Edit: So, it looks like the progress for this has stalled a bit (in part because we've been working on the Standard deck). Of course, the priority is to get going on the writeups for both decks, but in case someone wants something to chew on in the meantime, here are several things we could talk about:
1. We could try to get the gigastage (Tier 4 stage) deck to actually work. Obviously, this is much less damage than a deck that can compute the Busy Beaver function, but this pretty much belongs in a different category from those decks - perhaps a category for decks with a finite game tree, or at least no infinite lines of play.
Hyp Cos has looked at these decks and mentioned some problems, so we have something to build off of.
2. We could shoot for the moon, and try to get higher order Busy Beaver functions. Hyp Cos also linked a relevant paper:
The idea is that doing roughly the same thing as our Busy Beaver decks, except over a higher order arithmetic formula rather than a recursive formula, will allow us to compute higher order Busy Beaver functions. However, the basic gist of the implementation is that the two players will represent different quantifiers in the arithmetic formula. So, the power of the construction seems to rely on the opposing nature of the two players. Therefore, for the more powerful formulas, I imagine what would happen when we consider what happens with opponent cooperation, is that we would go infinite.
So, for this challenge I believe we would need to amend the no-infinites rule: instead of considering opponent cooperation, a deck is only invalidated if we can deal arbitrarily high amounts of damage regardless of the opponent's actions.
3. There are of course many other formats that we haven't tried yet: Modern, Pioneer, Commander, Historic, Pauper, etc.
Of course we should keep plugging away at the writeups. (I haven't worked on them in a while due to irl busyness.)
Yeah I put some work into the standard writeup, will add more over the next week.
Making the gigastage deck work would be nice, but it is SO outclassed by BB decks that it is even more niche. Though the Chalice of the Void tricks were pretty impressive if we could ever get them to work properly.
Allowing the opponent to cooperate with us to get more damage seems wrong, even if we can convert their setup to work in one turn.
My thought for a higher order BB function was to use Shahrazad, and somehow use going infinite on later subgame turns as an oracle for the TM in the main game. But there doesn't seem to be a good way to cast Shahrazad enough during computation, nor a good way to communicate the answer up to the main game.
Other formats have the difficulty of going off turn 1, also their banlist is much more fluid. Tibalt's trickery barely missed the banhammer in standard (and I think is still banned in MTGA best of 1?).
Legacy: just a slightly worse version of the vintage deck, we have only had like 2 banned cards most iterations. (currently just Flash?)
Modern: Might be the home for the gigastage?
Pioneer: Not very familiar with this format, at a wild uninformed guess: something like turn 2 second order stage?
Commander: This is mostly legacy with bigger decks, commander color identity restrictions basically force us to play an uninteresting 5 color commander.
Historic: No clue, maybe like 4-7 stages?
Pauper: Would be slow and lose basically every engine we've ever used, probably at least one stage, but I'm not sure at all.
Oh, for the higher order Busy Beaver function we would still assume the opponent does their best to reduce our final score, so that part is the same as it is now. The difference would be that we don't invalidate the deck if we can do arbitrarily high damage, but only with opponent cooperation. I think this keeps the spirit of the challenge - of course it would make some things easier, but I think the challenge is interesting enough nevertheless.
A modern gigastage would be amazing! I don't know if we could get that far, but even getting to a hyperstage or better would be nice.
Most of the gigastage cards are legal in modern, or have close equivalents. In particular the spellweaver cards are both legal, along with worldfire and friends.
The whole thing probably doesn't port over cleanly, but there's gotta be at least an order 3 stage possible in modern.
A modern BB computation seems to be more than a few printings away, Artificial Evolution and Dralnu's Crusade in particular seem unlikely to be reprinted. Now Dralnu's Crusade might not be strictly necessary as we can implement flooding waterfall with it, but that seems unlikely to be TC.
Clearly we need some sort of stage to do it and Toralf, God of Fury can get us one with only a few support cards. Orvar, the All-Form seems good for generating copies, but maybe that is just bias from working on the standard deck.
After a very rough outline I'm thinking we'd need about 12-15 cards total.
Interesting, I want to say that we've had smaller stage combos before, but im not sure off the top of my head. it'd be nice to get down to 7 cards to avoid needing to draw, but im not sure how realistic that is with also needing to make mana, as we are down to 4 after lotus/show and tell/omniscience or similar openers.
I want to say there was a potential bootstrap stage with drake familiar but can't remember.
Granted, I mostly worked with creature stages in here. If you remember something that does most of the support by itself I'd be delighted.
With bootstrapping I think we usually had to draw a few cards and then could get 2-4 layers going, leading into a massive draw before we get to the stage.
Play Black Lotus (6 cards in hand)
Play Channel. Always convert all but 1 life to mana (20 mana, 5 cards in hand)
Play Chromatic Orrery and tap it for mana (18 mana, 4 cards in hand)
Play Omniscience (8 mana, 3 cards in hand)
Play Elite Guardmage (11 mana, 3 cards in hand, 1 creature)
Play Orvar, the All-Form (11 mana, 2 cards in hand, 2 creatures)
Play Elvish Fury with buyback targeting Elite Guardmage 5 times. Everytime we create a token copy from Orvar's ability. That gains us 3 life back and draws us a card. It also increases our creature count (6 mana, 7 cards in hand, 7 creatures)
Play Toralf, God of Fury (6 mana, 6 cards in hand, 8 creatures)
Mutate Cubwarden onto Toralf. Put Cubwarden on top, so the resulting creature is non-legendary and has lifelink plus Toralf's ability (2 mana, 5 cards in hand, 8 creatures)
Play Hunted Phantasm. The opponent gets five 1/1s (2 mana, 4 cards in hand, 9 creatures)
Play Djinn Illuminatus (2 mana, 3 cards in hand, 10 creatures)
The cards in hand are: Elvish Fury, Outnumber, and Experimental Overload
Now it gets fun:
Cast Outnumber and use the rest of our mana to replicate it twice. The original and the two copies each target a different creature of the opponent.
Let the first Outnumber copy resolve. It deals 10 damage, 9 of which are overkill and trigger Toralf's ability. Target a fourth of the opponents creatures with that.
Resolve Toralf's ability. It deals 9 damage with lifelink, so we now have 9 mana again. 8 of the damage are overkill, so Toralf's ability triggers again. Target the fifth of the opponents creatures with that.
Cast Elvish Fury with buyback, targeting Hunted Phantasm. Orvar creates a token copy and the opponent gets five fresh 1/1s.
Cast Elvish Fury with buyback, targeting the Cupwarden/Toralf stack. This gets us a token copy. This token will have lifelink and Toralf's ability. And since it is non-legendary noone has to die for this.
Resolve Toralf's ability. It deals 8 damage with lifelink. 7 of which are overkill and trigger both instances of Toralf's ability.
...
We have a stage going. By the time we run out of Toralf's ability we have a few uparrows worth of our creatures. Call that number X for now.
Resolving the second Outnumber copy gets us f_w(X).
Resolving the original Outnumber gets f_w(f_w(X))
Then we cast Experimental Overload, replicating it that many times. Each copy will return Outnumber and we get f_w+2(f_w(f_w(X)))
To deal damage use a Toralf ability from the last big Outnumber to target the opponent.
Mull to 5
Play Black Lotus (4 cards in hand, storm 1)
Play Channel. Always convert all but 1 life to mana (20 mana, 3 cards in hand, storm 2)
Play Chromatic Orrery and tap it for mana (18 mana, 2 cards in hand, storm 3)
Play Elite Guardmage (17 mana, 2 cards in hand, 1 creature, storm 4)
Play Orvar, the All-Form (13 mana, 1 cards in hand, 2 creatures, storm 5)
Play Change of Heart with buyback targeting Elite Guardmage 6 times. Everytime we create a token copy from Orvar's ability. That gains us 3 life back and draws us a card. It also increases our creature count (7 mana, 7 cards in hand, 8 creatures, storm 11)
Play Thousand-Year Storm (1 mana, 6 cards in hand, 8 creatures, storm 12)
Play Festival of Trokin. TYS creates 12 copies. (0 mana, 5 cards in hand, 8 creatures, storm 13, 13 Festivals on the stack)
Resolve the first Festival, gain 16 life. (16 mana, 5 cards in hand, 8 creatures, storm 13, 12 Festivals on the stack)
Play Toralf, God of Fury (12 mana, 4 cards in hand, 9 creatures, storm 14, 12 Festivals on the stack)
Mutate Cubwarden onto Toralf. Put Cubwarden on top, so the resulting creature is non-legendary and has lifelink plus Toralf's ability (8 mana, 3 cards in hand, 11 creatures, storm 15, 12 Festivals on the stack)
Play Hunted Phantasm. The opponent gets five 1/1s (5 mana, 2 cards in hand, 12 creatures, storm 16, 12 Festivals on the stack)
Play Change of Heart with buyback targeting Hunted Phantasm. We get a copy, the opponent gets five 1/1s (1 mana, 2 cards in hand, 13 creatures, storm 17, 12 Festivals on the stack)
Resolve the next Festival, gain 26 life (27 mana, 2 cards in hand, 13 creatures, storm 17, 11 Festivals on the stack)
Play Change of Heart with buyback targeting one of our creatures 6 times (3 mana, 2 cards in hand, 19 creatures, storm 23, 11 Festivals on the stack)
Resolve the next Festival, gain 38 life (41 mana, 2 cards in hand, 13 creatures, storm 17, 10 Festivals on the stack)
Play Change of Heart with buyback targeting one of our creatures 10 times (1 mana, 2 cards in hand, 29 creatures, storm 33, 10 Festivals on the stack)
Resolve the next Festival, gain 58 life (59 mana, 2 cards in hand, 29 creatures, storm 33, 9 Festivals on the stack)
Play Change of Heart with buyback targeting one of our creatures 19 times (3 mana, 2 cards in hand, 48 creatures, storm 52, 9 Festivals on the stack)
...
repeat to get a bunch of storm and creature copies. Make sure that the opponent has more 1/1s than the storm count. Put the rest of the creature copies into the Cupwarden-Toralf mutate stack.
...
Resolve the last Festival, gain 2462 life (2465 mana, 2 cards in hand, 1231 creatures, storm 1235)
Play Change of Heart with buyback targeting one of our creatures 616 times (1 mana, 2 cards in hand, 1847 creatures, storm 1851)
(I might have made a mistake with the calculations here, but ballpark should be right, and getting it exact hardly matters in the end)
Now it gets fun:
Cast Outnumber and get 1851 copies from TYS. Each copy targets a different creature of the opponent.
Let the first Outnumber copy resolve. It deals 1847 damage, 1846 of which are overkill and trigger all Toralf's abilities. Target yet more different creatures of the opponent.
Resolve Toralf's ability. It deals 1846 damage with lifelink, so we now have 1846 mana again. 1845 of the damage are overkill, so the Toralf's abilities trigger again. Target even more different creatures of the opponent again.
Cast Change of Heart with buyback, targeting Hunted Phantasm. Orvar creates a token copy and the opponent gets five fresh 1/1s.
Repeat until there are enough fresh targets.
Cast Elvish Fury with buyback, targeting the Cupwarden/Toralf stack. This gets us a token copy. This token will have lifelink and Toralf's ability. And since it is non-legendary no one has to die for this.
Repeat until out of mana.
Resolve Toralf's ability. It deals 1845 damage with lifelink. 1844 of which are overkill and trigger all instances of Toralf's ability.
...
We have a stage going. By the time we run out of Toralf's ability we will have rougly f_w(1847) Toralfs
Resolving the next Outnumber copy gets us f_w(f_w(1847)
Resolving the next Outnumber gets f_w(f_w(f_w(1847))
...
Once we have resolved all 1851 Outnumber copies we have f_w+1(1852) creatures
To deal damage use a Toralf ability triggered by the original Outnumber to target the opponent.
regarding the growth of our creature numbers: each 2 damage Toralf ability gets us from X Toralfs with valid targets to (25/24)*X Toralfs with valid targets:
we destroy X targets and get X mana from that. we can use that to create X/4 new creature copies. X/5 Haunted Phantasms are needed to replenish the targets. That leaves X/20 creatures to make progress. A progress batch can have 5 Toralfs and 1 Haunted Phantasm for targets. so (5/6)*(X/20) new Toralfs with target.
So the 3 damage abilities are at the a^x level.
4 damage: a^^x
5 damage: a^^^x
and so on
Granted, I mostly worked with creature stages in here. If you remember something that does most of the support by itself I'd be delighted.
With bootstrapping I think we usually had to draw a few cards and then could get 2-4 layers going, leading into a massive draw before we get to the stage.
Play Black Lotus (6 cards in hand)
Play Channel. Always convert all but 1 life to mana (20 mana, 5 cards in hand)
Play Chromatic Orrery and tap it for mana (18 mana, 4 cards in hand)
Play Omniscience (8 mana, 3 cards in hand)
Play Elite Guardmage (11 mana, 3 cards in hand, 1 creature)
Play Orvar, the All-Form (11 mana, 2 cards in hand, 2 creatures)
Play Elvish Fury with buyback targeting Elite Guardmage 5 times. Everytime we create a token copy from Orvar's ability. That gains us 3 life back and draws us a card. It also increases our creature count (6 mana, 7 cards in hand, 7 creatures)
Play Toralf, God of Fury (6 mana, 6 cards in hand, 8 creatures)
Mutate Cubwarden onto Toralf. Put Cubwarden on top, so the resulting creature is non-legendary and has lifelink plus Toralf's ability (2 mana, 5 cards in hand, 8 creatures)
Play Hunted Phantasm. The opponent gets five 1/1s (2 mana, 4 cards in hand, 9 creatures)
Play Djinn Illuminatus (2 mana, 3 cards in hand, 10 creatures)
The cards in hand are: Elvish Fury, Outnumber, and Experimental Overload
Now it gets fun:
Cast Outnumber and use the rest of our mana to replicate it twice. The original and the two copies each target a different creature of the opponent.
Let the first Outnumber copy resolve. It deals 10 damage, 9 of which are overkill and trigger Toralf's ability. Target a fourth of the opponents creatures with that.
Resolve Toralf's ability. It deals 9 damage with lifelink, so we now have 9 mana again. 8 of the damage are overkill, so Toralf's ability triggers again. Target the fifth of the opponents creatures with that.
Cast Elvish Fury with buyback, targeting Hunted Phantasm. Orvar creates a token copy and the opponent gets five fresh 1/1s.
Cast Elvish Fury with buyback, targeting the Cupwarden/Toralf stack. This gets us a token copy. This token will have lifelink and Toralf's ability. And since it is non-legendary noone has to die for this.
Resolve Toralf's ability. It deals 8 damage with lifelink. 7 of which are overkill and trigger both instances of Toralf's ability.
...
We have a stage going. By the time we run out of Toralf's ability we have a few uparrows worth of our creatures. Call that number X for now.
Resolving the second Outnumber copy gets us f_w(X).
Resolving the original Outnumber gets f_w(f_w(X))
Then we cast Experimental Overload, replicating it that many times. Each copy will return Outnumber and we get f_w+2(f_w(f_w(X)))
To deal damage use a Toralf ability from the last big Outnumber to target the opponent.
I'm sure that can be optimized further
Oh wow, that's wild. Could we get away with cutting Omniscience and Guardmage for Genesis Ultimatum?
Is there a reason for going for Experimental Overload as the recursion card? Fossil Find is cheaper so we could replicate it more, although that wouldn't change the number of cards needed.
I just edited my post above with a 12 card version that cuts Omniscience and uses Festival of Trokin to gain a bunch of life to get started.
I didn't use Fossil Find because it would go infinite: create a Toralf, God of Fury copy and let the original die. Use Fossil Find to bring it back and play the other side: Toralf's Hammer. Use that as a repeatable damage source to trigger all the Toralf abilities without using Outnumber.
But for the new version that level of recursion is overkill anyway
Edit: Just noticed that I gave too many copies with TYS in my line of play above. It should only count instants and sorceries for "storm". But we still get 8 Festivals, and that should be enough life to still beat graham's number.
Edit2: Running the numbers I get 215 Outnumber copies.
I forgot about mutate as a way to solve the legendary problem while also giving lifelink, very nice.
The idea with Cormela was as a limited mana source and a way to recur spells (dying to legend rule). We can only get the mana by keeping the new one, which fizzles the TYS copies targeting the old one so each TYS trigger only makes mana once, while the other copies can make more TYS.
But there are a few pieces missing.
Edit: also: Hello to the newcomer, CaptainMarcia! Always nice to see new people. Seems like you have some experience with this?
I just edited my post above with a 12 card version that cuts Omniscience and uses Festival of Trokin to gain a bunch of life to get started.
I didn't use Fossil Find because it would go infinite: create a Toralf, God of Fury copy and let the original die. Use Fossil Find to bring it back and play the other side: Toralf's Hammer. Use that as a repeatable damage source to trigger all the Toralf abilities without using Outnumber.
But for the new version that level of recursion is overkill anyway
Edit: Just noticed that I gave too many copies with TYS in my line of play above. It should only count instants and sorceries for "storm". But we still get 8 Festivals, and that should be enough life to still beat graham's number.
Edit2: Running the numbers I get 215 Outnumber copies.
Hi Iijil!
Would we be able to activate Hammer in that scenario? It doesn't look like we have anything with haste.
Regarding the Festival of Trokin version, how are we casting Toralf/Cubwarden/Phantasm with Festival copies on the stack?
Oh, you're right. We can't activate Toralf's Hammer without haste. Now why does mtgsalvations "oracle text" not include the tap-symbol? >:(
And yeah, not having flash for those creatures is a problem. Not creating any new creatures between the Festival copies doesn't gain enough life to beat graham's number. And the creatures that are already out can't be copied with Change of Heart. Orvar, the All-Form because it doesn't trigger of itself and Elite Guardmage because that would cause too many draws.
That is quite annoying. I'm sure there is a workaround, but I'll have to think about it for a bit.
Oh, you're right. We can't activate Toralf's Hammer without haste. Now why does mtgsalvations "oracle text" not include the tap-symbol? >:(
And yeah, not having flash for those creatures is a problem. Not creating any new creatures between the Festival copies doesn't gain enough life to beat graham's number. And the creatures that are already out can't be copied with Change of Heart. Orvar, the All-Form because it doesn't trigger of itself and Elite Guardmage because that would cause too many draws.
That is quite annoying. I'm sure there is a workaround, but I'll have to think about it for a bit.
That makes sense. In the meantime, do you think the Genesis Ultimatum strategy is enough to get us to 12?
Lotus/Channel/Orrery, then Ultimatum. Flip Toralf/Orvar/Phantasm/Djinn and draw the remaining card. 11 mana.
Mutate Cubwarden onto Toralf. 7 mana.
Play Outnumber 7 times. Have the bottom two target the original Toralf, to have Orvar make more copies of him. Continue from there.
Alternatively, with TYS, use Change of Heart for two Toralfs and a Phantasm (to have enough targets) and then Outnumber x4.
Thinking about the Genesis Ultimatum strategy, I'm noticing that Mind's Desire seems like a more efficient alternative. One benefit is that it could allow freely upgrading from Outnumber to Kamahl's Will, although I'm not sure if its wording has any weird implications for Toralf's effect. My impression is that it would still count as one event, though.
I'm not entirely sure how the scaling of the Toralf damage works or how it compares to Graham's function. Do we have an idea of how many times we need to start the damage chain?
Edit: Wait, can we condense the token making into Cogwork Assembler?
With this, we're only getting one Kamahl's Will on the initial volley, with a base damage of 14, and our token generation is slower. But afterward, we can get a ton of Experimental Overload replications to reuse it. (Can't use Fossil Find this time since we do have a haste source now.)
The main function to create Graham's number (let's call it A(X)) takes X to 3 ^^^...^^^ 3, where the number of ^'s is X. We obtain Graham's number by applying this function 64 times starting from the number 4.
If Toralf deals 2 damage to an opponent's creature with 1 toughness, and we have X Toralfs in play at the time, then we will trigger X instances of 1 damage. Then it depends on what benefit we get from dealing that 1 damage, but to keep things simple, if dealing 1 damage allows us to create 1 new Toralf, then dealing 2 damage allows us to double our number of Toralfs: X -> 2X. Toralf dealing 3 damage will trigger X instances of Toralf doing 2 damage; since each instance of 2 damage allows us to double our Toralfs, X such instances doubles our Toralfs X times, taking us to 2^X Toralfs. ( Actually X*2^X, but the first X doesn't matter much) Similarly, dealing 4 damage will create X instances of 3 damage, allowing us to exponentiate X times, getting us to 2^^X. In general, Y damage will increase our Toralfs to 2^^^...^^^X with Y-2 up arrows.
If we call B(X) the number produced by the Toralf combo starting from dealing X damage, it won't be exactly A(X-2), but it's gonna be pretty close - like we'll probably have B(X+3) > A(X) > B(X) or something like that. Since the growth rate of these functions is so enormous, the minor differences aren't gonna come close to the impact of applying the function. So if we apply B(X) 63 times it won't be enough, if we apply it 65 times it will definitely be over. If we apply it 64 times, then we would want to look at the numbers more carefully. (But if we start with a number much more than 4 it should definitely go over.)
Yeah, Kamahl's Will deals damage as a single event, so it would trigger Toralf once.
The main function to create Graham's number (let's call it A(X)) takes X to 3 ^^^...^^^ 3, where the number of ^'s is X. We obtain Graham's number by applying this function 64 times starting from the number 4.
If Toralf deals 2 damage to an opponent's creature with 1 toughness, and we have X Toralfs in play at the time, then we will trigger X instances of 1 damage. Then it depends on what benefit we get from dealing that 1 damage, but to keep things simple, if dealing 1 damage allows us to create 1 new Toralf, then dealing 2 damage allows us to double our number of Toralfs: X -> 2X. Toralf dealing 3 damage will trigger X instances of Toralf doing 2 damage; since each instance of 2 damage allows us to double our Toralfs, X such instances doubles our Toralfs X times, taking us to 2^X Toralfs. ( Actually X*2^X, but the first X doesn't matter much) Similarly, dealing 4 damage will create X instances of 3 damage, allowing us to exponentiate X times, getting us to 2^^X. In general, Y damage will increase our Toralfs to 2^^^...^^^X with Y-2 up arrows.
If we call B(X) the number produced by the Toralf combo starting from dealing X damage, it won't be exactly A(X-2), but it's gonna be pretty close - like we'll probably have B(X+3) > A(X) > B(X) or something like that. Since the growth rate of these functions is so enormous, the minor differences aren't gonna come close to the impact of applying the function. So if we apply B(X) 63 times it won't be enough, if we apply it 65 times it will definitely be over. If we apply it 64 times, then we would want to look at the numbers more carefully. (But if we start with a number much more than 4 it should definitely go over.)
Yeah, Kamahl's Will deals damage as a single event, so it would trigger Toralf once.
Edit: Looks like that deck works!
That makes sense, thanks!
The list is getting increasingly tight, but with the first Kamahl's Will round now only needing 9 cards, I wonder if there's some way to condense the roles of Djinn Illuminatus and Experimental Overload to get us down to 10 while still hitting the target of ~65 rounds. Maybe with the help of changing up the Mind's Desire slot again somehow.
Edit: Ardent Dustspeaker could do it. After we finish resolving the first Kamahl's Will, we should be able to make a bunch of copies of it, with haste, then swing for a bunch of attack triggers. This gets a bit weird since it's interacting with the rest of our library, but whether or not we assume there's 50 blanks there, we eventually start hitting Kamahl's Will and can keep bringing it back with each successive one. Is that a legal move?
Edit: Doesn't matter if it's legal or not. It can't eat through our library because it doesn't exile if it whiffs.
Perhaps Djinn Illuminatus could be replaced by something with Replicate, Storm or similar mechanic. Stream of Thought looks like the most promising replicate spell. It would work if we switched Mind's Desire to Yawgmoth's Bargain - except then we don't have enough life to pay for everything. But it seems close.
Perhaps Djinn Illuminatus could be replaced by something with Replicate, Storm or similar mechanic. Stream of Thought looks like the most promising replicate spell. It would work if we switched Mind's Desire to Yawgmoth's Bargain - except then we don't have enough life to pay for everything. But it seems close.
For a reusable Mind's Desire, how about Urza, Lord High Artificer? The mana is tight, but I think it works out if we go back to Outnumber.
Still, 11 cards is amazing! Mind's Desire for the start is a nice improvement.
My gut feeling is that 4 copies of Outnumber gets us to higher numbers than a single Kamahl's Will, even with only 5 targets initially available. To do that only the last Outnumber needs to deal more damage than the Will would, and I think we can sufficiently increase the creature count with the first three copies. I'll look at the numbers later.
Still, 11 cards is amazing! Mind's Desire for the start is a nice improvement.
My gut feeling is that 4 copies of Outnumber gets us to higher numbers than a single Kamahl's Will, even with only 5 targets initially available. To do that only the last Outnumber needs to deal more damage than the Will would, and I think we can sufficiently increase the creature count with the first three copies. I'll look at the numbers later.
Good point. Any non-specific recursion seems likely to run into that.
Given that we can use Cogwork Assembler to duplicate any permanent, I wonder if there's a way we can incorporate Doubling Season or Primal Vigor, to dramatically increase the rate of Toralf generation. Could that do enough to meaningfully change the scaling?
Edit: I also did some looking for cards that could give higher upfront damage and found Burn at the Stake. Not sure how much it changes, but I think that and Primal Vigor should at least make for a new 10-card record.
So with Primal Vigor, instead of using X token creations to create X new Toralfs (doubling to 2X), we can create Primal Vigors X-1 times and then create a Toralf, taking the number to about 2^^X. That seems like a lot, but it basically improves the gain from deal X damage from 2 -> X -> X-2 to 2 -> X -> X. So it's like getting a bump of 2 on the initial damage.
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Edit: So, it looks like the progress for this has stalled a bit (in part because we've been working on the Standard deck). Of course, the priority is to get going on the writeups for both decks, but in case someone wants something to chew on in the meantime, here are several things we could talk about:
1. We could try to get the gigastage (Tier 4 stage) deck to actually work. Obviously, this is much less damage than a deck that can compute the Busy Beaver function, but this pretty much belongs in a different category from those decks - perhaps a category for decks with a finite game tree, or at least no infinite lines of play.
Hyp Cos has looked at these decks and mentioned some problems, so we have something to build off of.
2. We could shoot for the moon, and try to get higher order Busy Beaver functions. Hyp Cos also linked a relevant paper:
Magic is as Hard as Arithmetic
The idea is that doing roughly the same thing as our Busy Beaver decks, except over a higher order arithmetic formula rather than a recursive formula, will allow us to compute higher order Busy Beaver functions. However, the basic gist of the implementation is that the two players will represent different quantifiers in the arithmetic formula. So, the power of the construction seems to rely on the opposing nature of the two players. Therefore, for the more powerful formulas, I imagine what would happen when we consider what happens with opponent cooperation, is that we would go infinite.
So, for this challenge I believe we would need to amend the no-infinites rule: instead of considering opponent cooperation, a deck is only invalidated if we can deal arbitrarily high amounts of damage regardless of the opponent's actions.
3. There are of course many other formats that we haven't tried yet: Modern, Pioneer, Commander, Historic, Pauper, etc.
Of course we should keep plugging away at the writeups. (I haven't worked on them in a while due to irl busyness.)
Making the gigastage deck work would be nice, but it is SO outclassed by BB decks that it is even more niche. Though the Chalice of the Void tricks were pretty impressive if we could ever get them to work properly.
Allowing the opponent to cooperate with us to get more damage seems wrong, even if we can convert their setup to work in one turn.
My thought for a higher order BB function was to use Shahrazad, and somehow use going infinite on later subgame turns as an oracle for the TM in the main game. But there doesn't seem to be a good way to cast Shahrazad enough during computation, nor a good way to communicate the answer up to the main game.
Other formats have the difficulty of going off turn 1, also their banlist is much more fluid. Tibalt's trickery barely missed the banhammer in standard (and I think is still banned in MTGA best of 1?).
Legacy: just a slightly worse version of the vintage deck, we have only had like 2 banned cards most iterations. (currently just Flash?)
Modern: Might be the home for the gigastage?
Pioneer: Not very familiar with this format, at a wild uninformed guess: something like turn 2 second order stage?
Commander: This is mostly legacy with bigger decks, commander color identity restrictions basically force us to play an uninteresting 5 color commander.
Historic: No clue, maybe like 4-7 stages?
Pauper: Would be slow and lose basically every engine we've ever used, probably at least one stage, but I'm not sure at all.
A modern gigastage would be amazing! I don't know if we could get that far, but even getting to a hyperstage or better would be nice.
The whole thing probably doesn't port over cleanly, but there's gotta be at least an order 3 stage possible in modern.
A modern BB computation seems to be more than a few printings away, Artificial Evolution and Dralnu's Crusade in particular seem unlikely to be reprinted. Now Dralnu's Crusade might not be strictly necessary as we can implement flooding waterfall with it, but that seems unlikely to be TC.
Clearly we need some sort of stage to do it and Toralf, God of Fury can get us one with only a few support cards. Orvar, the All-Form seems good for generating copies, but maybe that is just bias from working on the standard deck.
After a very rough outline I'm thinking we'd need about 12-15 cards total.
I want to say there was a potential bootstrap stage with drake familiar but can't remember.
something like TYS + replication technique + Cormela, glamour thief seems close-ish?
Granted, I mostly worked with creature stages in here. If you remember something that does most of the support by itself I'd be delighted.
With bootstrapping I think we usually had to draw a few cards and then could get 2-4 layers going, leading into a massive draw before we get to the stage.
I don't understand what Replication Technique + Cormela, Glamour Thief would do for us here. What do you have in mind with them?
Edit: I think I have a working 13 card version:
2 Channel
3 Chromatic Orrery
4 Toralf, God of Fury
5 Orvar, the All-Form
6 Cubwarden
7 Elvish Fury
8 Hunted Phantasm
9 Outnumber
10 Djinn Illuminatus
11 Experimental Overload
12 Elite Guardmage
13 Omniscience
Play Black Lotus (6 cards in hand)
Play Channel. Always convert all but 1 life to mana (20 mana, 5 cards in hand)
Play Chromatic Orrery and tap it for mana (18 mana, 4 cards in hand)
Play Omniscience (8 mana, 3 cards in hand)
Play Elite Guardmage (11 mana, 3 cards in hand, 1 creature)
Play Orvar, the All-Form (11 mana, 2 cards in hand, 2 creatures)
Play Elvish Fury with buyback targeting Elite Guardmage 5 times. Everytime we create a token copy from Orvar's ability. That gains us 3 life back and draws us a card. It also increases our creature count (6 mana, 7 cards in hand, 7 creatures)
Play Toralf, God of Fury (6 mana, 6 cards in hand, 8 creatures)
Mutate Cubwarden onto Toralf. Put Cubwarden on top, so the resulting creature is non-legendary and has lifelink plus Toralf's ability (2 mana, 5 cards in hand, 8 creatures)
Play Hunted Phantasm. The opponent gets five 1/1s (2 mana, 4 cards in hand, 9 creatures)
Play Djinn Illuminatus (2 mana, 3 cards in hand, 10 creatures)
The cards in hand are: Elvish Fury, Outnumber, and Experimental Overload
Now it gets fun:
Cast Outnumber and use the rest of our mana to replicate it twice. The original and the two copies each target a different creature of the opponent.
Let the first Outnumber copy resolve. It deals 10 damage, 9 of which are overkill and trigger Toralf's ability. Target a fourth of the opponents creatures with that.
Resolve Toralf's ability. It deals 9 damage with lifelink, so we now have 9 mana again. 8 of the damage are overkill, so Toralf's ability triggers again. Target the fifth of the opponents creatures with that.
Cast Elvish Fury with buyback, targeting Hunted Phantasm. Orvar creates a token copy and the opponent gets five fresh 1/1s.
Cast Elvish Fury with buyback, targeting the Cupwarden/Toralf stack. This gets us a token copy. This token will have lifelink and Toralf's ability. And since it is non-legendary noone has to die for this.
Resolve Toralf's ability. It deals 8 damage with lifelink. 7 of which are overkill and trigger both instances of Toralf's ability.
...
We have a stage going. By the time we run out of Toralf's ability we have a few uparrows worth of our creatures. Call that number X for now.
Resolving the second Outnumber copy gets us f_w(X).
Resolving the original Outnumber gets f_w(f_w(X))
Then we cast Experimental Overload, replicating it that many times. Each copy will return Outnumber and we get f_w+2(f_w(f_w(X)))
To deal damage use a Toralf ability from the last big Outnumber to target the opponent.
I'm sure that can be optimized further
Edit2: Improved to 12 cards:
2 Channel
3 Chromatic Orrery
4 Toralf, God of Fury
5 Orvar, the All-Form
6 Cubwarden
7 Change of Heart
8 Hunted Phantasm
9 Outnumber
10 Elite Guardmage
11 Thousand-Year Storm
12 Festival of Trokin
Play Black Lotus (4 cards in hand, storm 1)
Play Channel. Always convert all but 1 life to mana (20 mana, 3 cards in hand, storm 2)
Play Chromatic Orrery and tap it for mana (18 mana, 2 cards in hand, storm 3)
Play Elite Guardmage (17 mana, 2 cards in hand, 1 creature, storm 4)
Play Orvar, the All-Form (13 mana, 1 cards in hand, 2 creatures, storm 5)
Play Change of Heart with buyback targeting Elite Guardmage 6 times. Everytime we create a token copy from Orvar's ability. That gains us 3 life back and draws us a card. It also increases our creature count (7 mana, 7 cards in hand, 8 creatures, storm 11)
Play Thousand-Year Storm (1 mana, 6 cards in hand, 8 creatures, storm 12)
Play Festival of Trokin. TYS creates 12 copies. (0 mana, 5 cards in hand, 8 creatures, storm 13, 13 Festivals on the stack)
Resolve the first Festival, gain 16 life. (16 mana, 5 cards in hand, 8 creatures, storm 13, 12 Festivals on the stack)
Play Toralf, God of Fury (12 mana, 4 cards in hand, 9 creatures, storm 14, 12 Festivals on the stack)
Mutate Cubwarden onto Toralf. Put Cubwarden on top, so the resulting creature is non-legendary and has lifelink plus Toralf's ability (8 mana, 3 cards in hand, 11 creatures, storm 15, 12 Festivals on the stack)
Play Hunted Phantasm. The opponent gets five 1/1s (5 mana, 2 cards in hand, 12 creatures, storm 16, 12 Festivals on the stack)
Play Change of Heart with buyback targeting Hunted Phantasm. We get a copy, the opponent gets five 1/1s (1 mana, 2 cards in hand, 13 creatures, storm 17, 12 Festivals on the stack)
Resolve the next Festival, gain 26 life (27 mana, 2 cards in hand, 13 creatures, storm 17, 11 Festivals on the stack)
Play Change of Heart with buyback targeting one of our creatures 6 times (3 mana, 2 cards in hand, 19 creatures, storm 23, 11 Festivals on the stack)
Resolve the next Festival, gain 38 life (41 mana, 2 cards in hand, 13 creatures, storm 17, 10 Festivals on the stack)
Play Change of Heart with buyback targeting one of our creatures 10 times (1 mana, 2 cards in hand, 29 creatures, storm 33, 10 Festivals on the stack)
Resolve the next Festival, gain 58 life (59 mana, 2 cards in hand, 29 creatures, storm 33, 9 Festivals on the stack)
Play Change of Heart with buyback targeting one of our creatures 19 times (3 mana, 2 cards in hand, 48 creatures, storm 52, 9 Festivals on the stack)
...
repeat to get a bunch of storm and creature copies. Make sure that the opponent has more 1/1s than the storm count. Put the rest of the creature copies into the Cupwarden-Toralf mutate stack.
...
Resolve the last Festival, gain 2462 life (2465 mana, 2 cards in hand, 1231 creatures, storm 1235)
Play Change of Heart with buyback targeting one of our creatures 616 times (1 mana, 2 cards in hand, 1847 creatures, storm 1851)
(I might have made a mistake with the calculations here, but ballpark should be right, and getting it exact hardly matters in the end)
Now it gets fun:
Cast Outnumber and get 1851 copies from TYS. Each copy targets a different creature of the opponent.
Let the first Outnumber copy resolve. It deals 1847 damage, 1846 of which are overkill and trigger all Toralf's abilities. Target yet more different creatures of the opponent.
Resolve Toralf's ability. It deals 1846 damage with lifelink, so we now have 1846 mana again. 1845 of the damage are overkill, so the Toralf's abilities trigger again. Target even more different creatures of the opponent again.
Cast Change of Heart with buyback, targeting Hunted Phantasm. Orvar creates a token copy and the opponent gets five fresh 1/1s.
Repeat until there are enough fresh targets.
Cast Elvish Fury with buyback, targeting the Cupwarden/Toralf stack. This gets us a token copy. This token will have lifelink and Toralf's ability. And since it is non-legendary no one has to die for this.
Repeat until out of mana.
Resolve Toralf's ability. It deals 1845 damage with lifelink. 1844 of which are overkill and trigger all instances of Toralf's ability.
...
We have a stage going. By the time we run out of Toralf's ability we will have rougly f_w(1847) Toralfs
Resolving the next Outnumber copy gets us f_w(f_w(1847)
Resolving the next Outnumber gets f_w(f_w(f_w(1847))
...
Once we have resolved all 1851 Outnumber copies we have f_w+1(1852) creatures
To deal damage use a Toralf ability triggered by the original Outnumber to target the opponent.
regarding the growth of our creature numbers: each 2 damage Toralf ability gets us from X Toralfs with valid targets to (25/24)*X Toralfs with valid targets:
we destroy X targets and get X mana from that. we can use that to create X/4 new creature copies. X/5 Haunted Phantasms are needed to replenish the targets. That leaves X/20 creatures to make progress. A progress batch can have 5 Toralfs and 1 Haunted Phantasm for targets. so (5/6)*(X/20) new Toralfs with target.
So the 3 damage abilities are at the a^x level.
4 damage: a^^x
5 damage: a^^^x
and so on
Oh wow, that's wild. Could we get away with cutting Omniscience and Guardmage for Genesis Ultimatum?
Is there a reason for going for Experimental Overload as the recursion card? Fossil Find is cheaper so we could replicate it more, although that wouldn't change the number of cards needed.
I just edited my post above with a 12 card version that cuts Omniscience and uses Festival of Trokin to gain a bunch of life to get started.
I didn't use Fossil Find because it would go infinite: create a Toralf, God of Fury copy and let the original die. Use Fossil Find to bring it back and play the other side: Toralf's Hammer. Use that as a repeatable damage source to trigger all the Toralf abilities without using Outnumber.
But for the new version that level of recursion is overkill anyway
Edit: Just noticed that I gave too many copies with TYS in my line of play above. It should only count instants and sorceries for "storm". But we still get 8 Festivals, and that should be enough life to still beat graham's number.
Edit2: Running the numbers I get 215 Outnumber copies.
The idea with Cormela was as a limited mana source and a way to recur spells (dying to legend rule). We can only get the mana by keeping the new one, which fizzles the TYS copies targeting the old one so each TYS trigger only makes mana once, while the other copies can make more TYS.
But there are a few pieces missing.
Edit: also: Hello to the newcomer, CaptainMarcia! Always nice to see new people. Seems like you have some experience with this?
Hi Iijil!
Would we be able to activate Hammer in that scenario? It doesn't look like we have anything with haste.
Regarding the Festival of Trokin version, how are we casting Toralf/Cubwarden/Phantasm with Festival copies on the stack?
And yeah, not having flash for those creatures is a problem. Not creating any new creatures between the Festival copies doesn't gain enough life to beat graham's number. And the creatures that are already out can't be copied with Change of Heart. Orvar, the All-Form because it doesn't trigger of itself and Elite Guardmage because that would cause too many draws.
That is quite annoying. I'm sure there is a workaround, but I'll have to think about it for a bit.
About a week of experience, lol. Started out with trying to prove a point on Reddit, then kept going.
Before that, I'd taken a few looks at stuff like the 2015 megacombo and this thread, but hadn't really tried working on this stuff myself.
That makes sense. In the meantime, do you think the Genesis Ultimatum strategy is enough to get us to 12?
Yeah that seems to work
Notice that Thousand-Year Storm is absent in the deck.
The only number keeper is Brain Freeze because of Storm. But how to regain your number with massive storm copies but few of other resources?
I'm not entirely sure how the scaling of the Toralf damage works or how it compares to Graham's function. Do we have an idea of how many times we need to start the damage chain?
Edit: Wait, can we condense the token making into Cogwork Assembler?
The main function to create Graham's number (let's call it A(X)) takes X to 3 ^^^...^^^ 3, where the number of ^'s is X. We obtain Graham's number by applying this function 64 times starting from the number 4.
If Toralf deals 2 damage to an opponent's creature with 1 toughness, and we have X Toralfs in play at the time, then we will trigger X instances of 1 damage. Then it depends on what benefit we get from dealing that 1 damage, but to keep things simple, if dealing 1 damage allows us to create 1 new Toralf, then dealing 2 damage allows us to double our number of Toralfs: X -> 2X. Toralf dealing 3 damage will trigger X instances of Toralf doing 2 damage; since each instance of 2 damage allows us to double our Toralfs, X such instances doubles our Toralfs X times, taking us to 2^X Toralfs. ( Actually X*2^X, but the first X doesn't matter much) Similarly, dealing 4 damage will create X instances of 3 damage, allowing us to exponentiate X times, getting us to 2^^X. In general, Y damage will increase our Toralfs to 2^^^...^^^X with Y-2 up arrows.
If we call B(X) the number produced by the Toralf combo starting from dealing X damage, it won't be exactly A(X-2), but it's gonna be pretty close - like we'll probably have B(X+3) > A(X) > B(X) or something like that. Since the growth rate of these functions is so enormous, the minor differences aren't gonna come close to the impact of applying the function. So if we apply B(X) 63 times it won't be enough, if we apply it 65 times it will definitely be over. If we apply it 64 times, then we would want to look at the numbers more carefully. (But if we start with a number much more than 4 it should definitely go over.)
Yeah, Kamahl's Will deals damage as a single event, so it would trigger Toralf once.
Edit: Looks like that deck works!
That makes sense, thanks!
The list is getting increasingly tight, but with the first Kamahl's Will round now only needing 9 cards, I wonder if there's some way to condense the roles of Djinn Illuminatus and Experimental Overload to get us down to 10 while still hitting the target of ~65 rounds. Maybe with the help of changing up the Mind's Desire slot again somehow.
Edit: Ardent Dustspeaker could do it. After we finish resolving the first Kamahl's Will, we should be able to make a bunch of copies of it, with haste, then swing for a bunch of attack triggers. This gets a bit weird since it's interacting with the rest of our library, but whether or not we assume there's 50 blanks there, we eventually start hitting Kamahl's Will and can keep bringing it back with each successive one. Is that a legal move?Edit: Doesn't matter if it's legal or not. It can't eat through our library because it doesn't exile if it whiffs.
For a reusable Mind's Desire, how about Urza, Lord High Artificer? The mana is tight, but I think it works out if we go back to Outnumber.
Still, 11 cards is amazing! Mind's Desire for the start is a nice improvement.
My gut feeling is that 4 copies of Outnumber gets us to higher numbers than a single Kamahl's Will, even with only 5 targets initially available. To do that only the last Outnumber needs to deal more damage than the Will would, and I think we can sufficiently increase the creature count with the first three copies. I'll look at the numbers later.
Good point. Any non-specific recursion seems likely to run into that.
Given that we can use Cogwork Assembler to duplicate any permanent, I wonder if there's a way we can incorporate Doubling Season or Primal Vigor, to dramatically increase the rate of Toralf generation. Could that do enough to meaningfully change the scaling?
Edit: I also did some looking for cards that could give higher upfront damage and found Burn at the Stake. Not sure how much it changes, but I think that and Primal Vigor should at least make for a new 10-card record.