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    posted a message on Most turn 1 damage in a deck with no infinite combos
    Hi clone103! Always glad to see a new person join the challenge.

    Adding Ionize doesn't go infinite, since in order to counter a spell, you need to take 2 damage, and you only have a finite amount of life. It's a good way to extend the "chain" of resources and effects that lead to big numbers.

    It's true that adding more of the "best possible effects" is a good idea, but only when you run out of ideas that can lead to better possible effects. Before we got to stage combos and beyond, the challenge was all about creating as many "layers" as possible, where a layer is basically a recursion loop over the previous layer. If you can come up with something that can add a layer (like FortyTwo's idea of adding Ionize), it pretty much automatically beats adding a bunch of cards that duplicate the previous layer.

    I will go over your deck and estimate the final damage in a little bit.


    Edit: Okay, let's start by talking about the notation that we mainly use to describe numbers in this challenge: the Knuth arrow notation. This notation technically uses the up-arrow symbol, but we just use the caret symbol "^". (both for convenience, and also because apparently the up-arrow symbol is not allowed on this forum)

    Just as multiplication is repeated addition (6*5 = 6+6+6+6+6) and exponentiation is repeated multiplication (6^5 = 6*6*6*6*6), we can continue this pattern and talk about tetration, which is repeated exponentiation:

    6^^5 = 6^6^6^6^6.

    Note that the operations are executed from right to left; this leads to much bigger numbers than if we executed the operations from left to right.

    Then we have pentation, or triple arrows, which is repeated tetration:

    6^^^5 = 6^^6^^6^^6^^6

    So to compute this number, we would start by computing 6^^6, which is an exponential tower of six 6's (6^6^6^6^6^6). Then to compute 6^^6^^6, we would take the result of the last computation (6^^6) and make an exponential tower with that many 6's (so 6^6^6^6^6^...^6 with 6^^6 6's), and the result of computing that entire tower would be 6^^6^^6. We continue this process until we get to 6^^6^^6^^6^^6.

    So, the process continues forever:

    6^^^^5 = 6^^^6^^^6^^^6^^^6

    7^^^^^6 = 7^^^^7^^^^7^^^^7^^^^7^^^^7

    and so on. Each new operation a^....^b means repeating the previous operation with b copies of a. So each new operation represents an extra recursive layer over the previous operation.

    Similarly, the "layers" of a magic combo refer to the stacking of recursive layers in the combo. "Layer 1" refers to some combo that performs exponentation; "Layer 2" refers to some combo that repeats Layer 1 X times, so it performs tetration; "Layer 3" refers to some combo that repeats Layer 2 X times, so it performs pentation; and so on. Not by accident, the layer numbers match up with the corresponding number of arrows; executing a Layer N combo will perform an N-arrow operation.


    So, let's talk about your deck. Each time we resolve an effect that will create a token of a Doubling Season when there are X Doubling Seasons on the battlefield, we will actually create 2^X tokens. So this is a Layer 1 effect. Each time we resolve a Precursor Golem trigger of a copy spell, we create a copy of the spell for each of the X Doubling Seasons, and each one will exponentiate the number of Doubling Seasons, so we will go from X to approximately 2^2^2^...^2^X Doubling Seasons, with X 2's. This is a little more than 2^^X, so this is Layer 2. Then, each time we cast a copy spell, we will get a Precursor Golem trigger for each of the X Precursor Golem triggers in play, so the number of Doubling Seasons will go from around X to more than 2^^2^^2^^...^^2^^X Doubling Seasons, or a little more than 2^^^X Doubling Seasons. Then, when we get around to copying the Precursor Golems we will get more than 2^^^X Precursor Golems thanks to the more than 2^^^X Doubling Seasons in play. So this is Layer 3.

    Of course, we will also have the effects of Kiki-Jiki, Mirror Breaker and Thousand-Year Storm. If we have X Kiki-Jikis, we can create X token creation abilities, enough to take X to 2^^X; but this is the same effect as a single Precursor Golem trigger, so the benefit of all the Kiki-Jikis is roughly equivalent to having one more Precursor Golem. Since we have a gajillion Precursor Golems, the effect of the Kiki-Jikis won't show up in our estimates.

    Similarly, Thousand-Year Storm gets us a bunch of token creation abilities from a copy spell. This time, if we have around X Thousand-Year Storms, and the number of instants/sorceries cast is around X, then we will get around X^2 copies rather than X, so the Thousand-Year Storms have the effect of taking X to 2^^(X^2). This is better than 2^^X, but only by a little bit; it will certainly be worse than the effect of two Precursor Golem triggers, which will take X to 2^^(2^^X). So the effect of the Thousand-Year Storms is more than one Precursor Golem trigger but less than two; again, since we have a gajillion Precursor Golems, the effect won't show up in the estimate.

    Next up, we have Echo of Eons. Each time we cast this, we get many copies from our Thousand-Year Storms; if we have more than X Thousand-Year Storms and a storm count of more than X, then we will be able to retrieve more than 7X^2 copy spells, taking X to more than 2^^^2^^^2^^^...^^^2^^^X with more than 7X^2 2's which is more than 2^^^^(7X^2), so this is Layer 4. Note that there isn't a big difference between 7X^2 and X here; if we had a different ability that only took X to 2^^^^X, getting N+1 copies of that ability would still be better than N copies of Echo of Eons, so it's not as big a difference as it might seem at first.

    Finally, we have our exile spells. Each time we cast Time Reversal or something similar, we get X^2 copies thanks to the Thousand-Year Storms. So for each of these spells, we get to redraw Echo of Eons X^2 times, taking X to approximately 2^^^^^(X^2). So this is Layer 5.

    Okay, now for the precise estimate. We have 26 spells that can copy creatures (Saheeli's Artistry is not as good, since it won't copy our Doubling Seasons, which is the main source of growth here). After the first copy spell, we will have more than 2^^^^3 Doubling Seasons, since 2^^^^3 = 2^^^4 = 2^^2^^4 = 2^^65536, and we can pass that; but, we will have much fewer than 2^^^^4 = 2^^^(2^^65536) Doubling Seasons, since that would require about 2^^65536 Precursor Golem triggers off a copy spell, and we only have 1 to 3. Each copy spell after the first will increase the second number by one, since it takes X to 2^^^X, and 2^^^(2^^^^N) = 2^^^^(N+1). So after 26 copy creature spells, we have between 2^^^^28 and 2^^^^29 Doubling Seasons.

    We then cast Echo of Eons, which takes us to between 2^^^^2^^^^28 and 2^^^^2^^^^29 Doubling Seasons. This is between 2^^^^^4 and 2^^^^^5, and therefore between 2^^^^^^3 and 2^^^^^^4. Then, each of our 16 exile cards will increase the rightmost number by one, so we will end up between 2^^^^^^19 and 2^^^^^^20 creatures by the end.

    So the final damage is between 2^^^^^^19 and 2^^^^^^20.
    Posted in: Magic General
  • 1

    posted a message on Most turn 1 damage in a deck with no infinite combos
    Good to hear from you Stakfish!

    Yeah, that seems to be a problem with Perpetual Timepiece. We can get by without losing any cards by replacing Cephalid Shrine with Broken Ambitions to keep our ability to mill, and then replace Perpetual Timepiece with some card that can shuffle cards in our graveyard into our library, like Canal Dredger.

    With Soul Foundry, we don't need to actually bounce or kill Godtoucher! It is enough to bring it from the graveyard/library into our hand, so that we can imprint it onto Soul Foundry. So Wirewood Herald will do the trick. I just noticed, though, that if we use the Herald to bring back Godtoucher, we won't be able to bring back the Herald as well. So I guess we can go back to using Verdant Succession, which will require rejuggling the end stage sequence. Using double Heralds plus Assembly Hall is another possibility.

    So how about:




    I'm certainly interested in what we can do in Standard, although I'm not quite familiar with exactly which cards are in Standard, and it's something of a pain to keep checking which cards are in Standard and which aren't. So I'll let you take the lead Stakfish, and I'll contribute what I can.
    Posted in: Magic General
  • 1

    posted a message on Most turn 1 damage in a deck with no infinite combos
    I have updated the OP with links to a few of the explanatory posts.


    EDIT: I've been wondering whether the "target artifact" combos like Arcum Dagsson or Goblin Trashmaster could be made to work after all. They would certainly go infinite in the old version of the combo, but in this new version we have a lot of new moving parts, and maybe those can't be refreshed as easily as the artifacts. But I'm not sure.

    If it still goes infinite, then I think we can still make it work by getting rid of Mirrorworks. Just replace Mirror of Fate and Mirrorworks with Izzet Guildmage and Pull from Eternity, I think. That resolves the problem of requiring more Psychic Battle triggers of our stage artifact, since we will need lots of mana for the stage transition and one copy of Mana Vault and a Keyrune won't be enough. However, we may have the Muzzio problem of being able to bring a copy of whatever we are using to target/sacrifice an artifact below the hyperstage transition, and getting an extra use out of them. Stakfish, do you know an artifact combo that would be safe from this happening?


    EDIT: It just occurred to me that getting rid of Mirrorworks can also solve the problem we had with the Soul Foundry deck. That deck also used Muzzio, so we have to switch that out as well, but perhaps we can squeeze out an extra stage or two with that deck.
    Posted in: Magic General
  • 1

    posted a message on Most turn 1 damage in a deck with no infinite combos
    Child of Alara can be retrieved with Goryo's Vengeance. For Muzzio, the idea was to have Scattershot Archer deal damage to it while it was dealing damage to Moggcatcher. But, either way.
    Posted in: Magic General
  • 1

    posted a message on Most turn 1 damage in a deck with no infinite combos
    Reality Spasm is very nice, because it allows us to create a hasted Disciple of Tevesh Szat token, which we need, and also contributes a layer. In our last deck Goblin Bushwhacker was ridiculous - it created hasted tokens, and managed to create three layers per card draw, with the aid of Black Lotus and Dual Nature; unfortunately, we don't have either of those. With a creature stage at the end, it looks like we need at least a token creator and an initial card drawer to get started, and probably more. Hopefully we can optimize the main combo list to allow for whatever cards we need to get started.

    One problem with generating mana is that we have to watch for infinities in each of the stage transitions. For the hyperstage transition, for instance, we have to watch out for being able to restart the hyperstage without any Metallurgeon tokens, just with mana and the artifacts that we can use, and generate more mana each time we repeat. Mana Echoes and Mana Reflection look very dangerous to me, since if the deck works we would be able to have gajillions of those, and could get a lot of mana without any Metallurgeons. Thran Dynamo / Mana Vault may be safe, since then there is a limit to how much mana we can get starting from no artifacts.


    When the Wizards forum was going to be shut down, I took the time to save all the pages on archive.org: http://web.archive.org/web/20150917153327/http://community.wizards.com/forum/cards-and-combos/threads/2230096

    Have fun!


    Edit: Hmm, since you have switched to Thorn Lieutenant, is Shrewd Negotation really necessary? You just need to copy Onward // Victory prior to copying Switcheroo to give Raging Swordtooth to the opponent.
    Posted in: Magic General
  • 1

    posted a message on Most turn 1 damage in a deck with no infinite combos
    Nice deck, CTNC!

    I don't see how you can tap the Mimic Vats, since they will be creatures with March of the Machines on the battlefield. We can't destroy the March of the Machines cards using Righteous Fury since we can't tap March of the Machines.

    I think Jace, Architect of Thought goes infinite. You can cast it with one Doubling Season on the battlefield, so that it gets 8 counters. Then, activate the ultimate ability, destroying it, and copy the ability many times with Rings of Brighthearth. Use Archangel's Light to shuffle Jace into the library, and use one of the ability copies to redraw Jace, along with many other cards like Kheru Spellsnatcher

    Similarly, Jace, the Mind Scupltor goes infinite; cast it with two Doubling Seasons on the battlefield, then use the +0 and +12 loyalty abilities.

    Assuming these problems are fixed, I get the following for the numbers:

    Each Kiki-Jiki, Mirror Breaker abiilty can copy a Doubling Season, taking X Doubling Seasons to 2^X Doubling Seasons.
    Each time you tap Kiki-Jiki with X Rings of Brighthearth, we get X copies of the ability, taking X to 2^^X.
    Each time we create a copy of Kiki-Jiki with X Doubling Seasons, we get 2^X Kiki-Jikis, taking X to 2^^^(2^X).
    Each time we tap a Mimic Vat imprinted with Kiki-Jiki with X Rings, we take X to 2^^^^X.
    Each time we cast a spell with X Paradox Engines, we can untap the Mimic Vats X times, taking X to 2^^^^^X.
    Each time we gain X life, we can do the Wydwen dance X times, taking X to 2^^^^^^X.
    Each time we cast Righteous Fury, with X Rikus/Meleks, we get X copies, taking X to 2^^^^^^^X.
    Each time we cast Archangel's Light with X Rikus, we can reshuffle in Righteous Fury X times, taking X to 2^^^^^^^^X.
    Each time we activate a Lesser Walker with X Rings, we can retrieve Kheru Spellsnatcher X times, taking X to 2^^^^^^^^^X.
    Each time we activate a Greater Walker with X Rings, we can draw a Lesser Walker X times, taking X to 2^^^^^^^^^^X.
    Each time we activate Garruk, Apex Predator with X Rings, we can destroy a Greater Walker X times and retrieve it, taking X to 2^^^^^^^^^^^X.
    Each time we have a combat phase, we can attack with X Bragos, flickering Garruk X times and taking X to 2^^^^^^^^^^^^X.

    So, first you attack with Brago 9 times, allowing 18 activations of Garruk, Apex Predator and generating about 2^^^^^^^^^^^^20. Then, we attack with 2^^^^^^^^^^^^20 Scourge of the Thrones, getting that many combat phases. So the final damage will be about 2^^^^^^^^^^^^^(2^^^^^^^^^^^^20) (13 and 12 arrows respectively).


    Concerning your loophole - is a Vintage game modifiable by a previously played Unglued game? If that were to be allowable, then I suppose one could further argue that we don't even need the modifying game to be an official M:tG game - we could just make up a game that says that in your next Magic game you start out with Rayo's number of life. Perhaps I should add another rule saying we are just playing a standard game starting fresh, unmodified by anything happening prior (This would eliminate someone's idea to use one of the Karns to restart the game with cards already in play.).
    Posted in: Magic General
  • 1

    posted a message on Most turn 1 damage in a deck with no infinite combos
    This thread may be of interest to you guys.


    [EDIT] I just had a thought. We previously rejected Argivian Find and one Fountain of Youth, because we couldn't find a way to get back three life after the Worldfire resolved, so we were bleeding a life each stage. But, that was before we realized that we could use a Retract in the higher hyperstage to bring Fountain of Youth back to our hand.

    So I think the hyperstage transition could go something like this:

    We are at 4 life and have a black mana. Cast Acorn Harvest, bringing us to one life, and get Spellweaver Volute triggers for both Battle Cry and Argivian Find. Resolve Battle Cry, and get lots of hasted untapped Metallurgeons. Put Argivian Find on the stack, but do not resolve it. Set up a stage with the Metallurgeon tokens, and use those tokens to transition to the higher hyperstage a bunch of times using Lingering Souls. When the Metallurgeon stage is exhausted, we are down to Argivian Find next on the stack. We make sure to bring back Fountain of Youth to our hand in the higher hyperstage, so we play it and gain a life. Then we play March of the Machines to put the Fountain of Youth in the graveyard, exile it with Mimic Vat, bounce March of the Machines, and use the created token to get another life. Then we resolve Argivian Find, bring Fountain of Youth back to our hand, play it, and use it to gain a third life. We then resolve some Psychic Battle triggers for Metallurgeon so that we can get some hasted tokens again, and then recast Acorn Harvest.

    So it looks like we don't actually have to bleed a life. The question is whether we can go infinite if we throw a Retract in there. But, as Iijil noted in the previous analysis, if we resolve Retract, whether before or after Battle Cry, prior to Argivian Find, then all the hasted Metallurgeon tokens will be destroyed, and we can't continue the combo when we go to the higher hyperstage. So this doesn't seem to be a problem.

    What does seem to be a problem: When we go to lower stages in the lower hyperstage, if we set them up the same way, we will have an Argivian Find on the stack, and if we resolve it that gives us the ability to gain 2 life. But we need three life in order to recast Acorn Harvest. So we will bleed a life each stage again if we do it this way. So, how about the following: for all the stages below the top stage, we will put the triggers in the order Battle Cry, Argivian Find, and the Retract. We spend 3 life to cast Acorn Harvest, and gain 2 life back when we resolve Argivian Find, so we lose a net 1 life and add one Retract to the stack each time we go up a stage. For the topmost stage we do as described in the beginning.

    Does this make sense?
    Posted in: Magic General
  • 3

    posted a message on Most turn 1 damage in a deck with no infinite combos
    The way the hyperhyperstage works is quite similar to how the hyperstage works, which is modeled itself modeled after the stage combo, but in enacted using multiple spells to get the needed effect.

    So our main stage combo (you can skip this paragraph if you understand how this works already) uses Metallurgeon tokens. If we have N metallurgeon tokens, we can cast and counter Metallurgeon (getting a bunch of Bloodbond March triggers), use one Bloodbond March trigger to bring it back to the battlefield, then tap a Metallurgeon token for a bunch of Psychic Battle triggers. Repeat N times for 2N alternating groups of Bloodbond March and Psychic Battle triggers. When we resolve the top stack of Psychic Battle triggers, we can destroy the original Metallurgeon with one of them, then use the rest to create more Psychic Battles. (In the latest deck, we can destroy Core Prowler to put a counter on Saltcrusted Steppe; once we have a bunch of counters on Saltcrusted Steppe, we can activate it and remove all but one counter to get a bunch of white and green mana, and then activate Vitu-Ghazi Guildmage a bunch of times to populate a Psychic Battle token a bunch of times.). The destruction of Metallurgeon allows us to imprint it on Mimic Vat, and create a new Metallurgeon token; we then use Mirror of Fate to bring it back from exile into the library, and Hair-Strung Koto to put it into the graveyard. Then, after the top group of Psychic Battle triggers are all gone, we use the topmost Bloodbond March trigger to bring Metallurgeon back to the battlefield, and we can use our newly created Metallurgeon token to target Metallurgeon and create a bunch more Psychic Battle triggers. We do this for each Bloodbond March trigger, until we get to the last one; after the last Bloodbond March trigger is used, we use the Psychic Battle triggers to create more Bloodbond March tokens, and we use the next Psychic Battle trigger on the stack (rather than tapping the latest Metallurgeon token) to bounce Metallurgeon back to our hand. This allows us to cast and counter Metallurgeon again, creating a bunch more Bloodbond March triggers, and we use each one to create a bunch of Psychic Battle triggers, and the process continues like this. Each group of X triggers in the stack iterates the group above it X times, so the top group adds some constant amount to the number of enchantments, the second group from the top multiplies by some constant amount, the third group exponentiates, the fourth applies tetration, and so on for 2N operations. This is what we call a "stage combo". A "second stage" will be similar, except the top group of Psychic Battle triggers will create X Metallurgeons, which allows us to increase the number of Psychic Battle tokens from X to F_w(X) in the fast-growing hierarchy. Then the second group increases from X to F_{w+1}(X), the third from X to F_{w+2}(X), and so on, so X tokens for the second group will take X to F_{w2}(X); N stages will take X to F_{wN}(X) in the fast-growing hierarchy.

    So next, a hyperstage. Here, the resource controlling the combo is red mana, supplied by Mox Ruby. We make use of three spells, Bump in the Night, Battle Cry, and Rebuild. We cast Battle Cry and Rebuild so that they are in the graveyard, and then cast Spellweaver Volutes on both of them. (One of the Spellweaver Volutes will be a copy so that we don't have to waste a card on a second Spellweaver Volute.) We also cast Bump in the Night so that it is in the graveyard, and from then on we use a red mana from Mox Ruby to flashback it. So, the combo proceeds as follows: We start M red mana. Also, we have N Metallurgeon tokens and the original Metallurgeon, which we cast and tap accordingly to set up a stack of 2N alternating groups of triggers, just like in the last paragraph. Then, we spend a red mana to flashback Bump in the Night, which triggers both Spellweaver Volutes. We resolve the Spellweaver trigger on Battle Cry first and resolve the Battle Cry; this untaps all N of the Metallurgeons. Resolve the Spellweaver trigger on Rebuild, but do not resolve the Rebuild; leave it on the stack, above the stack of 2N alternating Psychic Battle and Bloodbond March triggers. Next, we start casting and tapping Metallurgeons again to build up another stack of 2N alternating groups of triggers, then spend another red mana to flashback Bump in the Night again (we use Mirror of Fate and Hair-Strung Koto to put all our spells back in the graveyard, and use Copy Enchantment to replace missing Spellweaver Volutes as needed.) Repeat the process M times, creating M+1 copies of 2N alternating groups of triggers, separated by M copies of Rebuild. Now we start resolving our stack. After we resolve the topmost stage, we resolve the topmost Rebuild, which will destroy all the Metallurgeon tokens, and bring Metallurgeon and Mox Ruby back to our hand. The Mox Ruby gives us the mana to flashback Bump in the Night, but (fortunately) we cannot do so immediately; in order to reset our spell and Spellweaver Volute situation, we have to use Mirror of Fate a few times, and copy a Spellweaver Volute. To do this, we need to activate Vitu-Ghazi Guildmage a few times, and for this we need mana. So we need to use a few Psychic Battle triggers from the next stage down to get the needed mana, and then we can properly set up our spells, and use a red mana to flashback Bump in the Night, recreating the top stage. So, we get to apply the Ackermann function for a few Psychic Battle triggers in the second stage; all told the second stage gets us F_{w2}(X), the third stage F_w3(X), and the entire hyperstage gets us F_{w^2}(X).

    Next is the hyperhyperstage. This time, the resource that controls the hyperhyperstage is life. The flashback spell will be Flash of Defiance (which costs 3 life to flashback), and casting it triggers Spellweaver Helixes imprinting Fungal Sprouting and Worldpurge. The setup is precisely analogous to the hyperstage. Let's say we start with between 3P+1 and 3P+3 life, at least 2 red mana, and at least 1 Metallurgeon token. First, we setup the first hyperstage using whatever red mana we currently have more than 2(this might just be one stage if we have only 2 red mana); do not use the last Metallurgeon token on the topmost stage. Next, we use a red mana to cast Bump in the Night with a Spellweaver Volute on Time and Tide. (You will see why we do this later) Then we spend 3 life and a red mana to flashback Flash of Defiance, triggering the Spellweaver Helixes for both Fungal Sprouting and Worldpurge. Resolve the Fungal Sprouting first, after we generate a large Auratog by sacrificing enchantments. This creates a bunch of Saprolings, which we can sacrifice to Utopia Mycon for a bunch of red mana. Now we can implement a second hyperstage, with a lot more red mana to use. Repeat this process P times, creating P+1 groups of hyperstages. Resolve as above; when the topmost hyperstage is completely expended, make sure that we have put a bunch of storage counters on Saltcrusted Steppe by targeting Core Prowler a bunch of thimes. Next, play Urborg, Tomb of Yawgmoth and Shimmer, naming Swamps, and finally Time and Tide (triggered by Bump in the Night) so that Urborg and Saltcrusted Steppe phase out. Then we play Glimmerpost and Radiant Fountain, gaining 3 life. Then we play Portcullis, and then bounce a bunch of our nontoken permanents, until we are down to at most seven nontoken permanents. Finally, we let Worldpurge resolve, emptying our mana pool (but we have spent it all already), and moving the permanents still on the battlefield back to our hand, including Glimmerpost, Radiant Fountain, Mox Ruby, and the Portcullis currently exiling most of our permanents, which now come back to the battlefield. Then, Time and Tide resolves, so that Saltcrusted Steppe and Urborg, Tomb of Yawgmoth phase back in; we can now remove storage counters from Saltcrusted Steppe to generate a bunch of mana, and recreate a bunch of Psychic Battles and Bloodbond Marches. We now have the resources to get 3 more life and one more red mana, so we can flashback Flash of Defiance again - but again, to reset Flash of Defiance we need to expend some of our Psychic Battles from the next hyperstage down. So each few Psychic Battles from the next hyperstage down allows us to restart the hyperstage above it. Each hyperstage is with w^2 in the fast growing hierarchy, so P hyperstages is worth F_{w^2*P}(X), and the whole hyperhyperstage is worth F_{w^3}(X).

    Whew - well, if you have any questions on the above, or the deck in general, please feel free to ask!
    Posted in: Magic General
  • 1

    posted a message on Most turn 1 damage in a deck with no infinite combos
    Hi CTNC! It would be great to have another mind thinking about this.

    The combo you talk about is indeed nowhere close, at least not yet. Each time Paradox Engine untaps Selvala, Selvala gets X mana where X is the power of Heroes' Bane, which allows us to double Heroes' Bane's power X/4 times, to go from X to X * (2^(X/4)). This is basically exponentiation, or the first layer of recursion. Let's say you had something (call it resource A) that allowed you could expend to cast X spells, where X is the power of Heroes' Bane or the amount of mana that you have. Then you could untap Selvala X times, allowing you to pump Heroes' Bane to about X^X^X^...^X with X X's. This is called tetration, and is the second layer of recursion. It is denoted by X^^X in Knuth arrow notation. Then, suppose that you had something (call it resource B) that you could expend to gain X of resource A, where again X is one of the things that is growing bigger and bigger. Then you could apply X -> X^^X X times, which would get you to about X^^^X; this is called pentation, and is the third layer of recursion. So this is the pattern; each additional layer of recursion means that you can repeat the previous layer X times, where X represents the current output so far.

    To get a better idea of how the layer system works, you can take a look at at the following writeup of one of the earliest contestants in this challenge:

    Metroidcomposite's deck

    This achieved 36 layers, which seemed incredible at the time (and is more than enough to overpower everyone else's attempts at this challenge). Later, we came up with the following deck that achieved 408 layers:

    Our best layer deck

    But, shortly after that deck was written up, we came up with a combo that changed everything: a combo that could take X of something (mana or creature tokens) and produce X layers! You could then iterate the process, so if you started with say 10 mana, iterating the "stage" (that's what we're calling the combo that creates X layers) takes you to 10^^^^^^^^^^10, then a 10^^^...^^^10 where the number of ^'s is 10^^^^^^^^^^10, and so on.

    Iijil wrote an excellent post summarizing the details:

    Iijil's summary

    In particular, follow the first link in the above link to my post on page 2, where I describe the stage combo in some detail. Basically, you build up the stack with groups of alternating triggers, each group representing a layer; once one group runs out, one of the next group creates X of that group at the expense of one trigger in the next group, so it functions just like the layer system. The amount of mana / creature tokens keeps track of which group you are at in the stack, so as you get lower in the stack your mana/tokens builds up, allowing you to build the stack back up to the top.

    That was the cutting edge for a while, but then we figured out how to build a hyperstage; this is where you take X of a resource (mana or tokens) and build up X stages. That was the best we could do for a while; then later we figured out how to do two hyperstages. Finally, very recently we came up with ideas for how to implement a hyperhyperstage - as you may guess, this is where you take X of a resource and build up X hyperstages. So this is the current frontier. We are still in the midst of tinkering with the deck and checking for problems/infinities.

    Okay, I still haven't gone through the current deck step by step. That will take some doing. I will go over it generally in a later post; in the meantime, feel free to ask any questions you may have about what I have linked so far.
    Posted in: Magic General
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