ConwayLife.com

Edit 3: Here's another "Trivial p6", just for amusement. Right is new. x = 24, y = 14, rule = LifeHistory 11.A.2A$.2A6.2A3.A$A2.A9.A4.A$.2A2.2A4.A5.A.A2.2A$3.2A2.A4.A4.A2.A2.A $.2A3.A3.A.3A.2A.2A.A$A2.2A2.6A4.A4.2A$A.A3.3A6.2A2.A.A$.A.2A.A7.2A.A 4.2A$3.A2.A.A2.4A2.A.2A.A$3.A.A4.A4.A.A.A2.A$2.2A.A.2...

Found with Chris's script. Seems pretty elementary as a transformation. x = 33, y = 35, rule = B3/S23 bo$2bo$3o5$28bo$27bo$27b3o11$25bo$24bo$24b3o3$12bo$10bobo$11b2o4bo$17b obo$17b2o10bo2bo$27b6o$26bo$27b2obo$25bobob2o$25b2o! Two seconds in Seeds of Destruction: x = 23, y = 33, rule = B3/S23 21bo$2...

gameoflifemaniac wrote: ↑

November 4th, 2019, 11:24 am

Is there any lower bound on how quickly the Busy Beaver function grows?

Any computable function suffices.

no. Probably gamma_1 = sup{gamma_0+1,phi_gamma_0+1(0),phi_phi_gamma_0+1(0)(0),etc.} Probably? Yes. Remember, phi(1,0,a) enumerates fixed points of phi(0,a,0). Then phi(1,1,a) enumerates fixed points of phi(1,0,a), phi(1,2,a) the fixed points of phi(1,1,a), and so on until phi(2,0,a) enumerates the ...

Another promising catalysis:

toroidalet wrote: ↑

October 31st, 2019, 8:32 pm

This is sort of a stupid question, but why isn't ε_1=ε_0^ω?

Epsilon numbers are the fixed points of w^, so e_a = w^(e_a) for any a.

w^(e_0^w) = w^(e_0+e_0^w) = e_0^e_0^w > e_0^w, so e_1 cannot be e_0^w.

Great work, Goldtiger! And now for the obligatory round of improvements from the community. :wink: The fuse could be shortened with an activation like this: rle EDIT: Optimizing out the LWSS coming from the right: rle Another four cells off the boat, and one fewer cleanup glider: x = 62, y = 67, ru...

Can someone explain bashicu matrix systems? I think I understand what this file is trying to say, but not much else: https://googology.wikia.org/wiki/File:BMS_expansion.png I'll try my best to paraphrase what goes on for BM2.3: If your last column is all zeroes, your matrix is a successor, and you ...

testitemqlstudop wrote: ↑

October 13th, 2019, 11:39 pm

Is the diagonalization omega, epsilon, delta, gamma, etc. literally just w^^...n times...^w?

By most definitions, w ^{n} w is phi(n-1,0).
Gamma_0 is then w ^{w ^{w^{ ... } w} w} w, in keeping with the "standard" result f_(w+1) = s_(G0)

Feather to carrier: rle This will probably find a few things during the next transfer.py run. Stumbled upon a way to save four gliders in the middle of trying something more complicated: x = 46, y = 32, rule = Life 8bo$6bobo31bo$7b2o19bo9b2o$26b2o11b2o$4bo22b2o$5b2o$4b2o2$29bobo$29b2o $30bo3$24bo$2...

Here's a synthesis of xs19_03p6853z3543: x = 93, y = 19, rule = B3/S23 7b2o2b2o34b2o2b2o34b2o2b2o$7bobo2bo34bobo2bo34bobo2bo$9bobo37bobo23bob o11bobo$8bobo37bobo5bo19b2o10bobo$8bo39bo6bo20bo11bo$6b2obo36b2obo5b3o 28b2obo$5bo2bo36bo2bo36bo2b2o$5b2o37bobo37bobo$obo7bobo32bo39bo$b2o7b 2o$bo9bo2$78bo$9...

This should help with the various Chop Suey still lifes, though I'm not sure what an efficient way to build the scaffolding would be: x = 42, y = 40, rule = B3/S23 3bo$4bo12bobo4bobo$2b3o15bo3b2o$20bo4bo$17bo2bo$18b3o3$16bo$2bo12bo$3b o11b3o$b3o17bo3bobo$20bo4b2o$20b3o3bo3$9b2o$9b2o2$9b2o$9bobo2b2o...

Now that one was tough , but 90: x = 13, y = 14, rule = LifeHistory 3.E$.3E$E6.2E$.A.3A2.E$EA.A2.2E$.A.EDE$E.DA.A3.2E.E$E.A2.2A2.A.2E$.A. EA2.A.A$3.A2.A.2A$3.E.2A$4.A2.3A$6.E2.E$7.2E! EDIT: 88 x = 13, y = 14, rule = LifeHistory 2.2E$3.E$2.E4.2E$.A.3A2.E$.A.A2.2E$EA.EDE$E.DA.A3.2E.E$2.A2.2A2.A.2E$ .A...

Testitem's variant is very close to Hardy, which is related to FGH as I stated above, so dco's relation to Hardy is essentially dco's relation with that variant. dco(G0+n) is certainly weaker than f_(G0+1)(n), probably between H_(G0) and H_(G0+w). so dco(G0+n) would be between f_G0(n) and f_(G0+1)(...

A for awesome wrote: ↑

October 8th, 2019, 5:32 pm

103:

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x = 6, y = 10, rule = LifeHistory
3.2E$.3A.E$A2.D.E$.4A$2.A$E2.D$E4A$2.D2.A$2.3A$2.E!

I'd be surprised if this wasn't minimal.

I would certainly be inclined to agree, since I specifically designed it to be impossible with two OFFs, well done!
Next puzzle is yours.

85, as promised: New puzzle:

So how does dco(a+n) compare to the hierarchies? And where is dco(gamma_0+n) in the fgh? Does it eventually exceed f_a(n) for all n < gamma_0 or something? Or is it only fgh_(something in the Veblen hierarchy)-ish? As I've stated before, I think Goodstein sequences are less powerful than dco, makin...

Isn't it also stronger than the HH? Given: dco(a+1,n) = dco(a, n+1 )+1 In that case, dco(Γ_0+n) catches up to f_Γ_0(n), since the HH catches up to the fgh at e_0, and thus dco(Γ_0+n) = dco(Γ_0,n) >= HH(Γ_0,n) = f_Γ_0(n) (for large enough n) Incorrect. Hardy and FGH are related in that if f_a(n) = H...

dco(a) is slightly stronger than the SGH:

SGH wrote: s(a+1,n) = s(a,n)+1
s(a,n) = s(a[n],n)

dco wrote: s(a+1,n) = s(a,n+1)+1
s(a,n) = s(a[n]+a[n-1]+...+a[0],n+1)

Yes, that's right for unidimensional arrays. But I don't get anything above 2d Let the first non-one entry after b be the "active point" P and the separator before it S. Decrease S according to the following rules, and call the result S': {k #} -> {k-1 #} {# 1,k #} -> {# b,k-1} If this separator ha...

127: x = 19, y = 11, rule = LifeHistory 11.2E$11.E.E$9.2E2.E$E.EA.EA.A2.2E.AE$2E.A.AD2.2A2.ED.E$3.E.A.2A.DA.A 2.A$3.A.A.E2.2A.E2A$4.E2.E$7.2E6.3E$15.E2.E$17.2E! Can be trivially modified to save an OFF and get you to 94: x = 19, y = 11, rule = LifeHistory 11.2E$11.E.E$9.2E2.E$E.EA.EA.A2.2E.AE$2E.A....

Hdjensofjfnen wrote: ↑

October 3rd, 2019, 1:49 am

toroidalet wrote: ↑

October 2nd, 2019, 11:49 pm

176

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rle

I am pretty sure the above is minimal.
One above minimal (177):

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rle

I believe the minimum is 174 (5/14), you're almost there.

alternate x = 8, y = 9, rule = LifeHistory 2.2E.AE$.A.E.E$.A.E.E$2A2.E$2.2E$2.E2.D$3.E2.EA$4.A2.A$5.2A! The red cell is impossible 42 is the min (this pattern is actually 45, A for Awesome got 42) Incorrect, 41: x = 7, y = 7, rule = LifeHistory E.2A.EA$2E.A.A$3.A.A$3.ED2E$2.2A$2.A2.AE$3.AE.A! 39, w...

∂ |-> w∂ is the same as w^∂ |-> w^(1+∂), so f(i) is w^{i'th limit of successors} = w^(wi). The "last fixed point" does not exist because there is no greatest ordinal. There has to be something wrong with that logic, because if ∂ = w2 = w+w, then it is implied that w+w = w(w+w), or w+w = w^2+w^2. Of...

...This topic is taking over BOTH the LTCFC and TFNCAQ so now it deserves its own thread. Primarily I'm concerned about large ordinals here. For example, the first fixed point of ∂ |-> w∂ is w^w. Hence we can make a function: f(i) = the i-th fixed point of ∂ |-> w∂ What would f(2) be? f(w)? Would t...