$125bobo56bo4bo49bo4bo7b4o$126boo57b3obobboo45boo3bo8bobb3o$187boo3b
oo49bo10boobbo$242boo13bo$239bo18b3o$238bobo19bo$239bo$226boo18bobo$
140bobo83bo18bobbo$141boo84bo19boo$131bo9bo72bo11boo$132boo47bo32b3o
11boo12boo11boo$131boo48b3o10bo22bo8boobbo10bobo11bobo$184bo7b3o21boo
9bobo11bo15bo$183boo6bo19boo7bo4boboboo9boo15boo$191boo17bobo6bobo3boo
$156bo51b3obobo4bobbo$157bo49bo5boo5boo$155b3o49boo35bo$147bo94boo5bo$
145bobo82bo12boo3bobo$146boo82b3o16bo6b3o$199bo24boo7bo14bo7b3o$198bob
oo9boo10boo7boo13bobo6b3o$197bo4bo8boo12bo20bo6b3o$198bo48bobo3b3o$
199b4o45bo4b3o$188boo$160bobo16boo4boobo30bo$161boo16boo4bobbo29bobo
26boo$151bo9bo50boo4bobo4boo15boo3boo$152boo58bobbo3bo3bobbo8boo5bobo
35boobb3o$151boo30bobboo25b3o7b3o9boo7bo35bobobo6boo$183bobo14boo42boo
35bo10bo$184bo15bobo8b7o3b7o54bobo7bobo3bobboo$202bo8bo6bobo6bo3bo55bo
5boo3boobbo$176bo8boo15boo8boboboo3boobobo3boboboo46bob3obo7bobb3o$
177bo7bobo25boobo5boboo4bobobobo44bobboobbo7boo$175b3o6booboboo27bobo
8boobobobobbo42boobo10bobb3o$167bo16boobobbo26booboo8bobboob4o44bo12b
oobbo$165bobo16boobobo40bo4bo49bobo12bo$166boo19boo42b3obobboo42bo18b
3o$233boo3boo41bobo19bo$282bo$269boo19b3o$190bo78bo19bo3bo$191boo77bo
17bo5bo$190boo65bo11boo18bo3bo$180bobo41bo32b3o11boo12boo3b3o5boo$181b
oo41b3o10bo22bo8boobbo10bobo11bobo$171bo9bo45bo7b3o21boo9bobo11bo15bo$
172boo52boo6bo19boo7bo4boboboo9boo15boo$171boo61boo17bobo6bobo3boo$
251b3obobo4bobbo$250bo5boo5boo$250boo$196bo35bo59bo$197bo33b3o26bo12bo
17bobo$195b3o32boobbo25bo12b3o10boo4bo7boo$187bo42bo3bo20bo4bo15bo8bo
12boboo$185bobo42bobo21boboo8bo8boo13boo5bo$186boo66boboo7bobo22b3o7bo
$231booboo19b4obo6bo9b3o10b3o3boobo$233bo25boo5bo9bo3bo15boo$230bo44bo
5bo$210bo11boo4bo3bo29bo12bo5bo$211boo9boo9bo28bo12bo5bo8boo$210boo18b
oobo21boo5bo5boo6bo3bo4boo3boo$200bobo24bobobobbo20bobo4bo4bobo7b3o5bo
bo$201boo25boobobbo21bo4b3o4bo18bo42bo$191bo9bo28bobbo9boo15bobobo22b
oo36boo3bo$192boo36bobbo9bobo8b8ob8o54bobobbo5boo$191boo37b3o12bo8bo6b
obo6bo3bo51bobo8bo$245boo8boboboo3boobobo3boboboo48b3o7bobo6boo$256boo
bo5boboo4bobobobo49bobbo5boobb3obbo$261bobo8boobobobobbo44b3obobo10b3o
$216bo43booboo8bobboob4o44bo4boo7boo$217bo55bo4bo48boobboo8boob3o$215b
3o56b3obobboo47b3o10bobbo$207bo68boo3boo47boo13bo$205bobo119bo18b3o$
206boo118bobo19bo$327bo$314boo19boo$314bo19bobboo$230bo84bo18booboo$
231boo69bo11boo18booboo$230boo37bo32b3o11boo12boo4bo6boo$220bobo46b3o
10bo22bo8boobbo10bobo11bobo$221boo49bo7b3o21boo9bobo11bo15bo$211bo9bo
49boo6bo19boo7bo4boboboo9boo15boo$212boo65boo17bobo6bobo3boo$211boo83b
3obobo4bobbo$295bo5boo5boo$246bo48boo$244bobo90bo$236bo8boo64bo6bo10bo
bo4bobo6bo$237bo72boo6b3o8boo6bo$235b3o50bo21bobo8bo8bo12b3obo$227bo
56b6o9boo19boo12bo7boboo$225bobo55booboobbo8boo32bobo6boobo$226boo47b
3o5boo3boboo41bobbo3bob3o$274b5o6b4obo43b3o$273boo3bo7boo47bo6bo$267b
oo4b5o29bo$250bo16boo3b4o30bobo26boo$251boo18b3o26boo4bobo4boo15boo3b
oo$250boo19boo27bobbo3bo3bobbo8boo5bobo$240bobo29boo27b3o7b3o9boo7bo$
241boo30bobo12boo42boo36boobb3o$231bo9bo31b3o12bobo8b7o3b7o54bobo8boo$
232boo38boboo14bo8bo6bobo6bo3bo51bo10bo$231boo39boo4bo11boo8boboboo3b
oobobo3boboboo48boo8bobo6boo$274bo3boo21boobo5boboo4bobobobo48booboo5b
oobboo3bo$266bo6b3o3boo25bobo8boobobobobbo45bo4bo8bob3o$264bobo5boobob
o27booboo8bobboob4o44bo4bo8boo$256bo8boo4boobbobo40bo4bo48boo3bo9bob3o
$257bo13boo4bo41b3obobboo48bo11bobbo$255b3o18bo44boo3boo47boo13bo$247b
o124bo18b3o$245bobo123bobo19bo$246boo124bo$359boo19boo$279bobo77bo19bo
bbo$280boo78bo18booboo$270bo9bo66bo11boo19bobo$271boo41bo32b3o11boo12b
oo4bo6boo$270boo42b3o10bo22bo8boobbo10bobo11bobo$260bobo54bo7b3o21boo
9bobo11bo15bo$261boo53boo6bo19boo7bo4boboboo9boo15boo$251bo9bo62boo17b
obo6bobo3boo$252boo87b3obobo4bobbo$251boo87bo5boo5boo$340boo$286bo88bo
6bo7boo$284bobo35bo40bo10bo6bobo$276bo8boo34b3o25boo12b3o8b3o5bo6bo3bo
$277bo42bobboo23boboo14bo9bo11bo4bo$275b3o42bo3bo19b3o3boo13boo13bo6bo
bobo$267bo53boobo19b3obobo5boo22b3o3bobobo$265bobo55boo24boo3bobo27bo
4bo$266boo81bo5bo11b3o14bo3bo$319bobo44bo3bo$299bobo10boo4bobbo43bo5bo
14boo$300boo10boo4bo3bo29bo13bo3bo9boo$290bo9bo16boobo24boo3booboo3boo
7b3o5boo3boo$291boo25bo3boo21bobb4ob4obbo15bobo$290boo31bo22b4obbobb4o
18bo42bo$280bobo36bobbo10boo42boo36boo3bo$281boo37b3o10bobo8b7o3b7o54b
o4bo5boo$271bo9bo53bo8bo6bobo6bo3bo51boo9bo4bo$272boo61boo8boboboo3boo
bobo3boboboo58bobo6boo$271boo73boobo5boboo4bobobobo48booboo5boo3bo3bo$
351bobo8boobobobobbo45boobobo8b5o$306bo43booboo8bobboob4o47b3o9bo$304b
obo56bo4bo48boobo11bob3o$296bo8boo57b3obobboo44booboo11bobbo$297bo68b
oo3boo45b3o14bo$295b3o119bo18b3o$287bo128bobo19bo$285bobo129bo$286boo
116boo19boo$404bo19bobbo$319bobo83bo19bobo$320boo70bo11boo19b3o$310bo
9bo38bo32b3o11boo12boo11boo$311boo46b3o10bo22bo8boobbo10bobo11bobo$
310boo50bo7b3o21boo9bobo11bo15bo$300bobo58boo6bo19boo7bo4boboboo9boo
15boo$301boo66boo17bobo6bobo3boo$291bo9bo84b3obobo4bobbo$292boo41bo49b
o5boo5boo$291boo43bo48boo$334b3o84bo5bo7bo$326bo81bo10boo5bobo6boo$
324bobo74boo5b3o9boo5bo6boobo$316bo8boo73boo9bo21bobb3o$317bo56bobobb
oo8boo11bo7boo20bobobo$315b3o56bobboo10boo32b3o5bobobo$307bo66bobboobb
o41boobobb3obbo$305bobo57b3o7boobbobo43boo3boboo$306boo56booboo62boo$
357boo4bobbo30bo34bo$339bobo15boo4boo31bobo26boo$340boo21boo25boo4bobo
4boo15boo3boo$330bo9bo22boo25bobbo3bo3bobbo8boo5bobo$331boo30bo27b3o7b
3o9boo7bo42bo$330boo29bo3bo12boo42boo36boob3o$320bobo41boo12bobo8b7o3b
7o54bob3o6boo$321boo39bo17bo8bo6bobo6bo3bo51boo9bo5bo$311bo9bo40boo16b
oo8boboboo3boobobo3boboboo58bobobobobboo$312boo41bo7bo3boo22boobo5bob
oo4bobobobo49boboo5boobo5bo$311boo43bo6boo3boo26bobo8boobobobobbo45boo
bobo9b4o$354b3o6bo3boo26booboo8bobboob4o43bo5bo9bo$346bo14booboobo40bo
4bo48boobo11bob3o$344bobo15b4obo41b3obobboo45b4o11bobbo$336bo8boo17b3o
44boo3boo62bo$337bo124bo18b3o$335b3o123bobo19bo$327bo134bo$325bobo41bo
79boo19boo$326boo42boo77bo19bobbo$369boo79bo19b3o$359bobo75bo11boo20bo
$360boo42bo32b3o11boo12boo11boo$350bo9bo43b3o10bo22bo8boobbo10bobo11bo
bo$351boo54bo7b3o21boo9bobo11bo15bo$350boo54boo6bo19boo7bo4boboboo9boo
15boo$340bobo71boo17bobo6bobo3boo$341boo88b3obobo4bobbo$331bo9bo88bo5b
oo5boo$332boo41bo54boo$331boo43bo88bo6bo$374b3o76bo11bobo3bobo5b3o$
366bo45bo26boo3boo7b3o9boo5bo9bo$364bobo44b3o24bobo3b3o9bo21bo3bo$356b
o8boo43boobbo19boobboo5bobo7boo11b3o6bobobbo$357bo54b3o19boobboo5boo
20b5o3bobbobo$355b3o81bo5boo20boobbo3bo3bo$347bo92booboo12boo16bo$345b
obo41bo20bo45bobbo16b3o$346boo42boo10boo5boo45booboo$389boo11boo6bo46b
obo10boo$379bobo26b3o6bo17boo3booboo3boo8bo6boo3boo$380boo26bo3boo3bo
17bobbo3bo3bobbo15bobo$370bo9bo28bo3bo22b3o7b3o18bo$371boo37b3o10boo
42boo$370boo51bobo8b7o3b7o$360bobo62bo8bo6bobo6bo3bo$361boo62boo8bobob
oo3boobobo3boboboo$351bo9bo74boobo5boboo4bobobobo$352boo41bo45bobo8boo
bobobobbo$351boo43bo43booboo8bobboob4o$394b3o56bo4bo$386bo67b3obobboo$
384bobo69boo3boo$376bo8boo$377bo$375b3o$367bo$365bobo41bo$366boo42boo$
409boo$399bobo$400boo$390bo9bo$391boo$390boo$380bobo$381boo42bo$371bo
9bo41bobo$372boo41bo8boo$371boo43bo$414b3o$406bo$404bobo$396bo8boo$
397bo$395b3o$387bo$385bobo41bo$386boo42boo$429boo$419bobo$420boo$410bo
9bo$411boo$410boo$400bobo$401boo42bo$391bo9bo41bobo$392boo41bo8boo$
391boo43bo$434b3o$426bo$424bobo$416bo8boo$417bo$415b3o$407bo$405bobo
41bo$406boo42boo$449boo$439bobo$440boo$430bo9bo$431boo$430boo$420bobo$
421boo42bo$411bo9bo41bobo$412boo41bo8boo$411boo43bo$454b3o$446bo$444bo
bo$436bo8boo$437bo$435b3o$427bo$425bobo41bo$426boo42boo$469boo$459bobo
$460boo$450bo9bo$451boo$450boo$440bobo$441boo42bo$431bo9bo41bobo$432b
oo41bo8boo$431boo43bo$474b3o$466bo$464bobo$456bo8boo$457bo$455b3o$447b
o$445bobo41bo$446boo42boo$489boo3$470bo16boo$471boo13boo$470boo16bo$
460bobo$461boo42bo$451bo9bo41bobo$452boo50boo$451boo4$476bo$477bo38b3o
$475b3o38bo$467bo49bo$465bobo$466boo7$480bobo$481boo$471bo9bo$472boo$
471boo5$536b3o$536bo$487bo49bo$485bobo$486boo9$491bo$492boo$491boo5$
556b3o$556bo$557bo18$576b3o$576bo$577bo18$596b3o$596bo$597bo11$551bo$
552boo$551boo5$616b3o$616bo$567bo49bo$565bobo$566boo7$580bobo$581boo$
571bo9bo$572boo$571boo4$596bo$597bo38b3o$595b3o38bo$587bo49bo$585bobo$
586boo4$610bo$611boo$610boo$600bobo$601boo$591bo9bo$592boo$591boo$$
626bo$624bobo$616bo8boo$617bo38b3o$615b3o38bo$607bo49bo$605bobo$606boo
$$639bobo$640boo$630bo9bo$631boo$630boo$620bobo$621boo$611bo9bo$612boo
41bo$611boo43bo$654b3o$646bo$644bobo$636bo8boo$637bo$635b3o$627bo$625b
obo41bo$626boo42boo$669boo$659bobo$660boo$650bo9bo$651boo$650boo$640bo
bo$641boo$631bo9bo$632boo41bo$631boo43bo$674b3o$666bo$664bobo$656bo8b
oo$657bo38b3o$655b3o38bo$647bo49bo$645bobo$646boo$$679bobo$680boo$670b
o9bo$671boo$670boo$660bobo$661boo$651bo9bo$652boo$651boo$$686bo$684bob
o$676bo8boo$677bo38b3o$675b3o38bo$667bo49bo$665bobo$666boo4$690bo$691b
oo$690boo$680bobo$681boo$671bo9bo$672boo$671boo4$696bo$697bo38b3o$695b
3o38bo$687bo49bo$685bobo$686boo7$700bobo$701boo$691bo9bo$692boo$691boo
5$756b3o$756bo$707bo49bo$705bobo$706boo9$711bo$712boo$711boo5$776b3o$
776bo$777bo18$796b3o$796bo$797bo18$816b3o$816bo$817bo11$771bo$772boo$
771boo5$836b3o$836bo$787bo49bo$785bobo$786boo7$800bobo$801boo$791bo9bo
$792boo$791boo4$816bo$817bo38b3o$815b3o38bo$807bo49bo$805bobo$806boo4$
830bo$831boo$830boo$820bobo$821boo$811bo9bo$812boo$811boo$$846bo$844bo
bo$836bo8boo$837bo38b3o$835b3o38bo$827bo49bo$825bobo$826boo$$859bobo$
860boo$850bo9bo$851boo$850boo$840bobo$841boo$831bo9bo$832boo41bo$831b
oo43bo$874b3o$866bo$864bobo$856bo8boo$857bo$855b3o$847bo$845bobo41bo$
846boo42boo$889boo$879bobo$880boo$870bo9bo$871boo$870boo$860bobo$861b
oo$851bo9bo$852boo41bo$851boo43bo$894b3o$886bo$884bobo$876bo8boo$877bo
38b3o$875b3o38bo$867bo49bo$865bobo$866boo$$899bobo$900boo$890bo9bo$
891boo$890boo$880bobo$881boo$871bo9bo$872boo$871boo$$906bo$904bobo$
896bo8boo$897bo38b3o$895b3o38bo$887bo49bo$885bobo$886boo4$910bo$911boo
$910boo$900bobo$901boo$891bo9bo$892boo$891boo4$916bo$917bo38b3o$915b3o
38bo$907bo49bo$905bobo$906boo7$920bobo$921boo$911bo9bo$912boo$911boo5$
976b3o$976bo$927bo49bo$925bobo$926boo9$931bo$932boo$931boo5$996b3o$
996bo$997bo8$947boo4boo4boo4boo4boo4boo4boo4boo$947bo5bo5bo5bo5bo5bo5b
o5bo$948b3o3b3o3b3o3b3o3b3o3b3o3b3o3b3o$950bo5bo5bo5bo5bo5bo5bo5bo$$
994boo$994boobboo$998bobo$999bo$$996boo4b3o11b3o$997bo4b3o11bo$994b3o
5b3o12bo31boo$994bo10b3o31bo8bobo$1005b3o30bobobb3obobo$1005b3o22bo8bo
3boboboo$1030b3o12b3o$1033bo7boo6b3o$1032boo7b3obo$1038bo3b3obo$1026b
3o7boobo4boo$1026bo8bo$1014boo3b3o5bo7boboobbo$1014boo3b3obbo11bobbobb
o6bo$1019boobbobo14boo5b3o$1024bo21bo$1046boo$$1018boo23b3o$1018boobo
20bobo$1022bo9boo8bo3boo$1019bo13bo9boboo$1020boboo6b3o8bobobo$1022boo
6bo9boboobobo$1040bo5boo$1039boo!
I am amazed by this pattern. It's not easy to see how adding another stream at the left edge of the existing stream can change the period of the rightmost stream.
