Yeah definitely, this was hard enough. Its kind of fun to learn more about how these notations work, but explaining it is a bit of a chore, and the vintage deck is a whole 'nother beast. I tried to puzzle it out on my own, and its just incredible how fast these things get completely intractable.
You might want to improve my estimates a bit, and currently its at 6->2^(700,000)->20->12 but it should really probably be something like 6->(6->2^(700,000)->X->6)->20->12. Or some further nested chain nonsense.
Edit: Hmm, just double checking the combat triggers, which seem like they need to be really good to be worth it. I think maybe Hadana's Climb is the closest? It looks like it would definitely be better than a Concoct if we didn't already use Sarkhan precombat to let Raging Regisaur kill him. It is an extra black mana from combat though, so it is probably still good.
Another thought is using gideon6 and Role reversal to kill sarkhan in combat by attacking him, without using Raging Regisaur at all. One bonus to this is that we can equip gideon and then wrath in combat, though we still can't Purify, so it's probably not worth it. (Edit2: oops gideon doesnt become indestructible himself, so this is definitely a no go)
Hmm with 1 black from Hadana's climb and the 3 mana from dowsing dagger, is Nikya of the old ways better than a connive // concoct? Just for the 2 extra black before the first real azor's gateway flip?
EDIT: ABORT, UGH
I found an infinite:
Activate Azor's Gateway, draw bankrupt in blood, exile an artifact (like dowsing dagger)
use Karn to return the artifact
Repeated reverberation Sarkhan, dragonsoul kills both karn and himself
cast bankrupt in blood
combo off.
use excess draws to redraw the walkers.
I don't see any sort of simple fix.
Cut repeated reverberation? loses the Sarkhan layer, Also makes the start much less explosive.
Cut Azor's gateway? Can't make much black then.
Cut Karn? This seems the best, we can still exile enough for a few flips, but more like 4 than 20
Ah, good catch. It does look like cutting Karn is the way to go. We still want [c}Azor's Gateway[/c], since we don't have Spark Double for Nikya of the Old Ways. And I guess Squee, the Immortal will go infinite with the card drawing.
Without the artifact retrieval, it looks like we have 25 exilable cards without Gravitic Punch, or 24 with. That means that if we are required to send the opponent to a low negative life total, adding Saheeli, Sublime Artificer won't add another Sanctum of the Sun. It won't lose one either though, so we might as well keep it in, since we can still have Dowsing Daggers and Hadana's Climb.
We can exile 24 cards, so we can get 4 Sanctum of the Suns that way, and we can get one more by using the Saheeli trick. That takes us to more than 10 -> X1 -> 6 -> 12 final damage.
X1 is how much life we can get before we tap the first Sanctum of the Sun. The main component of this is the number of times we can activate Sarkhan, Dragonsoul. We can activate it once when we play it, then we can recycle it once using Raging Regisaur, then another time for each black mana that we can produce prior to activating the forst SotS. We can use Nikya so that each of the four Dowsing Daggers can add four black mana each, then Hadana's Climb can add two more. So we can activate Sarkhan 20 times, so that X1 > 10 -> X2 -> 21 -> 11.
In turn, X2 is how great a TYS count we can get prior to the first Sarkhan activation. THe biggest component of that comes from using Whispering Snitch to gain the opponent a lot of life. We can Switcheroo a Citywatch Sphinx to the opponent, then destroy it (with Cleansing Nova, say). Then in response to the surveil trigger, we can Switcheroo a lot of Whispering Snitches to the opponent, and the surveil will trigger a lot of life gains from the Whispering Snitches. So that gives us X2 > 10 -> 10 -> X3 -> 9, where X3 is how many Whispering Snitches we can transfer over to the enemy.
The main component of X3 comes from the fact that we can resolve 19 copies of Vicious Rumors, making use of the opponent's initial 20 health. So X3 > 10 -> X4 -> 19 -> 8, where X4 is how high we can pump our TYS count before the first Vicious Rumors.
The main component of X4 comes using our 16 remaining life after our initial start to cast Gaea's Blessing followed by Ionize. We can do this 7 times, and each time we can get many Repeated Reverberation triggers, and four Thousand-Year Storm triggers, so we get many copies of Gaea's Blessing 35 times. So X4 > 10 -> X5 -> 36 -> 7.
The main component of X5 comes from the Bankrupt in Blood that we can draw and cast during the initial start, for 5 big batches of copies. So X5 > 10 -> X6 -> 6 -> 6.
The main component of X6 comes from the four Blood for Bones, which each get 5 big batches of copies, so X6 > 10 -> X7 -> 21 -> 5.
The biggest component of X7 copies from putting many copies of World Shaper into play, and destroying them, each of which creates a trigger to refetch [c/]Lotus Field[/c]. So X7 > 10 -> 10 -> X8 -> 3.
Before we put World Shaper into play, we can use Wayward Swordtooth to allow us to play Emergence Zone with more than 2^700,000 Evolution Sages in play, so X8 > 10 -> 10 -> 2^700,000 -> 2.
Note that X1 > 10 -> X2 -> 21 -> 11 > 2 -> 4 -> 2 -> 12. So after 5 Sanctum of the Suns, we get a final damage of more than 2 -> 4 -> 7 -> 12. So that's a simpler lower bound, but it is harder to expand like we did above.
Yeah, I checked for that infinite before but that was before sarkhan+repeated reverberation was a layer. Oh well, its still ->12.
We cant use karn for an early animation so there's no ministage before the Emergence Zone. So we make more like 10^13 Evolution sages.
Our exileable cards are a bit clogged at 4, with 2x leyline 3x blood for bones, kykar, and raging regisaur. Good thing connive // concoct is basically the same card for this challenge.
Being one card short of 5 flips is kind of frustrating, but I don't see any way to squeeze an extra slot, and still kill them.
I think we can cut Militia bugler, Kaya's Ghostform, and one The Antiquities War for Mystic Forge and Tezzeret's Gatebreaker. Hmm, no this is still short a way to recycle Teshar.
The Gatebreaker is unfortunately not naturally a creature to replicate.
Edit: i tried a bunch of lines, but cant seem to get to another card for 5 flips. Oh well, will rewrite karn out tomorrow.
Edit2: the start needs an extra untap effect, to have enough blue for Precognition field, TYS, and replicate. Making the last Blood for bones a Stony Strength
Edit3: isn't Connive // Concoct just better than Blood for bones? we get a free reshuffle with each resolution instead of a free guy to hand. If we had an extra creature worth returning (worldshaper#2?), that could be worth it, but we don't.
Also, we are actually 2 cards away from 5 flips, we need to keep a Stony Strength to actually make the guy we Gravitic punch them with giant.
edit4: rereading your estimation and I think I'm missing something?
Are you sure we get to ...->7->12?
I agree that we get to 7->X1->6->12 where X1 is 7->x2->21->11 but I don't see how that can possibly be big enough to give us something like 7->c->7->12? where c is some small non chained arrow constant like 3x10^18 or 4.
By definition, 7->c->7->12 = 7->c->(7->c->6->12)->11 which seems difficult to compare to 7->(7->x2->21->11)->6->12
Yeah, maybe Connive // Concoct is better - I didn't really look at it carefully before just choosing Blood for Bones. So make both the main combo card and the two extras Connive // Concoct?
Good point about needing Stony Strength - we could also use Leyline of Abundance, but either way we would need to keep another card around. I imagine there is some other way to kill the opponent with just one card needed though. If we find a way to save a card, that would definitely worth looking for. I guess right now it's not so pressing.
So, before the first Sanctum of the Sun we get to more than a -> b -> 21 -> 11 for nongigantic a and b. This will be more than a -> b -> 2 -> 12 = a -> b -> a^b -> 11, so long as a^b <= 21. So for example we have more than 2 -> 4 -> 2 -> 12. Then each Sanctum of the Sun will increase the third number by one, so we wind up with more than 2 -> 4 -> 7 -> 12. But we need the 2 and 4 at the beginning.
Concoct is almost the same exact card as blood for bones here, and either way it doesnt really show in the estimate, but Connive is also a blue instant we can cast in the stage too, which contributes to the leading 7 in my estimate.
Stony Strength is better to keep than Leyline as it can make white mana from an animated key. I actually couldn't find a better card to kill them with, though I didn't look very hard.
Oh I see, the 2 -> 4 -> 2 -> 12 is the base storm before doing any flips and we are increasing that by the 5 flips we get to come to 2 -> 4 -> 7 -> 12. Kind of a tricky way to get the extra number, the first two numbers seem like they really shouldn't matter so much, but when the numbers are small it works out like that.
That a -> b -> 2 -> 12 identity seems odd, but actually checks out a -> b -> 2 -> 12 = a -> b -> (a -> b -> 1 -> 12) -> 11 = a -> b -> (a -> b) -> 11 = a -> b -> a^b -> 11
I guess by that logic we get past 2->2->2->13 (which is just 4), and are one flip shy of getting to 2->3->2->13 (which is much more than 4).
It does seem like these would not be nearly as tight of a bound as the 7 -> (7 -> x2 -> 21 -> 11) -> 6 -> 12 estimate, even though they have bigger ending numbers.
Edit: Well, I refactored Karn out, redid the beginning with an extra stony strength for storm, and updated all of the changed images. I'm gonna give us one more day to find any other errors then post the view only link to r/magictcg.
Yeah, its hard to get a picture of how the different estimations are all linked, especially with things like Whispering Snitch and extra TYS.
How big does x2 need to be for 7 -> (7 -> x2 -> 21 -> 11) -> 6 -> 12 to be greater than 2 -> 4 -> 7 -> 12?
My estimate isn't great either,
Because of not factoring in the extra sphinxes each top level wrath actually squares the storm and we get a huge number in the front as well, but actually saying that involves some more fiddling with exponents in an already fiddly section.
So the better estimate would be X1->X1->6->12.
where X1= X2->X2->21->11 (extra sarkhans)
and X2= x3->x3->x3->9 (extra life gain triggers from the snitch)
and so on... with their extra life, our extra life, the extra cards from hand from the setup... and extra TYS triggers the whole way, making everything more powerful.
Edit: sent you the edit link again, good catch on the free stony strength I missed.
2 -> 4 -> 7 -> 12 is approximately 2 -> (2 -> 4 -> 16 -> 11) -> 6 -> 12. (The latter is a little bit greater; for the former, we can start with 2 -> 4 -> 16 -> 11 = 2 -> 4 -> 2 -> 12, and then apply the function X to 2 -> 4 -> X -> 11 five times; for the latter, we can start with 2 ^ (2 -> 4 -> 16 -> 11) = 2 -> (2 -> 4 -> 16 -> 11) -> 1 -> 12, and then apply the function X to 2 -> 4 -> X -> 11 five times. So the latter is a little bit greater since we start with a larger number.)
Switching the initial 2 to a 7 makes almost no difference, so 2 -> 4 -> 7 -> 12 will be just a little bit less than 7 -> (7 -> 4 -> 16 -> 11) -> 6 -> 12 as well. So 2 -> 4 -> 7 -> 12 will be greater than 7 -> (7 -> 3 -> 16 -> 11) -> 6 -> 12, for instance.
The first number doesn't make a whole lot of difference; really the biggest significance of a and b in a -> b -> c -> d is the fact that when you reduce down to a -> b -> 1 -> d, that equals a^b. So it is the value of a^b that is important; in case of a tie, b is the tiebreaker. (As is typical, this rule of thumb does not always hold, for example if d = 1, then we just have a -> b -> c, and we know that b is much more important then a then, if a >= 3.) Since we are dealing with X's that are length 4 Conway chains themselves, generally speaking if X1 is a Conway Chain such that X > 2^X1, then X will in all likelihood will be greater than X1^X1 as well. So we could choose X1 = 2 -> X2 -> 21 -> 11, or X1 = X2 -> X2 -> 21 -> 11, and either will be valid. I've been using 10 as a default number, since 10 is a commonly used "base" number.
I got the extra red back by fizzling Ionize after Omniscience, so everything after stays the same.
Added a tappedout link to the decklist.
Thanks for the explanation, these corner cases with relatively small numbers really help.
Yeah I'm just using 7 because we add 6 storm each top level reshuffle and have X of those so we bring X to 7X, and it'll be easier to follow that than switching the base halfway through.
Edit: hmm, it looks like we can use disdainful stroke to fizzle ionize before the first gaea's blessing too, and get an extra storm.
Edit2: I know this doesn't impact any of the estimations, but someone will probably ask. Do you have any clue about how to optimally make/spend our red mana once we have flash? Other than Kykar its generally better as white mana from mox amber, instead, but we can get Kykar back with a connive, and will have to to cast Repeated Reverberation for Sarkhan. Though we can't save it in hand like we could with blood for bones, so we dont get a ton of mana, it is still more than enough to cast a few extra repeated reverberations. Actually no, we can replicate the elemental tokens, so we can get X red.
I kinda think we want to concoct back kykar as little as possible, as Worldshaper is a better target all the time. but we do want to still repeated reverberate the card draw and the concoct itself. But do we want to use the red in the stage? Do we want to get RR on RR for the next layer card? Do we want to just spend all the red we can ASAP to make the stage combo stronger?
Hmmm... As you say, it's not worth getting Kykar with Concoct, except when we absolutely need it for Sarkhan, I think. So we wouldn't be using red mana for the stage.
When we do actually fetch Kykar with Concoct, I imagine it will be the last Concoct before we activate Sarkhan. So we don't get any more stages, we just use our red mana to cast maybe a couple of Repeated Reverberations (since we get a reshuffle from Concoct), and then activate Sarkhan. We can have some leftover red mana after the Sarkhan activation, I imagine we want to use it as quickly as possible, since generally earlier is better. I don't think we'll be saving any red mana to use during a stage.
Oh, we don't actually need to ever get kykar with concoct, we can just make the needed red at the very end of a stage. That seems much more efficient. Kykar gets exiled anyway too.
As for Throne of Eldrane, rotation will kill the speed of the deck, but Danse of the Manse finally lets us animate Thousand Year Storm, and Mirrormade is also very good for this.
Edit: Post rotation, assuming we want to be able to Replicate or Quasiduplicate TYS, reshuffling looks tricky. Midnight Clock and Enhanced Surveillance look too cheap, we can animate them once and use tokens to reshuffle. Any sweeper will also likely kill the TYS and need another dance of the manse to animate it, which we could also use to animate the reshuffler.
Spells like Finale of Revelation and Emergency powers would eliminate a lot of countermagic, which we probably need to be able to recast whatever spells make the stage.
One crazy idea I had early on for this deck was to use Clear the Mind+Experimental Frenzy to get our deck stuck in our hand as we go, and then we's need to pay R to sacrifice the frenzy to continue. But this seems unlikely to work.
Probably best to wait for at least the next set before trying to make anything actually work.
Ah, having a way to animate enchantments will be really nice. As for overcheap reshuffling cards, I guess we don't even know yet whether we will use Precognition Field for our stage combo (assuming we can come up with one), since Cleansing Nova is going away. Precognition Field is certainly a good card though; it lets us cast instants/sorceries (which are very good with TYS) without having to draw them. But, I guess the advantage of still requiring mana in the presence of Omniscience goes away, since we won't have Omniscience any more. Losing Omniscience is actually pretty painful; if we use mana for our stage resource or higher, we have to make sure that we can get everything else we need without using that color of mana. Stakfish used Unclaimed Territory to help out with that, but we won't have access to that anymore either.
Yeah, we should probably wait for at least the next set to really think about it.
Precognition field is also leaving. We lose both Ixalan sets, Dominaria and M19. Ixalan had a few useful cards, and won't be too terrible of a loss, but losing the other two sets really hurts a lot. But Experimental Frenzy will still be legal, though it will be harder to use.
Angrath's Rampage almost does a half passable Cleansing Nova impression, though I hope we get something that works better with animated artifacts like a cmc based sweeper. Maybe along the lines of XBW destroy all nonland permanents with CMC X.
I don't think losing Omniscience is all that bad, we barely needed it in the stage combo. It will hurt for layer extensions, and for getting a second stage in there anywhere. (still sad we couldn't get crafty cutpurse to work in here.)
And as for recursion, if a set with "Beyond Death" in the title doesn't have a good way to get things back I'd be shocked.
I'll probably start looking into Modern, though that will require a better understanding of higher order stages. But I can at least look into openings. (though the banlist eliminates some of the obvious lines.)
I posted it to reddit but it got buried under spoilers, (though one user did point out an error: TYS triggers' # is set when the spell is cast, only nerfs the initial storm by 20% or so). Gonna wait until they are done to try again.
Ah, I see. In that case, having extra Thousand-Year Storms isn't worth it, since it's not giving us extra batches of creature retrieval or card drawing; it just multiplies the one batch by up to four, which is miniscule. So instead of 3 Connive // Concocts and 3 extra Thousand-Year Stroms, it would be better to have 3 Aid the Fallens and 3 creature fetching Planeswalkers, (like Vivien, Nature's Avenger) which will allow us to fetch many creatures 15 times before we start with the Bankrupt in Bloods.
We mainly take advantage of the creature fetching Planeswalkers before we first cast Bankrupt in Blood, since that is when the stack is clear. But that's the most important part anyway.
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You might want to improve my estimates a bit, and currently its at 6->2^(700,000)->20->12 but it should really probably be something like 6->(6->2^(700,000)->X->6)->20->12. Or some further nested chain nonsense.
Edit: Hmm, just double checking the combat triggers, which seem like they need to be really good to be worth it. I think maybe Hadana's Climb is the closest? It looks like it would definitely be better than a Concoct if we didn't already use Sarkhan precombat to let Raging Regisaur kill him. It is an extra black mana from combat though, so it is probably still good.
Another thought is using gideon6 and Role reversal to kill sarkhan in combat by attacking him, without using Raging Regisaur at all. One bonus to this is that we can equip gideon and then wrath in combat, though we still can't Purify, so it's probably not worth it. (Edit2: oops gideon doesnt become indestructible himself, so this is definitely a no go)
Hmm with 1 black from Hadana's climb and the 3 mana from dowsing dagger, is Nikya of the old ways better than a connive // concoct? Just for the 2 extra black before the first real azor's gateway flip?
EDIT: ABORT, UGH
I found an infinite:
Activate Azor's Gateway, draw bankrupt in blood, exile an artifact (like dowsing dagger)
use Karn to return the artifact
Repeated reverberation
Sarkhan, dragonsoul kills both karn and himself
cast bankrupt in blood
combo off.
use excess draws to redraw the walkers.
I don't see any sort of simple fix.
Cut repeated reverberation? loses the Sarkhan layer, Also makes the start much less explosive.
Cut Azor's gateway? Can't make much black then.
Cut Karn? This seems the best, we can still exile enough for a few flips, but more like 4 than 20
This sucks, but at least I caught it.
Without the artifact retrieval, it looks like we have 25 exilable cards without Gravitic Punch, or 24 with. That means that if we are required to send the opponent to a low negative life total, adding Saheeli, Sublime Artificer won't add another Sanctum of the Sun. It won't lose one either though, so we might as well keep it in, since we can still have Dowsing Daggers and Hadana's Climb.
So, perhaps we can go with:
1 Leyline of Abundance
2 Leyline of Abundance
3 Breeding Pool
4 Llanowar Elves
5 Plains
6 Stony Strength
7 Stony Strength
8 Emergency Powers
9 Kykar, Wind's Fury
10 Samut, Tyrant Smasher
11 Mox Amber
12 Gyre Engineer
13 Precognition Field
14 Thousand-Year Storm
15 Omniscience
16 Emergence Zone
Stage Combo:
17 Manifold Key
18 The Antiquities War
19 The Antiquities War
20 Cleansing Nova
21 Cleansing Nova
22 Disdainful Stroke
23 Teshar, Ancestor's Apostle
24 Militia Bugler
25 Tragic Poet
26 Gaea's Blessing
27 Citywatch Sphinx
28 Kaya's Ghostform
29 Repudiate // Replicate
Proliferation:
30 Evolution Sage
Recycling Lands:
31 World Shaper
32 Lotus Field
Drawing:
33 Blood for Bones
34 Bankrupt in Blood
35 Ionize
Life Gain:
36 Sarkhan, Dragonsoul
37 Repeated Reverberation
39 Dreadmalkin
39 Imperial Ceratops
40 Angel of Vitality
41 Vicious Rumors
42 Dowsing Dagger
43 Switcheroo
44 Azor's Gateway
Putting Emergence Zone into play:
45 Wayward Swordtooth
The Final Hit:
46 Gravitic Punch
Disposable Cards:
47 Saheeli, Sublime Artificer
48 Dowsing Dagger
49 Dowsing Dagger
50 Dowsing Dagger
51 Hadana's Climb
52 Nikya of the Old Ways
53 Raging Regisaur
54 Whispering Snitch
55 Thousand-Year Storm
56 Thousand-Year Storm
57 Thousand-Year Storm
58 Blood for Bones
59 Blood for Bones
60 Blood for Bones
We can exile 24 cards, so we can get 4 Sanctum of the Suns that way, and we can get one more by using the Saheeli trick. That takes us to more than 10 -> X1 -> 6 -> 12 final damage.
X1 is how much life we can get before we tap the first Sanctum of the Sun. The main component of this is the number of times we can activate Sarkhan, Dragonsoul. We can activate it once when we play it, then we can recycle it once using Raging Regisaur, then another time for each black mana that we can produce prior to activating the forst SotS. We can use Nikya so that each of the four Dowsing Daggers can add four black mana each, then Hadana's Climb can add two more. So we can activate Sarkhan 20 times, so that X1 > 10 -> X2 -> 21 -> 11.
In turn, X2 is how great a TYS count we can get prior to the first Sarkhan activation. THe biggest component of that comes from using Whispering Snitch to gain the opponent a lot of life. We can Switcheroo a Citywatch Sphinx to the opponent, then destroy it (with Cleansing Nova, say). Then in response to the surveil trigger, we can Switcheroo a lot of Whispering Snitches to the opponent, and the surveil will trigger a lot of life gains from the Whispering Snitches. So that gives us X2 > 10 -> 10 -> X3 -> 9, where X3 is how many Whispering Snitches we can transfer over to the enemy.
The main component of X3 comes from the fact that we can resolve 19 copies of Vicious Rumors, making use of the opponent's initial 20 health. So X3 > 10 -> X4 -> 19 -> 8, where X4 is how high we can pump our TYS count before the first Vicious Rumors.
The main component of X4 comes using our 16 remaining life after our initial start to cast Gaea's Blessing followed by Ionize. We can do this 7 times, and each time we can get many Repeated Reverberation triggers, and four Thousand-Year Storm triggers, so we get many copies of Gaea's Blessing 35 times. So X4 > 10 -> X5 -> 36 -> 7.
The main component of X5 comes from the Bankrupt in Blood that we can draw and cast during the initial start, for 5 big batches of copies. So X5 > 10 -> X6 -> 6 -> 6.
The main component of X6 comes from the four Blood for Bones, which each get 5 big batches of copies, so X6 > 10 -> X7 -> 21 -> 5.
The biggest component of X7 copies from putting many copies of World Shaper into play, and destroying them, each of which creates a trigger to refetch [c/]Lotus Field[/c]. So X7 > 10 -> 10 -> X8 -> 3.
Before we put World Shaper into play, we can use Wayward Swordtooth to allow us to play Emergence Zone with more than 2^700,000 Evolution Sages in play, so X8 > 10 -> 10 -> 2^700,000 -> 2.
Note that X1 > 10 -> X2 -> 21 -> 11 > 2 -> 4 -> 2 -> 12. So after 5 Sanctum of the Suns, we get a final damage of more than 2 -> 4 -> 7 -> 12. So that's a simpler lower bound, but it is harder to expand like we did above.
We cant use karn for an early animation so there's no ministage before the Emergence Zone. So we make more like 10^13 Evolution sages.
Our exileable cards are a bit clogged at 4, with 2x leyline 3x blood for bones, kykar, and raging regisaur. Good thing connive // concoct is basically the same card for this challenge.
Being one card short of 5 flips is kind of frustrating, but I don't see any way to squeeze an extra slot, and still kill them.
It looks like the rebuild section is the most likely to have a cuttable card, but we need to have a way to get Gyre engineer, Citywatch sphinx, tragic poet and evolution sage, without getting World shaper. Oh, and still have a white legend
I think we can cut Militia bugler, Kaya's Ghostform, and one The Antiquities War for Mystic Forge and Tezzeret's Gatebreaker. Hmm, no this is still short a way to recycle Teshar.
Edit: i tried a bunch of lines, but cant seem to get to another card for 5 flips. Oh well, will rewrite karn out tomorrow.
Edit2: the start needs an extra untap effect, to have enough blue for Precognition field, TYS, and replicate. Making the last Blood for bones a Stony Strength
Edit3: isn't Connive // Concoct just better than Blood for bones? we get a free reshuffle with each resolution instead of a free guy to hand. If we had an extra creature worth returning (worldshaper#2?), that could be worth it, but we don't.
Also, we are actually 2 cards away from 5 flips, we need to keep a Stony Strength to actually make the guy we Gravitic punch them with giant.
edit4: rereading your estimation and I think I'm missing something?
Are you sure we get to ...->7->12?
I agree that we get to 7->X1->6->12 where X1 is 7->x2->21->11 but I don't see how that can possibly be big enough to give us something like 7->c->7->12? where c is some small non chained arrow constant like 3x10^18 or 4.
By definition, 7->c->7->12 = 7->c->(7->c->6->12)->11 which seems difficult to compare to 7->(7->x2->21->11)->6->12
Yeah, maybe Connive // Concoct is better - I didn't really look at it carefully before just choosing Blood for Bones. So make both the main combo card and the two extras Connive // Concoct?
Good point about needing Stony Strength - we could also use Leyline of Abundance, but either way we would need to keep another card around. I imagine there is some other way to kill the opponent with just one card needed though. If we find a way to save a card, that would definitely worth looking for. I guess right now it's not so pressing.
So, before the first Sanctum of the Sun we get to more than a -> b -> 21 -> 11 for nongigantic a and b. This will be more than a -> b -> 2 -> 12 = a -> b -> a^b -> 11, so long as a^b <= 21. So for example we have more than 2 -> 4 -> 2 -> 12. Then each Sanctum of the Sun will increase the third number by one, so we wind up with more than 2 -> 4 -> 7 -> 12. But we need the 2 and 4 at the beginning.
Stony Strength is better to keep than Leyline as it can make white mana from an animated key. I actually couldn't find a better card to kill them with, though I didn't look very hard.
Oh I see, the 2 -> 4 -> 2 -> 12 is the base storm before doing any flips and we are increasing that by the 5 flips we get to come to 2 -> 4 -> 7 -> 12. Kind of a tricky way to get the extra number, the first two numbers seem like they really shouldn't matter so much, but when the numbers are small it works out like that.
That a -> b -> 2 -> 12 identity seems odd, but actually checks out a -> b -> 2 -> 12 = a -> b -> (a -> b -> 1 -> 12) -> 11 = a -> b -> (a -> b) -> 11 = a -> b -> a^b -> 11
I guess by that logic we get past 2->2->2->13 (which is just 4), and are one flip shy of getting to 2->3->2->13 (which is much more than 4).
It does seem like these would not be nearly as tight of a bound as the 7 -> (7 -> x2 -> 21 -> 11) -> 6 -> 12 estimate, even though they have bigger ending numbers.
Edit: Well, I refactored Karn out, redid the beginning with an extra stony strength for storm, and updated all of the changed images. I'm gonna give us one more day to find any other errors then post the view only link to r/magictcg.
2 -> 4 -> 7 -> 12 is actually greater than 7 -> 4000000 -> 6 -> 12, although I suppose 7 -> 4000000 -> 6 -> 12 might LOOK bigger. Of course, 7 -> (7 -> x2 -> 21 -> 11) -> 6 -> 12 is bigger still.
Edit: I currently don't have access to edit the doc, so I will post some comments here:
Are the [ deck ] and [ /deck ] in the deck lists intentional?
After Precognition Field we should have 1G,1R, so that we can cast Stony Strength next. So we have:
Precognition Field - 1G,1R
L - Stony Strength - 3G,2R,1U
L - Stony Strength - 5G,3R,2U
Thousand-Year Storm - 1G,4R,1U
So it looks like everything has one less R from there on.
How big does x2 need to be for 7 -> (7 -> x2 -> 21 -> 11) -> 6 -> 12 to be greater than 2 -> 4 -> 7 -> 12?
My estimate isn't great either,
Because of not factoring in the extra sphinxes each top level wrath actually squares the storm and we get a huge number in the front as well, but actually saying that involves some more fiddling with exponents in an already fiddly section.
So the better estimate would be X1->X1->6->12.
where X1= X2->X2->21->11 (extra sarkhans)
and X2= x3->x3->x3->9 (extra life gain triggers from the snitch)
and so on... with their extra life, our extra life, the extra cards from hand from the setup... and extra TYS triggers the whole way, making everything more powerful.
Edit: sent you the edit link again, good catch on the free stony strength I missed.
Switching the initial 2 to a 7 makes almost no difference, so 2 -> 4 -> 7 -> 12 will be just a little bit less than 7 -> (7 -> 4 -> 16 -> 11) -> 6 -> 12 as well. So 2 -> 4 -> 7 -> 12 will be greater than 7 -> (7 -> 3 -> 16 -> 11) -> 6 -> 12, for instance.
The first number doesn't make a whole lot of difference; really the biggest significance of a and b in a -> b -> c -> d is the fact that when you reduce down to a -> b -> 1 -> d, that equals a^b. So it is the value of a^b that is important; in case of a tie, b is the tiebreaker. (As is typical, this rule of thumb does not always hold, for example if d = 1, then we just have a -> b -> c, and we know that b is much more important then a then, if a >= 3.) Since we are dealing with X's that are length 4 Conway chains themselves, generally speaking if X1 is a Conway Chain such that X > 2^X1, then X will in all likelihood will be greater than X1^X1 as well. So we could choose X1 = 2 -> X2 -> 21 -> 11, or X1 = X2 -> X2 -> 21 -> 11, and either will be valid. I've been using 10 as a default number, since 10 is a commonly used "base" number.
Added a tappedout link to the decklist.
Thanks for the explanation, these corner cases with relatively small numbers really help.
Yeah I'm just using 7 because we add 6 storm each top level reshuffle and have X of those so we bring X to 7X, and it'll be easier to follow that than switching the base halfway through.
Edit: hmm, it looks like we can use disdainful stroke to fizzle ionize before the first gaea's blessing too, and get an extra storm.
Edit2: I know this doesn't impact any of the estimations, but someone will probably ask. Do you have any clue about how to optimally make/spend our red mana once we have flash? Other than Kykar its generally better as white mana from mox amber, instead, but we can get Kykar back with a connive, and will have to to cast Repeated Reverberation for Sarkhan. Though we can't save it in hand like we could with blood for bones, so we dont get a ton of mana, it is still more than enough to cast a few extra repeated reverberations. Actually no, we can replicate the elemental tokens, so we can get X red.
I kinda think we want to concoct back kykar as little as possible, as Worldshaper is a better target all the time. but we do want to still repeated reverberate the card draw and the concoct itself. But do we want to use the red in the stage? Do we want to get RR on RR for the next layer card? Do we want to just spend all the red we can ASAP to make the stage combo stronger?
When we do actually fetch Kykar with Concoct, I imagine it will be the last Concoct before we activate Sarkhan. So we don't get any more stages, we just use our red mana to cast maybe a couple of Repeated Reverberations (since we get a reshuffle from Concoct), and then activate Sarkhan. We can have some leftover red mana after the Sarkhan activation, I imagine we want to use it as quickly as possible, since generally earlier is better. I don't think we'll be saving any red mana to use during a stage.
Edit: Post rotation, assuming we want to be able to Replicate or Quasiduplicate TYS, reshuffling looks tricky. Midnight Clock and Enhanced Surveillance look too cheap, we can animate them once and use tokens to reshuffle. Any sweeper will also likely kill the TYS and need another dance of the manse to animate it, which we could also use to animate the reshuffler.
Spells like Finale of Revelation and Emergency powers would eliminate a lot of countermagic, which we probably need to be able to recast whatever spells make the stage.
One crazy idea I had early on for this deck was to use Clear the Mind+Experimental Frenzy to get our deck stuck in our hand as we go, and then we's need to pay R to sacrifice the frenzy to continue. But this seems unlikely to work.
Probably best to wait for at least the next set before trying to make anything actually work.
Yeah, we should probably wait for at least the next set to really think about it.
Angrath's Rampage almost does a half passable Cleansing Nova impression, though I hope we get something that works better with animated artifacts like a cmc based sweeper. Maybe along the lines of XBW destroy all nonland permanents with CMC X.
I don't think losing Omniscience is all that bad, we barely needed it in the stage combo. It will hurt for layer extensions, and for getting a second stage in there anywhere. (still sad we couldn't get crafty cutpurse to work in here.)
And as for recursion, if a set with "Beyond Death" in the title doesn't have a good way to get things back I'd be shocked.
I'll probably start looking into Modern, though that will require a better understanding of higher order stages. But I can at least look into openings. (though the banlist eliminates some of the obvious lines.)
Are you satisfied with the writeup? Should we publicize it?
Hmm, does being sorcery speed matter there? It looks fine because Aid the Fallen still gets the World Shaper when we want it.