How to deal way too much damage in Magic: the Gathering (work in progress)

Section 1: Introduction

About a decade ago, a question occurred to me: how much damage can you deal in a game of Magic: the Gathering? There's a simple answer to this: as much as you want, do to the presence of "infinite combos" like [Nest of Squirrels] and [Earthcraft]. But, if you forbid the usage of such combos, then the question becomes interesting. After working on it for a while I was able to construct a 60 card deck that was capable of dealing around 10↑↑↑↑1000 using [Knuth's Arrow Notation]. I was duly impressed at the incredible size of this number, being much larger than any quantity I had heard of (aside from the infamous [Graham's Number]). I figured adding one or two more arrows was possible, but the most number of arrows you needed was probably not more than 10.

So, I was flabbergasted when I later read an article by metroidcomposite, describing a deck that could deal 2↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑30 damage on turn 1! This was the state of the art for a few years. Then in 2013, SadisticMystic (with the help of metroidcomposite) revised the deck, jumping the number of arrows to 113. At this point I became interested again, and joined in the discussion along with forum member plopfill. With successive attempts, we jumped the number of arrows to 210, then 223, then 234, then 263, then 408. (The big jump at the beginning was caused by the introduction of [Cowardice], and the huge jump at the end comes from the power of [Mimic Vat].) That final damage number of 2 ↑408 20 (here I'm using ↑408 as shorthand for writing 408 up arrows) is the largest damage deck (with no infinite combos, of course) described in an article... until now.

This deck is the work of Deedlit and Iijil, created in the MTG Salvation forums.


Section 2: The challenge

The precise rules for this challenge were formulated by SadisticMystic. I am taking the liberty of quoting them verbatim:

  • Start with a Vintage-legal Magic deck of exactly 60 cards. Certainly no less, of course, and if going over 60 was allowed then this challenge would simply devolve into an extremely inelegant mission to fill the deck up with as many dummy cards as you could possibly get your hands on. No cards may be brought in from outside the game either--once you count to 60, that's all you get.
  • Take the first turn of the game, using any 7-card hand you like. (Or fewer, if you have reason to take a mulligan. Amazingly enough, one of the ideas that came up during brainstorming really did involve a mulligan, with a purpose behind it.)
  • Conditions caused by randomness can be resolved in the most favorable manner you like. Even beyond the opening hand, if you can draw or otherwise utilize more cards on the first turn, those can be in any order you like. Additionally, you can win any coin flips that might come up. It's an all-out Magical Christmasland scenario: as long as there is some nonzero probability of your actions playing out, make the most of it! Just beware of the "nonzero" term. For example, if you put a card on the bottom of a non-empty library and then draw a card without using a shuffle effect first, it's guaranteed not to be the same card you just put back. (It might, however, be a duplicate copy of the card, as long as you keep the other rules about deck construction in mind.)
  • The opponent is a goldfish, with a deck of nothing but basic lands*, so their cards (probably) can't interfere with you at all. The goldfish will never concede the game, since this could trivially disrupt any action by ending the game before a single damage could be dealt. However, if your cards invite the opponent to make any other choice, such as Trade Secrets, you'll have to assume their decision is the one that's more effective at preventing your goal. Sometimes this could end up being extremely complicated: in the case of Fact or Fiction, there are 16 possible ways to split it, and any analysis involving that card would have to involve going over all 16 and figuring out which one leaves you the worst off. In short, you probably don't want to use cards like that.
  • Beyond that, all standard Magic rules apply, including conditions for ending the game. For example, if you have a setup that can deal 20 damage to target player a million times in a row, the game will end after the first such ability deals damage to the opponent (unless you take further countermeasures against that), and any subsequent "potential damage" may as well not exist. Be careful.
  • With all those rules laid out, the goal of this exercise, and the metric on which results are judged, is to deal as much damage as you can to the opponent by the end of that first turn. Even if you can take an extra turn before your "opponent" would get one, only the first turn itself is under consideration, so those effects are useless.
    • Of course, there is one more proviso that needs to be added such that this doesn't become trivial. There must be a definite, finite upper bound to the damage. If, for every integer N, a line of play exists that would allow your deck to deal at least N damage within the one-turn window, subject to the previous rules, then your deck is said to contain an unlimited loop (in other words, it "goes infinite"), and does not qualify for consideration here.

*: In the original challenge, the opponent's cards were assumed to be all [Island]s, but with the introduction of [Waste]s, that seemed like a more appropriate default card, so we will assume the opponent has 60 [Waste]s in his deck.

An important note: According to the above rules, it is not illegal to have a deck capable of producing unlimited amounts of a resource (such as mana, life, or creature tokens), so long as that resource cannot be converted into unlimited damage. So some "infinite combos" are legal. Happily, the following deck does not contain an infinite loop of any kind.

As mentioned in the introduction, there have been a number of articles written describing previous entries in this challenge:

version 1 "Mana Reflection" (July 2009, approximate damage: 2 ↑23 40; written by Kaitlyn Burnell)

version 2 "Soulshift" (August 2009, approximate damage: 2 ↑36 30; written by Kaitlyn Burnell)

version 3 "Sinking Feeling" (April 2013, approximate damage: 2 ↑113 5; written by Russell Jones)

version 4 "Cowardice" (February 2014, approximate damage: 2 ↑263 4; written by Russell Jones)

version 5 "Mimic Vat" (June 2015, approximate damage: 2 ↑408 20; written by Russell Jones)

You may be wondering at this point: how many Knuth arrows does the latest deck achieve? Well, I can't tell you that, because it's not a number that you can write down. The number of arrows is not even a number that can itself be described with Knuth arrow notation. Nor with the much stronger Conway chained arrow notation. No, we will need something a whole lot more powerful.


Section 3: Introducing the fast-growing hierarchy

(Note: This section is rather mathematical, so if you want to get to the Magic part of the article, or are already familiar with the fast-growing hierarchy, you can skip this section. However, you will need to somewhat understand the fast-growing hierarchy to make heads or tails of the numbers we will be using in the remainder of the article.)

As the title says, the powerful notation we will use is known as the [fast-growing hierarchy]. This is a family of functions Fα(n), where n is a positive integer, and α is what is known in mathematics circles as an [ordinal]. The technical definition of an ordinal is rather complicated, and we don't really need it here; (You can read the Wikipedia article if you are interested in ordinals.) instead, we will simply think of an ordinal as a polynomial over the symbol ω, with nonnegative integer coefficients.

So, the smallest ordinals are the nonnegative integers:

0,1,2,3,...

But, there is a smallest ordinal bigger than all the nonnegative integers, namely ω. So we continue:

ω, ω+1, ω+2, ω+3,...

ω2, ω2+1, ω2+2, ω2+3,...

ω3, ω3+1, ω3+2, ω3+3,...

...

ωn, ωn+1, ωn+2, ωn+3,...

...

...

ω2, ω2+1, ω2+2, ω2+3,...

ω2+ω, ω2+ω+1, ω2+ω+2, ω2+ω+3,...

ω2+ω2, ω2+ω2+1, ω2+ω2+2, ω2+ω2+3,...

ω2+ω3, ω2+ω3+1, ω2+ω3+2, ω2+ω3+3,...

...

...

ω22, ω22+1, ω22+2, ω22+3,...

ω22+ω, ω22+ω+1, ω22+ω+2, ω22+ω+3,...

ω22+ω2, ω22+ω2+1, ω22+ω2+2, ω22+ω2+3,...

ω22+ω3, ω22+ω3+1, ω22+ω3+2, ω22+ω3+3,...

...

...

...

ω3, ω3+1, ω3+2, ω3+3,...

ω3+ω, ω3+ω+1, ω3+ω+2, ω3+ω+3,...

ω3+ω2, ω3+ω2+1, ω3+ω2+2, ω3+ω2+3,...

ω3+ω3, ω3+ω3+1, ω3+ω3+2, ω3+ω3+3,...

...

and so on through all the polynomials of ω. (The actual ordinals don't stop there, but thankfully this is enough for our purposes.) Note that I wrote the coefficients on the right - for technical reasons, that is the correct place to write coefficients. But it means what you would think it means; ω3 = ω+ω+ω.

Okay, now we are ready for the definition of the fast-growing hierarchy:

1. F0(n) = n+1.

2. Fα+1(n) = Fαn(n) (this is shorthand for Fα(Fα(Fα(...(n)...))) with n copies of Fα).

3. For m > 0, Fα+ωm(n) = Fα+ωm-1n(n).

Note: For rules 2 and 3, it is important to always write every ordinal as a sum of decreasing powers of ω. For example, it would be incorrect to say:

Fω3+ω3+ω2(n) = Fω3+ω3+ωn(n). (WRONG!)

because we have incorrectly placed the ω3 before the ω2.

Technically, this is all that you need to know; however, since it is hard to grasp the magnitude of the numbers from just the dry definition, I will try to give some examples.

The first few levels

So, we can calculate the first few Fα pretty easily:

F0(n) = n+1.

F1(n) = F0(F0(F0(...(n)...))) = n+1+1+1... = 2n.

F2(n) = F1(F1(F1(...(n)...))) = 2(2(2(...(2n)...))) = n*2n.

Then, F3 gets somewhat complicated:

F3(1) = F2(1) = 1*21 = 2.

F3(2) = F2(F2(2)) = F2(2*22) = F2(8) = 8*28 = 2048.

F3(3) = F2(F2(F2(3))) = F3(F3(24)) = F3(402,653,184) = 402,653,184 * 2402,653,184.

F3(4) = F2(F2(F2(F2(4)))) = F2(F2(F2(64))) = F2(F2(270)) > F2(2270) > 22270.

There is no simple formula for F3(n) using elementary operations. But, we do know that F2(n) = n*2n is greater than 2n; so, since F3(n) is F2(n) applied n times, the result will be more than 222...n with n 2's. So we can think of F3(n) as somewhat bigger than the "tower" function; it takes n to more than an exponential tower of n 2's with an n at the top.

Next comes F4(n), which is of course F3(n) repeated n times. So, since F3(n) exceeds the tower function, F4(n) will be more than the tower function iterated n times. So for instance:

F4(5) is greater than 222...5, where the number of 2's is 222...5, where the number if 2's is 222...5, where the number of 2's is 222...5, where the number if 2's is 222225.

F5(n) is then F4(n) repeated n times, and so on up to infinity.

As you can imagine, we can generate unimaginably large numbers quite quickly! For example:

F3(4) is bigger than a googolplex.

F4(3) is bigger than googolplexgoogolplexgoogolplex...googolplex with a googolplex googolplexes in the tower.

F5(2) is bigger than "googolplexgoogolplexgoogolplex...googolplex, where the number of googolplexes is googolplexgoogolplexgoogolplex...googolplex, where the number of googolplexes is googolplexgoogolplexgoogolplex...googolplex,...(phrase repeated a googolplex times) ..., where the number of googolplexes is a googolplex".

To compare with Knuth arrow notation, we have that Fm(n) lies between 2 ↑m-1 n+1 and 2 ↑m-1 2n. For example, the 2 ↑408 20 damage of the previous deck entry is about F409(17).

Into the infinite ordinals

For Fω, we apply rule 3:

Fω(n) = F0+ω1(n) = F0+ω0n(n) = Fn(n).

At first glance, this is underwhelming; we can write F6912(6912) just as easily as we can write Fω(6912). But this is a major step! You first notice the difference when you start to evaluate Fω+1(n):

Fω+1(3) = Fω(Fω(Fω(3))) = Fω(Fω(F3(3))) = Fω(FF3(3)(F3(3))) = FFF3(3)(F3(3))(FF3(3)(F3(3))).

Note that we now we are iterating, not just over the numerical input, but over the ordinal subscript as well! So to describe Fω+1(n), we have to nest the ordinal subscript n times. In terms of Knuth arrows, we have:

Fω+1(n) is approximately than n↑↑↑...↑↑↑n, where the number of arrows is n↑↑↑...↑↑↑n, where the number of arrows is n↑↑↑...↑↑↑n, where the number of arrows is n↑↑↑...↑↑↑n,... (repeat the phrase n times)... where the number of arrows is n.

For those of you familiar with Graham's number G(64), that last sentence will sound familiar. In fact, since Graham's number is the function 3↑↑↑...↑↑↑3 repeated 64 times, we have that

Fω+1(64) > Graham's number.

And, since Fω+2(n) is Fω+1(n) repeated n times, we have

Fω+2(n) > G(G(G(...G(n)...))) with n G's.

And so we construct a new sequence of functions Fω+1(n), Fω+2(n), Fω+3(n), Fω+4(n), etc., each one iterating over the previous, except that we started from a much faster growing function as the base.

The next step is of course

Fω2(n) = Fω+n(n).

Again, not particularly overwhelming - until you get to Fω2+1(n), and see that it iterates over the new sequence of functions Fω+n(n). Then we construct a yet faster growing sequence of functions Fω2+m(n), which gets iterated over by Fω3+1(n), and so on and so on.

Now is a good time to compare the fast-growing hierarchy to the second most well-known large number notation, [Conway chained arrow notation]. In fact,

Fωa+b(n) is roughly n → n → n → ... n → b+1, where there are a+2 n's.

Continuing on,

Fω2(n) = Fωn(n) is roughly n → n → n → ... n where there are n+2 n's.

Fω2+1(n) is roughly n → n → n → ... n, where the number of n's is n → n → n → ... n, where the number of n's is n → n → n → ... n, ... (repeat the phrase n times) ... where the number of n's is n.

So as you can see, once we get a little past Fω2(n) the numbers become too unwieldy even for Conway chained arrow notation.


And now, our feature presentation:


Section 4: Getting Started

You can see the cards in our deck [here], and also at [Tapped Out]. {Black Lotus} {Show and Tell} {Omniscience} {Opalescence} {Dual Nature} {Consecrated Sphinx} {Words of Wisdom}

In our opening hand, we draw [Black Lotus], [Show and Tell], [Omniscience], [Opalescence], [Dual Nature], [Consecrated Sphinx], and [Words of Wisdom]. Play the cards in that order. When Dual Nature is played, it will enter the battlefield as a creature due to Opalescence, so its enter the battlefield ability will trigger and create a token copy of Dual Nature. Then, when Consecrated Sphinx is played, both Dual Natures will trigger, causing two token copies of Consecrated Sphinx to be created. Finally, Words of Wisdom is cast; we will draw two cards, then the opponent will draw a card, which will trigger all three Consecrated Sphinxes. This causes us to draw six more cards, or eight total.

{Leyline of Anticipation} {Copy Enchantment} {Cowardice} {Allay} {Drake Familiar} {Evacuation} {Spellweaver Volute} {Flash of Defiance}

The eight cards we draw are [Leyline of Anticipation], [Copy Enchantment], [Cowardice], [Allay], [Drake Familiar], [Evacuation], [Spellweaver Volute], and [Flash of Defiance]. Play Leyline of Anticipation; this gets two Dual Nature copies, but we don't really care about that. Play Copy Enchantment, and have it enter the battlefield as Dual Nature; both it and the previous two Dual Natures trigger, causing three more Dual Nature tokens to be created. Play Spellweaver Volute, enchanting Words of Wisdom in the graveyard. Play Cowardice, followed by Allay, targeting the Copy Enchantment card that is currently a Dual Nature. This triggers Cowardice, causing the Copy Enchantment card to bounce back to our hand. Since the Copy Enchantment was a nontoken creature, this will cause all the other Dual Natures to trigger and want to exile all token copies of Dual Nature. Do not let these triggers resolve! Since the triggers are the only abilities on the stack, there will be no need to resolve them until we need to enter the combat phase, at which point we won't care about nonhasted tokens. For this section, whenever we play a Copy Enchantment that is going to be bounced by an ability that is at the bottom of the stack (except for Dual Nature exile triggers), we can safely have it copy Dual Nature, since when it gets bounced the Dual Nature triggers will go on the bottom of the stack and we won't need to resolve them until just before combat.

So, we respond to the Dual Nature exile triggers by replaying Copy Enchantment. This time we do not have it enter as a copy of Dual Nature, since it will be bounced by an ability that will be above other abilities that we need to resolve. So we have Copy Enchantment enter copying nothing. This triggers the remaining five Dual Natures, causing five token copies of Copy Enchantment to be created; have them all enter as Dual Natures. We are now up to ten Dual Natures.

Play Drake Familiar. This causes all ten Dual Natures to trigger, and we will get ten token copies of Drake Familiar. So we will be able to return an enchantment back to our hand eleven times. The first nine times we will return Copy Enchantment and replay it. The first eight times we have the original Copy Enchantment copy nothing, and have all the created tokens copy Dual Nature. This doubles the total number of Dual Natures each time, so we wind up with 10 * 28 = 2560 Dual Natures by the end of it. For the ninth time, we will have Copy Enchantment and all its token copies enter the battlefield as copies of Dual Nature; this results in 5122 Dual Natures in total. For the tenth bounce, we return Opalescence to our hand. Before the final bounce, we play Evacuation. This will bring both Drake Familiar and Consecrated Sphinx back to our hand. We then replay Opalescence. We then resolve the final bounce, and we return Copy Enchantment and replay it, copying nothing. We get 5121 token copies, of which we use 5119 to copy Dual Nature, and the remaining two to copy Spellweaver Volute, using one of them to enchant Allay and the other to enchant Evacuation. We now have 10,240 Dual Natures

Next, we cast [Flash of Defiance]; this will trigger all three Spellweaver Volutes to cast the three instants in our graveyard. Order the triggers in the following order: Allay, Evacuation, Words of Wisdom. Cast the Allay targeting the nontoken Copy Enchantment, returning it to our hand. Replay it, have it copy nothing, and have all the generated tokens copy Dual Nature, resulting in 20,480 Dual Natures. Next, replay Drake Familiar; this triggers 20,481 bouncings of enchantments, and for all but one of them we return and replay Copy Enchantment, copy nothing, and copy Dual Nature with the token copies. This doubles the number of Dual Natures 20,480 times, resulting in 20,480 * 220,480 Dual Natures, or more than 220,494. Use the last bounce to bounce Opalescence.

We now resolve the Evacuation, bringing Drake Familiar back to our hand. Replay Opalescence, and replay Drake Familiar, creating more than 220,494 Drake Familiars and bouncing Copy Enchantment more than 220,494 times. This results in more than 2220,494 Dual Natures. Play Consecrated Sphinx, creating more than 2220,494 token copies. Finally, we resolve Words of Wisdom; we draw two cards, then the opponent draws one card and all our more than 2220,494 Consecrated Sphinxes trigger, each one allowing us to draw two cards.

{Perpetual Timepiece} {Mirror of Fate} {March of the Machines} {Black Lotus} {Allay}

Draw the remaining creatures, artifacts, and enchantments in the deck first. Play [Psychic Battle], creating a gajillion token copies. Play [Perpetual Timepiece] and [Mirror of Fate]. Activate the Perpetual Timepiece to mill two cards, [Saltcrusted Steppe] and [Volcanic Island]. Play [Titania, Protector of Argoth], getting Dual Nature copies and bringing both lands onto the battlefield. Tap each land for a mana. Spend the two mana and exile Perpetual Timepiece to reshuffle cards from our graveyard into our library, targeting Black Lotus, Show and Tell, Spellweaver Volute, and Flash of Defiance. In response, sacrifice the Mirror of Fate; it goes to the graveyard. We exile our library, and put the cards Perpetual Timepiece, Allay, Evacuation, and Words of Wisdom into our library. Before the Perpetual Timepiece ability resolves, we have to resolve a bunch of Psychic Battle abilities that trigger off the targeting in the Perpetual Timepiece ability. Use those to switch the target of Flash of Defiance to Mirror of Fate. Reshuffle Black Lotus, Show and Tell, Mirror of Fate, and Spellweaver Volute into our library. Next, resolve three Consecrated Sphinx triggers, drawing Black Lotus, Mirror of Fate, Spellweaver Volute, Perpetual Timepiece, Allay, and Evacuation. Play [March of the Machines], then play Mirror of Fate and Perpetual Timepiece, creating many token copies of each. Cast Black Lotus, triggering all the copies of Dual Nature, but do not resolve all those triggers yet. Black Lotus goes to the graveyard as a state-based effect, being a 0/0 creature. Cast Allay, targeting any enchantment; this causes all the Psychic Battles to trigger, and each one causes Cowardice to trigger after we win the contest. (Remember, the opponent's deck is entirely Wastes.) So we can bounce and replay Copy Enchantment more than 2220,494 times, resulting in more than 22220,494 Dual Natures. Most importantly though, we use one of the Cowardice triggers to bounce March of the Machines. Then we resolve all the Dual Nature triggers, creating more than 2220,494 Black Lotus tokens. At this point, we have everything we need; we have complete control over our library, graveyard, and exile zones thanks to all the Perpetual Timepiece and Mirror of Fate tokens, we have all the mana we need, and all our permanent cards are on the battlefield or in our hand (except for Black Lotus, which we don't need anymore.). Use Mirror of Fate and Perpetual Timepiece so that [Reality Spasm], Allay, and Show and Tell are in our library, and the remaining cards from the exile zone are in our graveyard. Draw Reality Spasm and Allay. Use all but three of our mana to repeatedly cast Allay with buyback; each time we get X Psychic Battle triggers, and can bounce Copy Enchantment X times, resulting in about X*2X Dual Natures and Psychic Battles by the end. So each time we apply F2 to our number of Dual Natures and Psychic Battles, and we can cast Allay more than 22220,494 times, so we wind up with more than F3(22220,494) Dual Natures after all the Allays are cast. (Note: Each time we cast Allay with buyback, when we get down to our second to last Psychic Battle trigger, we target Cowardice and bring it back to our hand. Then we use the last Psychic Battle trigger to target an inessential enchantment, which gets destroyed. The purpose of doing this is so that Allay is successfully cast, rather than fizzling due to the lack of a target - this would cause Allay to be placed in our graveyard rather than put back in our hand via buyback.) Cast [Precursor Golem], getting more than F3(22220,494) Dual Nature copies, and twice as many vanilla Golem tokens as well. Cast all our other permanents as well. Finally, we cast Reality Spasm for three blue mana, and target a single Golem (trying to untap it), triggering all of our Precursor Golems. Each time we resolve a Precursor Golem trigger, we get a copy of Reality Spasm for each Golem on the battlefield, and each copy of Reality Spasm will trigger all of our Psychic Battles, each one triggering either Cowardice or [Horobi, Death's Wail] (We can choose which one to resolve first.). We can generate mana at will by targeting Dual Nature copies of [Goblin Gardener] to destroy Volcanic Island, then bouncing and replaying [Titania, Protector of Argoth] to bring it back to the battlefield untapped. We can also use a copy of Reality Spasm to untap Saltcrusted Steppe, which we can now tap to add a counter, so that we can start bouncing and destroying [Core Prowler] to add more counters. At this point we have the ability to implement the full combo, so let us now give a complete description of it.


Section 5: Implementing layers

(Note: For this section only, I will be using cards that are not in our deck. This section is just to explain layers; I thought it would be best to use somewhat less complicated combos, to make things more clear.)

During the making of this deck, we came up with some quirky names for the various recursive procedures:

A procedure that takes X of a resource and repeats the previous procedure X times is called a "layer".

A procedure that takes X of a resource and creates X consecutive layers is called a "stage". (By this we don't mean implement the same layer X times; that would just be another layer. Instead, we mean X consecutive layers where each successive layer repeats the previous layer many times.)

A procedure that takes X of a resource and creates X consecutive stages is called a "hyperstage".

A procedure that takes X of a resource and creates X consecutive hyperstages is called a "megastage".

These procedures naturally correspond to ordinals in the fast-growing hierarchy. If a procedure implements the function Fα, then:

-adding a layer will implement the function Fα+1

-adding a stage will implement the function Fα+ω

-adding a hyperstage will implement the function Fα+ω2

-adding a megastage will implement the function Fα+ω3

So first, let's give examples of layers. {Opalescence} {Doubling Season} {Precursor Golem} {Xenograft} {Cackling Counterpart} {Firemind's Foresight} {Supply // Demand} {Sage's Knowledge} {Djinn Illuminatus} {Holistic Wisdom} {Rings of Brighthearth} {Phyrexian Altar} {Reito Lantern}

Let's say we have [Opalescence], [Doubling Season], [Precursor Golem], [Xenograft] (choosing creature type "Golem"), [Djinn Illuminatus], [Rings of Brighthearth], [Reito Lantern], and [Phyrexian Altar] on the battlefield. If we resolve the spell [Cackling Counterpart], we can make a token copy of a Doubling Season; if there are X Doubling Seasons in play, that will change the one token to 2X tokens, so the number of Doubling Seasons becomes 2X + X. So, this procedure implements exponentiation; we can call this the "starting layer".

When we cast Cackling Counterpart on any creature (which is a Golem due to Xenograft), we will trigger every Precursor Golem currently on the battlefield. And each time we resolve a Precursor Golem trigger, we create a copy of the Cackling Counterpart spell for each Golem on the battlefield, which includes all the Doubling Seasons. If we have X Doubling Seasons on the battlefield, we get X Cackling Counterparts targeting Doubling Seasons, so we exponentiate X times; this gives us roughly F3(X) Doubling Seasons in the end. So the Precursor Golem triggers represent the second layer.

If we have X Precursor Golems on the battlefield, casting Cackling Counterpart gets us X Precursor Golem triggers, which will allow us to take Y to F3(Y) X times. So this implements F4(X). We can use the last Cackling Counterpart copy to create F4(X) Precursor Golems. (This is important; if we could only turn X Precursor Golems into F4(X) Doubling Seasons, but had no way to create more Precursor Golems, then iterating the procedure would not have nearly the effect that we want.) So this is the third layer.

Next, we can cast [Firemind's Foresight], and replicate it X times through Djinn Illuminatus, where X depends on the amount of mana we can generate through Phyrexian Altar. Each copy of Firemind's Foresight allows us to draw and recast Cackling Counterpart, so we implement the function F4(Y) X times. This gives us F5, and represents the fourth layer.

This continues with [Supply // Demand] and [Sage's Knowledge], each of which can retrieve the previous spell X times thanks to Djinn Illuminatus, and so they implement F6 and F7. Finally, [Holistic Wisdom]'s ability can be copied with Rings of Brighthearth to retrieve Sage's Knowledge X times, implementing F8. If we can pitch say 30 sorceries to Holistic Wisdom, we can implement F8 30 times, so our final number will be about F9(30).

So that's the basic idea of how layers work. Note that we can use layers for much stronger functions: If a procedure implements Fω3, adding a layer will allow us to repeat that procedure X times, implementing Fω3+1(X). So layers represent a significant increase, no matter how high up we go.

Note, however, that it does nothing significant to precede a larger procedure with a smaller procedure. For example, if I add a layer, then add a stage, the stage will turn into X layers, where the X keeps growing tremendously. So each time we implement the stage we will take X to Fα+X+1(X) rather than Fα+X(X) - an insignificant difference. So, we need to implement our procedures top-down, from the strongest to the weakest.


Section 6: The main stage

{Opalescence} {Grip of Chaos} {Cowardice} {Horobi, Death's Wail} {Bloodbond March} {Mimic Vat} {Omniscience} {Vedalken Orrery} {Mirror of Fate} {March of the Machines} {Dual Nature} {Copy Enchantment} {Azorius Signet} {Metallurgeon} {Izzet Guildmage} {Kiora's Dismissal} {Broken Ambitions}

Here is how we implement the main stage using the above cards. First, we need to have at least one of each of the spells [Kiora's Dismissal] and [Broken Ambitions] on the stack (for Broken Ambitions we will have X=0). As we have seen in the "Getting Started" section, each time we accumulate a blue and two other mana, we can copy Kiora's Dismissal using [Izzet Guildmage]. When Kiora's Dismissal goes on the stack, it triggers each copy of [Grip of Chaos] that we have on the battlefield, and all those triggers, plus Kiora's Dismissal itself, will trigger each copy of [Cowardice], due to the presence of [Opalescence] on the battlefield. (Note: We won't be worrying about how many copies of Cowardice get triggered per Grip of Chaos, since only one can return the creature being targeted.) Now, Grip of Chaos selects a random target; but, since we are in a Magical Christmasland scenario, we can assume that any random event goes the way we want it to go. Essentially, we get to choose the new target of each Grip of Chaos. In general, we will be targeting [Copy Enchantment] with all of the Grip of Chaoses, which gets bounced back to our hand. We recast Copy Enchantment between successive targetings, (we can do this because [Vedalken Orrery] is on the battlefield) copying nothing; all X of the [Dual Nature]s on the battlefield trigger (as [Opalescence] will be on the battlefield), allowing us to create X token copies of Copy Enchantment, and we make all of them copy Dual Nature. Thus, each successive targeting doubles the number of Dual Natures; if we have X Dual Natures and X Grip of Chaoses to start with, we will wind up with F2(X) = X * 2X Dual Natures after the Grip of Chaos triggers are resolved. Then, we have the Cowardice triggers for the original Kiora's Dismissal, which we will usually use to copy Grip of Chaos; this adds F2(X) = X * 2X more Grip of Chaoses as well. However, if we are about to cast a creature spell, we will instead copy [Bloodbond March], getting F2(X) copies of those instead. Occasionally we will need to make a copy of [Spellweaver Volute] or some other enchantment, and occasionally we will need to remove Opalescence, Cowardice, or [March of the Machines] from the battlefield; the latter is done by bouncing the nontoken enchantment, then letting Dual Nature destroy all the token copies.

Next, we will talk about the main character of the primary stage, [Metallurgeon]. Lets say we have X hasted Metallurgeon tokens on the battlefield (created via [Mimic Vat]), and the Metallurgeon card in our hand. We will have copies of the rest of the permanents pictured above on the battlefield. Also, we have at least nine mana, including at least one white and at least two blue. Cast the Metallurgeon; this triggers all the Bloodbond Marches on the battlefield. That won't help us at the moment though, since Metallurgeon is stuck on the stack below all the Bloodbond March triggers. So, we spend a blue and two mana to copy Broken Ambitions, targeting Metallurgeon. We are offered the opportunity to pay 0 mana to allow Metallurgeon to be cast; decline this payment! Then Metallurgeon goes to the graveyard, where it is in position to be returned to the battlefield by all the Bloodbond March triggers below it. Use one Bloodbond March trigger to bring it back, then tap one of the Metallurgeon tokens and pay two mana to activate its ability, targeting the original Metallurgeon. This creates a bunch of Grip of Chaos triggers, which are now placed on top of all the Bloodbond March triggers we had before, and a Cowardice trigger at the very top. The Cowardice trigger will bounce the original Metallurgeon back to our hand; we are now in our starting position, except one of our Metallurgeon tokens is tapped, and we have a stack of Grip of Chaos triggers on top of a stack of Bloodbond March triggers. When we are too low on mana, we can resolve a Grip of Chaos trigger to bounce the original Azorius Signet and replay it, allowing us to generate white and blue mana equal to our number of Dual Natures. Repeat the process for each hasted Metallurgeon token.

By the end of this process, we will have the following abilities on our stack:

Grip of Chaos trigger
Grip of Chaos trigger
...
Grip of Chaos trigger
Bloodbond March trigger
Bloodbond March trigger
...
Bloodbond March trigger
Grip of Chaos trigger
Grip of Chaos trigger
...
Grip of Chaos trigger
Bloodbond March trigger
Bloodbond March trigger
...
Bloodbond March trigger

...

Grip of Chaos trigger
Grip of Chaos trigger
...
Grip of Chaos trigger
Bloodbond March trigger
Bloodbond March trigger
...
Bloodbond March trigger

In total, we have X groups of Grip of Chaos triggers alternating with X groups of Bloodbond March triggers. We now have the primary stage set up, and we are ready to reap the rewards.

We start of course from the very top of the stack, where we have a group of Grip of Chaos triggers from Metallurgeon to resolve. We will use one of them to target the original Metallurgeon. We will have both Cowardice and [Horobi, Death's Wail] on the battlefield, so both will be triggered by the targeting of a Grip of Chaos ability. We can choose the order we place those triggers on the stack, and whichever type we place on top will be the one that actually moves the target; all the others will simply fizzle with the target now having changed zones. So, each time we target a creature, we will have a choice of whether to destroy or bounce it, so long as both Cowardice and Horobi, Death's Wail are on the battlefield. In this case, we choose to destroy the original Metallurgeon. Also, we will occasionally need to bounce [Mimic Vat] and [Mirror of Fate] when we run out of token copies to use. For the rest of the Grip of Chaos triggers, we will bounce [Azorius Signet]. Each bounce allows us to replay it with [March of the Machines] on the battlefield, getting as many token copies as we have Dual Natures. We can then bounce March of the Machines with Kiora's Dismissal, and tap all those copies for mana, which we use mostly to cast Kiora's Dismissal (and occasional for other mana costs such as Metallurgeon's activated ability). Each Kiora's Dismissal takes X Dual Natures/Grip of Chaoses to F2(X), so bouncing Azorius Signet will allow us to implement F3(X). We have X Grip of Chaos triggers in the top group, so when we are done resolving the group we will have taken X to F4(X).

Now, there is one minor issue that we have to deal with: the destruction triggered abilities of Dual Nature. When we destroy the original Metallurgeon, if Dual Nature is on the battlefield, all Metallurgeon tokens will get exiled, including hasted tokens created by Mimic Vat. We definitely want to avoid this. The solution is to always bounce Dual Nature back to our hand before destroying Metallurgeon or any creature for which we want to keep the token copies. But, bouncing the original Dual Nature back to our hand will itself set off triggered abilities to exile all the token copies of Dual Nature. This will in turn set off the other destruction trigger of Dual Nature, and exile all tokens created by those Dual Natures, which has the potential to wipe out all our progress. To deal with this, we adopt the following strategy: Right before we need to destroy a creature, we first copy Kiora's Dismissal to bounce the original cards for the enchantments we want to keep multiple copies of, namely Grip of Chaos and Bloodbond March. Allow their token copies to get exiled; we'll get more soon enough. Replay those enchantments, getting Dual Nature triggers, but do not resolve those triggers yet. Copy another Kiora's Dismissal to bounce the original Dual Nature, destroying the token copies, and all tokens created by those copies. Finally, we can resolve the Dual Nature triggers for Grip of Chaos and Bloodbond March, getting as many as we had Dual Natures before. Now we are free to destroy the creature. Afterwards, we can replay Dual Nature. We are reduced to two Dual Natures, but we have many Grip of Chaoses (call the number X), and the next Kiora's Dismissal will give us F2(X) Dual Natures and Grip of Chaoses, and progress is restored.

Once we have resolved the top group, we come across the first Bloodbond March ability in the second group. After we destroyed the original Metallurgeon, it triggered the abilities of all our Mimic Vats; we imprint the Metallurgeon on one of them. We then tap the Mimic Vat to create a hasted token copy of Metallurgeon. This puts the original Metallurgeon card in the exile zone. We use a Mirror of Fate token to bring the Metallurgeon card to the top of the library (bouncing March of the Machines if it is on the battlefield), and copy Broken Ambitions to mill it back into the graveyard. Then, after we are done resolving the top group of Grip of Chaos triggers, we resolve the first Bloodbond March trigger to bring it back to the battlefield. We can now tap our newly created hasted Metallurgeon token, triggering a bunch of Grip of Chaoses, and creating a new top group, but this time with F4(X) of them instead of X, and we repeat the previous paragraph. We do this for every Bloodbond March in the second group, and by the end of that group we will have taken X to F5(X).

After a great long while, we will have reached the first Grip of Chaos ability in the third group. At this point, we will have the original Metallurgeon on the battlefield, as well as an untapped hasted token copy of it. This time, instead of tapping the token copy, we will resolve the Grip of Chaos ability from the third group to target the original Metallurgeon; and instead of destroying it, we will bounce it back to our hand with Cowardice. This allows us to replay it, creating a new second group of Bloodbond March triggers, with F5(X) of them if the previous group had X triggers. Again, we repeat everything that comes before for each Grip of Chaos trigger in the third group, and by the end of the group we will have taken X to F6(X).

At this point, the pattern should be clear. We use the top group to bounce Azorius Signet (and other artifacts as necessary) many times. We use the Bloodbond March groups to bring the original Metallurgeon back to the battlefield, so that it can be destroyed by a hasted Metallurgeon token to create a new one and incidentally create a bunch of Grip of Chaos triggers, forming the next group up. The Grip of Chaos groups below the topmost one will bounce the original Metallurgeon back to our hand, allowing it to be recasted to create a bunch of Bloodbond March triggers, forming the next group up as well. The Nth group implements the function FN+3(X), and if we have X hasted Metallurgeons to start with, we will have 2X groups, so when we are done we will have implemented F2X+3(X). In particular, this is more than Fω(X) = FX(X), so we have reached the ωth level of the fast-growing hierarchy, which we call the "first stage".

The reader may be wondering at this point why this mechanism can't be repeated forever. Each time we restore the top group, our board state is what it was before, with more resources - except for the fact that there is one less Bloodbond March trigger in the second group. Similarly, each time we refresh a particular group, it comes at the cost of one ability from the next group down. So, we have a system of payment that goes one way, from bottom to top; therefore, we can't have a loop, and eventually the combo must run out.

Another interesting question is how, no matter how deep down we resolve the triggers, the combo "knows" to rebuild the groups back up to 2X groups of triggers and no more. The answer lies in the number of hasted Metallurgeon tokens we have on the battlefield. When we are resolving the topmost group, we will have only the newly created Metallurgeon token resulting from the recent destruction of the original Metallurgeon and the imprinting on a Mimic Vat. When we are at the third group though, we will have one hasted token in reserve, since we use the Grip of Chaos trigger from that group to target the Metallurgeon rather than the token. (This is why we don't use the token, since that will permanently decrease the number of tokens and our number of stages will be diminished.) When we are at the fifth group, we will have two tokens in reserve, and so on. So the number of hasted tokens we have on the battlefield dictates how many more groups of triggers we can build, which is why we always build up to exactly 2X groups every time, until the combo peters out.

Now that we have built the first stage, achieving Fω(X), how do we build up later stages, for higher multiples of ω? In fact, we use the same or similar combo for all stages. This particular first stage combo was based on the fact that the topmost group bounced Azorius Signets, which implemented F3(X). But, if we can have to topmost group do something which implements some stronger function, than this combo will simply build 2X layers on top of that. To phrase it in terms of the fast-growing hierarchy, if the topmost group of this combo implements the function Fα(X), then this function will add ω layers and implement Fα+ω(X). So, in fact the above Metallurgeon stage will comprise most of the stages in this deck, not just the first stage. It is this generality in the combo that allows us to go to the next step, the hyperstage.


Section 7: The hyperstage

{Metallurgeon} {Deep Reconnaissance} {Battle Cry} {Rebuild} {Spellweaver Volute} {Mox Emerald} {Natural Affinity}

So, to get started on the hyperstage we need a bunch of green mana, at least one untapped hasted Metallurgeon token, and a few colorless mana, which we can convert to white thanks to Farrelite Priest. (When we need more colorless or white mana, we can use a Psychic Battle trigger from Metallurgeon to destroy a Su-Chi.) We have the Metallurgeon card in our hand, both the original Vedalken Orrery and a Vedalken Orrery token on the battlefield, and [Deep Reconnaissance], [Battle Cry], [Rebuild], and [Natural Affinity] in the graveyard. Rebuild will be enchanted by a [Spellweaver Volute] token; Battle Cry will be enchanted by as many Spellweaver Volute tokens as we can manage. Natural Affinity will be enchanted by the original Spellweaver Volute.

We start by casting the original Metallurgeon. This triggers a bunch of Bloodbond Marches, and as usual we counter the Metallurgeon spell, and allow the first Bloodbond March ability to bring it back to the battlefield. We then tap and activate the untapped Metallurgeon token, causing a bunch of Psychic Battles to trigger. Use the first Psychic Battle trigger to bounce the original Metallurgeon (using Cowardice).

Now we are ready to perform what we call the "hyperstage transition". First, we need to something for which the necessity will not become apparent until later: We must put a Dual Nature trigger for Vedalken Orrery on the stack. We start by using a Psychic Battle trigger of the Metallurgeon token to bounce the original Vedalken Orrery back to our hand. This triggers Cowardice (and Horobi, Death's Wail, but we will resolve Cowardice first). Before we resolve the Cowardice, we cast Allay, and bounce March of the Machines back to our hand; Vedalken Orrery is no longer a creature. Cowardice will still bounce Vedalken Orrery back to our hand, as it has already triggered, but this action will not trigger Dual Nature's ability to destroy all token copies of Vedalken Orrery, as it only applies to nontoken creatures. We now replay Vedalken Orrery and get a bunch of Dual Nature triggers. Next, we respond by spending a green mana and four colorless mana to flashback Deep Reconnaissance. Right now, we don't actually care about what Deep Reconnaissance does, so we counter it with Cephalid Shrine; what matters is that it is a sorcery, and therefore casting it (even using flashback) will activate all the Spellweaver Volutes. Most of the time we will ignore the Spellweaver Volute trigger for Natural Affinity, but we want all the other Spellweaver Volutes to resolve successfully. This is something of a problem for all the Spellweaver Volutes on Battle Cry, since after we resolve the first Spellweaver Volute the Battle Cry card will be sent to the exile zone, and the other Spellweave Volutes will no longer enchant anything. Fortunately, there is a quirk in the rules that gets around this; after all the Spellweaver Volutes are triggered, we destroy them using Allay. Then, when the time comes to resolve a particular trigger, the Spellweaver Volute in question is no longer on the battlefield, so we use last known information to determine what it enchanted, which happens to be Battle Cry. Thus it doesn't matter whether the Battle Cry card is still in the graveyard or not.

We place the Spellweaver Volute trigger for Rebuild on the stack first, and then all the Spellweaver Volute triggers for Battle Cry. We resolve the topmost Spellweaver Vollute trigger for Battle Cry, which casts Battle Cry and untaps our tapped Metallurgeon token. Next, repeat the same procedure with Metallurgeon to create a stack of Bloodbond March triggers, then a bunch of Psychic Battle triggers on top. We then resolve this two-layer stage, increasing the number of Dual Natures accordingly. After the last Bloodbond March trigger is resolved, we tap the last created Metallurgeon token to bounce the original Metallurgeon card to our hand. We now no longer have any untapped hasted Metallurgeon tokens, but we have a lot of tapped hasted Metallurgeon tokens, one for each Bloodbond March trigger. So, we resolve the topmost Battle Cry and untap all the tapped Metallurgeon tokens. We now have a bunch of untapped hasted Metallurgeon tokens, which we use to create a very large stage. Once the stage has been created, we repeat the hyperstage transition, create an even larger stage above it, and repeat the process until we run out of green mana.

So, in the end our stack will look like this:

Group of Psychic Battle triggers
Group of Bloodbond March triggers
Group of Psychic Battle triggers
Group of Bloodbond March triggers
...
Group of Psychic Battle triggers
Group of Bloodbond March triggers

Group of Spellweaver Volute triggers for Battle Cry
Spellweaver Volute trigger for Rebuild
Group of Dual Nature triggers for Vedalken Orrery

Group of Psychic Battle triggers
Group of Bloodbond March triggers
Group of Psychic Battle triggers
Group of Bloodbond March triggers
...
Group of Psychic Battle triggers
Group of Bloodbond March triggers

Group of Spellweaver Volute triggers for Battle Cry
Spellweaver Volute trigger for Rebuild
Group of Dual Nature triggers for Vedalken Orrery

...
...

Group of Psychic Battle triggers
Group of Bloodbond March triggers
Group of Psychic Battle triggers
Group of Bloodbond March triggers
...
Group of Psychic Battle triggers
Group of Bloodbond March triggers

Group of Spellweaver Volute triggers for Battle Cry
Spellweaver Volute trigger for Rebuild
Group of Dual Nature triggers for Vedalken Orrery

Group of Psychic Battle triggers
Group of Bloodbond March triggers

So, we have a bunch of alternating groups of Psychic Battle and Bloodbond March triggers, but now they are divided by these hyperstage transitions, consisting of Spellweaver Volute triggers for Battle Cry, a Spellweaver Volute trigger for Rebuild, and Dual Nature triggers for Vedalken Orrery. This is the hyperstage; just as the stage consists of of a bunch of trigger groups, the hyperstage consists of a bunch of stages. And, just as each trigger group in a stage is replenished by the group below it, each stage in a hyperstage is replenished by the stage below it.

The hyperstage has been completed. Now, we can reap the rewards and start resolving it. The topmost stage resolves as explained in the previous section; after it is completely exhausted, the next Battle Cry untaps all the tapped Metallurgeon tokens that have been created, and we can set up an even larger stage. Eventually, all the Spellweaver Volute triggers for Battle Cry will be resolved. At this point we bounce March of the Machines back to our hand if it is on the battlefield, and play [Mox Emerald]. We tap it to get a green mana, returning the mana that was used to flashback the last Deep Reconnaissance. We then resolve Rebuild; this will eliminate all the Metallurgeon tokens and get rid of all copies of Vedalken Orrery, but it will also return Mox Emerald back to our hand. We then resolve the Dual Nature triggers for Vedalken Orrery, restoring our ability to cast noninstants while the stack is nonempty. (This is why we needed Dual Nature triggers for Vedalken Orrery, else we would be helpless to continue the combo after Vedalken Orrery got bounced back to our hand.)

Next, we go about recreating the topmost stage again. There are a few things to take care of; Rebuild and Battle Cry are now in the exile zone, so we need to put them back in the graveyard and reenchant them with Spellweaver Volutes. This can be done with the Psychic Battle triggers for Metallurgeon that are conveniently at the top of the stack; they can destoy Su-Chi tokens to provide colorless mana to cast Allay with buyback, which in turn can bounce March of the Machines so that the Mirror of Fate and Perpetual Timepiece tokens can be used. So we can use Mirror of Fate to bring the instants into the library, and then Perpetual Timepiece can put them back in the graveyard. Allay can bounce Copy Enchantment, which can be replayed to create many copies of Spellweaver Volute. Next, we replay Metallurgeon, and use a Psychic Battle trigger to destroy it, allowing Mimic Vat to create a token copy. After resolving the rest of the Psychic Battle triggers and returning the original Metallurgeon to the graveyard, we use the first Bloodbond March trigger to return Metallurgeon to the battlefield. We now activate the Metallurgeon token, getting lots of Psychic Battle triggers, and use one trigger to bounce the Metallurgeon back to our hand. Finally, we play Vedalken Orrery to get Dual Nature triggers again. We can now spend a green and four colorless mana to flashback Deep Reconnaissance, triggering Rebuild and Battle Cry and allowing us to recreate the topmost stage, but this time with many more Battle Cries.

Since we can get our green mana back and recreate the topmost stage, why doesn't the combo go infinite? The explanation is similar to why the stage combo doesn't go infinite: In order for the recreated stage to get started, we need at least one hasted Metallurgeon token, tapped or untapped. But Rebuild gets rid of all Metallurgeon tokens, so in order to create another one, we needed to use a Psychic Battle trigger from the stage below (and also a Bloodbond March trigger to bring the original Metallurgeon back to the battlefield). So each time we recreate a higher stage, we need to slightly expend the next stage down. But there is no way for a higher stage to replenish a lower stage, so as in the previous section, everything goes one way, from lower stages to higher stages, and there can be no infinite loop.

In order for things to work out this way, it is important that our mechanism for getting the green mana back (Rebuild) also gets rid of the Metallurgeon tokens; if we allowed the Metallurgeon tokens to stay around, we would have an easy infinite. Also, if we were to replace Su-Chi and Farrelite Priest with say, [Tooth of Ramos], then we would get an infinite without the need to recreate the higher stage; we could keep casting Deep Reconnaissance via flashback, and we could reset the artifacts and instants using just the mana from replaying Tooth of Ramos with Dual Natures on the battlefield. But if we tried to do the same thing with Su-Chi, we would be unable to get any mana (Su-Chi requires a Metallurgeon to destroy it), and the combo would fizzle out.

The reader may be wondering why we do not put more Spellweaver Volutes on Rebuild, to cast it multiple times. We could do so, but there is no benefit to doing it. When we resolve the first Rebuild, the Vedalken Orrery and all its token copies are returned back to our hand, so there is no longer a Vedalken Orrery on the battlefield, and we cannot replay any artifacts. So any ensuing Rebuilds will do nothing; in particular, we cannot return Mox Emerald back to our hand more than once per flashback of Deep Reconnaissance, so our green mana does not spiral out of control. This is why we have Vedalken Orrery in our deck instead of the more convenient [Leyline of Anticipation]; the latter would not be returned to our hand by Rebuild, and we could repeatedly cast and return Mox Emerald, accumulating green mana without end.

So, what kind of numbers are we reaching? As we explained in the previous section, each group of triggers in the topmost stage contributes a layer: the top group of Psychic Battles generates f4(X), the next group of Bloodbond Marches generates f52X+3(X), or more than fω(X). Each Battle Cry will untap a number of tapped Metallurgeons equal to the number of Bloodbond Marches resolved, which will also be more than fω(X); so X Battle Cries will generate more than fω+1(X). So we are already beyond Graham's number, once the number of Battle Cries is at least 64.

Then, each Bloodbond March trigger in the topmost group of the second stage can recreate the first stage, so the topmost group of Bloodbond Marches generates fω+2(X). Then the next group of Psychic Battle triggers generates fω+3(X), and so on, and finally the Battle Cries at the bottom of the second stage will generate fω2+1(X). In general, the Battle Cries in the nth group will generate more than fωn+1(X), so the entire hyperstage starting from X green mana will generate more than fω2(X). This is enough to take us beyond Conway chained-arrow notation.

An important note: We need an untapped hasted Metallurgeon to get the stage started, so it is important that, when the last Metallurgeon token in the bottom stage is created, we do not tap it; instead we proceed to the next abilities in the the stack, which will replenish our green mana and allow us to rebuild our hyperstage again. This brings us to our next section.

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