So a friend of mine raised an interesting question/challenge that I'm hoping someone here might be able to answer:
Using a 60-card vintage-legal deck, with all the cards in the deck stacked, playing against an opponent with an infinite life total who is running a 60 basic lands deck, what is the highest finite amount of damage you can do, without using any infinite combos, by having your deck perform the Ackermann function in a single turn if you were on the play and this was turn one? Note: You are not allowed to use any of your opponent's cards.
In listed form, here's the challenge:
What is the highest amount of damage you can do using the Ackermann function?
Constraints:
-No infinite combos at any time (no infinite mana, turns, combat, etc.)
-Your deck must contain only 60 cards--no more, no less
-Your deck must be vintage legal
-You are on the play
-Your deck is stacked optimally
-Your opponent's deck has nothing but basic lands
-Your opponent's life total is infinite
-You are not allowed to use any of your opponent's cards
So a friend of mine raised an interesting question/challenge that I'm hoping someone here might be able to answer:
Using a 60-card vintage-legal deck, with all the cards in the deck stacked, playing against an opponent with an infinite life total who is running a 60 basic lands deck, what is the highest finite amount of damage you can do, without using any infinite combos, by having your deck perform the Ackermann function in a single turn if you were on the play and this was turn one? Note: You are not allowed to use any of your opponent's cards.
In listed form, here's the challenge:
What is the highest amount of damage you can do using the Ackermann function?
Constraints:
-No infinite combos at any time (no infinite mana, turns, combat, etc.)
-Your deck must contain only 60 cards--no more, no less
-Your deck must be vintage legal
-You are on the play
-Your deck is stacked optimally
-Your opponent's deck has nothing but basic lands
-Your opponent's life total is infinite
-You are not allowed to use any of your opponent's cards
Belcher could deal 112 damage. That is the first thing that popped into my head when I read this. Mull to 4. Opening hand: Belcher, Black Lotus, Mox, LED. Last card in Deck is a mountain. No other lands.
I like this challenge... time to start thinking about other decks and how they can work out.
Googling Ackermann, I found out it's math related, could you explain what it means more clearly, as the thought that goes thru my head is non-infinite loop where you Tendrils and or Minds Desire over and over looping them over and over with Regrowth and Timetwister this is not an infinite combo but if we are going to draw exactly what we need then the damage would probably be infinite.
More clarification on what you are looking for would be great.
Again using non-infinite methods you could reach any arbitrarily large number. Once you have a large surplus of mana you could even play some ridiculous stuff that would help you get there quicker like Furnace of Rath.
Spawnwrithe + Breath of Fury. Cast a bunch of timewalk effects to get tetrational scaling, A(4,n) vs the A(3,n) of Exponential. There are a bunch of cards to add to get stuff like A(4,f(n)) but I can't figure out a way to reach A(5,n).
I was going to mull this over a bit before drawing up a deck, but it's 4am and I can't get to sleep so I may as well just get this out of my system. Credit to Irini for the first line of play. This is basically the deck I would use to deal the most amount of damage possible on the first turn:
3. Cast a Riku. Use the 4 from Mana Echoes to put 2 counters on Gemstone Array. Cast a Doubling Season. Use the counters on Gemstone to copy it with Riku. 3 total Doubling Seasons, 1 Riku. At this point mana becomes a non-issue as you only need 4 for each copy from a Riku and you're adding more than that for each creature that enters the battlefield. However this is not "going infinite" because you can't use the mana to generate tokens without having nontoken creatures enter the battlefield, which you don't do an infinite number of times. It's just an arbitrarily large amount of mana.
4. Cast 2nd Riku, copy him with the first Riku: 3 Doubling Seasons, 10 Rikus.
5. Cast 2nd Doubling Season. Each Riku triggers.
1st: makes 16 tokens (4 doubling seasons in play).
2nd: makes 1046576 tokens (20 doubling seasons in play).
3rd: makes 7.07 x 10^315658. (1048596 doubling seasons in play).
Unfortunately, after this point WolframAlpha wouldn't give me exact numbers so I had to stop calculating there. However, it's pretty easy to see how you'd proceed: Cast all your enchantments, copy everything a million times with Riku, give everything haste with Maelstrom Wanderer, cast Craterhoof Behemoth for extra value and deal more damage than you can count in a lifetime. I'm sure there's better ways to optimize both the deck construction and the line of play itself, but without some serious processing power it'll be hard to figure it out.
Edit: With all the extra mana from the Mana Echoes otherwise going to waste, it's also possible something like an Aggravated Assault would be worth a slot.
3. Cast a Riku. Use the 4 from Mana Echoes to put 2 counters on Gemstone Array. Cast a Doubling Season. Use the counters on Gemstone to copy it with Riku. 3 total Doubling Seasons, 1 Riku. At this point mana becomes a non-issue as you only need 4 for each copy from a Riku and you're adding more than that for each creature that enters the battlefield. However this is not "going infinite" because you can't use the mana to generate tokens without having nontoken creatures enter the battlefield, which you don't do an infinite number of times. It's just an arbitrarily large amount of mana.
Opalescence doesn't turn enchantment cards into creature cards, it turns enchantment permanents into creature permanents. Which means that Doubling Season isn't cast as a creature and can't trigger Riku.
Spawnwrithe + Breath of Fury. Cast a bunch of timewalk effects to get tetrational scaling, A(4,n) vs the A(3,n) of Exponential. There are a bunch of cards to add to get stuff like A(4,f(n)) but I can't figure out a way to reach A(5,n).
I'm sorry I don't know what A(5,n) and so forth means, but I'd like to. Could you explain?
A(m,n) is the Ackermann function. A(1,n) = n and represents addition. A(2,n) is iterated addition, otherwise known as multiplication. A(3,n) is iterated multiplication, otherwise known as exponentiation. A(4,n) is iterated exponentiation otherwise known as tetration.
Opalescence doesn't turn enchantment cards into creature cards, it turns enchantment permanents into creature permanents. Which means that Doubling Season isn't cast as a creature and can't trigger Riku.
Riku doesn't trigger when you cast creatures, he triggers on nontoken creatures entering the battlefield under your control...
I like it. Have you estimated the damage output? What would the rest of your deck include? Urabrask (or whatever) to give haste and a bunch of clone effects to continue the exponential growth from Kiki-Jiki + Doubling Season. Maybe Craterhoofs. What else?
Kiki tokens already have haste.
So I plick up the call on Mana Echoes and Xenograft, as this solves my mana problems. Play Mycosynth Lattice so I can spent it everywhere.
1st KJ: makes 16 tokens for a total of 20 doubling seasons in play.
2nd KJ: makes 1046576 tokens for a total of 20 doubling seasons in play.
3rd KJ: makes 2^1046596 tokens.
4th KJ: makes 2^(2^1046596)tokens.
Play Djinn Illuminatus. Play Twiddle and replicate it with the Djinn for as many times as we have Doubling Seasons on the battle field to untap KJ to make more Doubling Seasons.
Play Regrowth and replicate it for as many times as we have Doubling Seasons on the battle field on Twiddle. Let one replication of Twiddle resolve, play Twiddle on KJ and replicate it as in the step above.
Play Leyline of Anticipation
Play Recollect and replicate it for as many times as we have Doubling Seasons on the battle field to get Regrowth back. Let one replication of Recollect resolve, play Regrowthas in the step above.
Do the same with Elven Cache on Recollect.
Play Eternal Witness. Play Momentary Blink on the Witness and replicate it for as many times as we have Doubling Seasons on the battle field to get the Elven Cache back from the step above.
Play Flicker on the Witness, replicate it, and get Momentary Blink back.
Play Ghostly Flicker on the Witness, replicate it, and get Flicker back.
So basically we just need more effects to add new layers to the recursion from above until we will hit the 60 cards in the deck.
I like it. Have you estimated the damage output? What would the rest of your deck include? Urabrask (or whatever) to give haste and a bunch of clone effects to continue the exponential growth from Kiki-Jiki + Doubling Season. Maybe Craterhoofs. What else?
Kiki tokens already have haste.
So I plick up the call on Mana Echoes and Xenograft, as this solves my mana problems. Play Mycosynth Lattice so I can spent it everywhere.
1st KJ: makes 16 tokens for a total of 20 doubling seasons in play.
2nd KJ: makes 1046576 tokens for a total of 20 doubling seasons in play.
3rd KJ: makes 2^1046596 tokens.
4th KJ: makes 2^(2^1046596)tokens.
Play Djinn Illuminatus. Play Twiddle and replicate it with the Djinn for as many times as we have Doubling Seasons on the battle field to untap KJ to make more Doubling Seasons.
Play Regrowth and replicate it for as many times as we have Doubling Seasons on the battle field on Twiddle. Let one replication of Twiddle resolve, play Twiddle on KJ and replicate it as in the step above.
Play Leyline of Anticipation
Play Recollect and replicate it for as many times as we have Doubling Seasons on the battle field to get Regrowth back. Let one replication of Recollect resolve, play Regrowthas in the step above.
Do the same with Elven Cache on Recollect.
Play Eternal Witness. Play Momentary Blink on the Witness and replicate it for as many times as we have Doubling Seasons on the battle field to get the Elven Cache back from the step above.
Play Flicker on the Witness, replicate it, and get Momentary Blink back.
Play Ghostly Flicker on the Witness, replicate it, and get Flicker back.
So basically we just need more effects to add new layers to the recursion from above until we will hit the 60 cards in the deck.
You ended up pretty similar to the line of play that I was using. I didn't think of Mycosynth Lattice, it's obviously better than Gemstone Array (like I was using) as you don't lose any mana on converting colourless to coloured. I also didn't think of using stuff like Eternal Witness or possibly Izzet Chronarch to rebuy token making effects.
First card I would want to add to your method would be Riku of Two Reflections since he basicaly fills a similar role to Djinn Illuminatus, but would also be able to make tokens of your Doubling Seasons, Parallel Lives and your recursion in Eternal Witness. I wonder what the optimum number of regrowth effects, token makers, etc. would be for the maximum damage output.
A fairly old but classic answer to this. It gets larger than you can comprehend.
Pretty awesome to read. It'd be interesting to see an updated version, ie. with an easier method of drawing your library (Enter the Infinite) and ways of casting them (Omniscience), and also since cards like Parallel Lives and Riku of Two Reflections have been printed since it was written.
A fairly old but classic answer to this. It gets larger than you can comprehend.
Pretty awesome to read. It'd be interesting to see an updated version, ie. with an easier method of drawing your library (Enter the Infinite) and ways of casting them (Omniscience), and also since cards like Parallel Lives and Riku of Two Reflections have been printed since it was written.
Yeah, a lot of the opening hand can be substituted out for Show and Tell into Omniscience and then Enter the Infinite. That saves you slots on all the rituals, Wheel and Deal, and Dream Salvage, which probably lets you eke out a few more layers here. I'm not sure Parallel Lives really helps (it's just another Doubling Season, so it's like starting with two instead of one which barely affects the outcome). Ditto Riku (he's basically another Mirari/Mirrorworks)
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A fairly old but classic answer to this. It gets larger than you can comprehend.
Pretty awesome to read. It'd be interesting to see an updated version, ie. with an easier method of drawing your library (Enter the Infinite) and ways of casting them (Omniscience), and also since cards like Parallel Lives and Riku of Two Reflections have been printed since it was written.
Yeah, a lot of the opening hand can be substituted out for Show and Tell into Omniscience and then Enter the Infinite. That saves you slots on all the rituals, Wheel and Deal, and Dream Salvage, which probably lets you eke out a few more layers here. I'm not sure Parallel Lives really helps (it's just another Doubling Season, so it's like starting with two instead of one which barely affects the outcome). Ditto Riku (he's basically another Mirari/Mirrorworks)
Yes, this was the combo that actually spurred the challenge. Apparently, a certain friend of mine boasted that he could come up with a vintage legal combo that could surpass that one. I proposed a combo to him that was of O(n!), but he said the number he had in mind was far larger than that. Wanting to one-up him, I then consulted another friend of mine who knew quite a bit about math and asked him what kind of function would be bigger than O(n!), and he said "Ackermann."
Hence, why there is this challenge. I believe the key to surpassing that combo is being able to replicate the steps of the Ackermann recursion via Magic cards, THEN devising a damage combo out of those steps. Currently, most of these posts I feel are disregarding imitating the Ackermann function and are just generating combos for damage. C'mon people, technique! Technique!
Yeah, a lot of the opening hand can be substituted out for Show and Tell into Omniscience and then Enter the Infinite. That saves you slots on all the rituals, Wheel and Deal, and Dream Salvage, which probably lets you eke out a few more layers here. I'm not sure Parallel Lives really helps (it's just another Doubling Season, so it's like starting with two instead of one which barely affects the outcome). Ditto Riku (he's basically another Mirari/Mirrorworks)
That's true about Parallel Lives if you're going by the original combo. Also you wouldn't want to use Riku going by from the one you linked, since it's largely focused on returning Minion Reflector to your hand with Hanna, Ship's Navigator. But I imagine there's more efficient combos in terms of the number of cards used now, and that might change the nature of the combo such that Riku is something you want. For example, Xenograft and Mana echoes along with Mycosynth Lattice solve all of your mana issues straight away.
That article was interesting. Something people need to really focus on from that article is that you want narrow cards and ideally you will be running mostly 1 ofs with very few cards having more than 1 copy. Things such as arbitrary amounts of mana actually limit you since they enable infinite combos too readily.
The thread in the WOTC forums from this explanation is still active, and they seem to have optimized it far more than anything posted so far here... It's a nightmare to try and figure out exactly how it all works without an explanation first though.
I gave it a try because why not using just storm cards for damage dealing + beacon of destruction. Just trying to calculate out the storm count is mind boggling though in terms of what it deals...here's the decklist:
If I had to wager a guess I'd say the above list isn't optimal, but changing things around affects the outcome dramatically in a challenge such as this. Obviously you don't start with leyline in your opener/on the field since it reduces the storm count by 1. Enter the infinite puts back beacon of course for mind's desire to cast thousands of times. Since gauntlet, furnace, and swath are all replacement effects you apply the order in which they apply/you can apply furnace last so that a single copy of grapeshot will deal 272 points of damage (first doubling goes to 34. Double that 3 times more and you get 272.) For beacon a single copy deals 336 points of damage. Tendrils is the weakest card in the deck in comparison as it isn't doubled by furnace as it's loss of life/not damage and it isn't helped out by swath or gauntlet. An idea would be to swap tendrils for remands. Remand draws beacon after remanding desire/grapeshot and then you beacon for free essentially. I'm far too lazy to calculate all the damage out though.
EDIT: Just realized regrowth and call to mind are cards/tendrils is worse than those when they just regrab grapeshot as well as ramp storm up more. -3 Tendrils +3 Regrowth. Another change is -4 venser +4 remand as remand deals free damage via drawing beacon and then casting beacon for free to shuffle back in when all venser did was bounce spells anyways to up storm count and remand works double duty in this way as it provides a storm count as well as beacon providing another storm count and more damage.
Just realized that with just 6 artifacts to bounce really retract is worse than venser at generating storm as venser basically doubles storm when it bounces the storm spell and allows you to recast it effectively doubling storm versus just retracting 6 artifacts and recasting them. On second thought there are more than 6 artifacts to bounce AND strionic resonator quadruples storm essentially. It is correct to go 4 ofs for retract, recall, and rebuild in this case then.
On third thought, since beacon deals more damage than grapeshot per copy essentially via mind's desire grapeshot is inferior to more regrowths and call to mind's (for desire). So -4 Grapeshot +2 Regrowth +2 Call to mind
Final edit: So the number was way too high for me to count in terms of how much damage would be dealt if I went through and did the entire spell chain. Let's just say it's probably more than 10 times the number of atoms in the entire universe or something absurd like that in terms of what I would deal damage wise.
Final edit #2 as dumb as that sounds: Forgot 4 regrowth in the decklist making it 56 cards so I added those as well as changed one call to mind to tendrils.
4. Praetor's Counsel and do it all again! This time you get 9 times the storm triggers because of being able to Rebuild your Strionic Resonators between Mind's Desires. 5.1 x 10^16 storm and 1.7 x 10^19 damage after the first counsel, 5.1 x 10^27 storm and 1.7 x 10^30 damage after the second, 5.1 x 10^38 storm and 1.7 x 10^41 damage after the third, and 5.1 x 10^49 storm and 1.7 x 10^52 damage after the fourth. After the final Yawgmoth's Will I'd put the total storm count at about 5.1 x 10^60 and the total damage output at about 1.7 x 10^63. These numbers are pretty approximate but you don't need to be too accurate as long as it's only a couple of factors of 10 when you're talking about numbers this big.
Unfortunately it falls short of number of atoms in the universe by a factor of about 10^15. It's a pretty decent amount of damage but not even close to the possible amount that you could deal. Please let me know if I screwed up my numbers very badly at some point.
The rough part to combo'ing out with the list is that you want to desire and copy it with resonator/key resonator each time. To activate each resonator you need 8 mana, 4 to untap them with key's, and then activate them all again. All in all that costs 20 each time you cast mind's desire. It also adds a ton of storm count. You can generate 14 mana each time via lotuses and the mana crypt. Although to make up the 6 other mana I realized you need some form of lifegain so you can channel more. So tendrils of agony gets reintroduced as a 1 of as after 1 tendrils you're going to have tons of mana via channel paying life that you gained through tendrils. The most replaceable card is probably 1 copy of call to mind. +1 tendrils -1 Call to mind.
The chain goes like this:
Lotus, show and tell in omniscience, enter the infinite put back beacon of destruction.
Lay down all the permanents, channel, storm count is 30. Channel down to 1 life, tap mana crypt, float 12 blue with gilded lotus. Mind's desire copying it 8 times via resonator so 1 colorless floating and 12 U floating. 30x9=270 along with the original copy of desire so you cast beacon 271 times adding 271 to the storm count. 336x271=91,056 damage.
Cast retract and replay all the artifacts adding 18 to the storm count. Storm at this point is 319.
Cast tendrils of agony for 320 copies copying it zero times with resonator to deal 640 damage and gain 640 life/mana via channel/have no mana problems ever.
Call to mind/regrowth desire storm 321.
Cast mind's desire copying it 8 times with resonators. 321x9=2889 copies + original desire. So 2890 copies of beacon to deal 971,040 damage and add 2890 to the storm count, bringing the storm count up to 3,211.
Retract artifacts upping the storm count by 18 floating green off of the lotuses and 2 colorless because why not. Storm is 3,229.
Regrowth desire storm count is 3,230. Cast it copying it 8 times. 3,230x9=29,070 + 1 original desire so add 29071 to the storm count via beacon casting. 29,071x336=9,767,856 damage via beacons. Add 29,071 to the storm count bringing the storm count up to 32,301.
Retract artifacts floating some green and colorless yet again adding another 18 to storm count. Storm count is 32,319.
Regrowth desire storm count is 32,320. Cast it copying it 8 times. 32,320x9=290,880 copies + original. Add 290,881 to the storm count and beacon them that many times. 290,881x336=97,736,016 damage. Storm count is 323,201.
Retract artifacts floating green off lotuses adding 18 to storm. Storm is 323,219.
Regrowth desire storm count is 323,220. Cast desire copying it 8 times. 323,220x9=2,908,980 + original so 2,908,981 copies of desire/beacon. 2,908,981x336=977,417,616 damage. Add 2,908,981 to the storm count bringing it up to 3,232,201.
Retract artifacts and regrowth desire adding 19 to storm count to bring it up to 3,232,220.
Cast desire copy it 8 times storm count is 3,232,221. 3,232,220x9=29,089,980 + original = 29,089,981. Add that much to storm count and beacon deals 9,774,233,616 damage. Storm count is 32,322,202.
Retract artifacts floating green and regrowth desire adding 19 to storm count. Storm count is 32,322,221.
Cast desire and copy it 8 times and then remand the original copy and cast the beacon for 336 damage. Storm count is 32,322,224. 32,322,221x9=290,899,989 copies of beacon to deal 97,742,396,304 damage and add 290,889,989 to storm count bringing storm up to 323,212,213.
Recall all artifacts to hand and float green off of lotuses recast them all upping storm by 18 bringing it to 323,212,231.
Desire, remand original, beacon them upping storm to 323,212,234 and dealing 336 damage. Copy it 8 times via resonators. 323,212,231x9=2,908,910,079 copies of beacon. 2,908,910,079x336=977,393,786,544 damage. Add 2,908,910,079 to storm count bringing storm count to 3,232,122,313.
Recall all artifacts to hand floating green off of lotuses. Recast upping storm by 18. Storm is 3,232,122,331.
Desire, remand, beacon again upping storm to 3,232,122,334 copy it 8 times. 336 damage off beacon. 3,232,122,331x9=29,089,100,979 copies of beacon multiplied by 336 = 9,773,937,928,944 damage. Add 3,232,122,331 to storm count bringing storm count up to 32,321,223,313.
This is where it get's so complicated that it might make your brain explode (you have been warned.) Desire, remand the original, beacon them, copy it 8 times. 32,321,223,313 x 9 = 290,891,009,817 copies of beacon. Add 290,891,009,820 to storm count. 290,891,009,817 x 336 = 97,739,379,298,512 damage. Storm count is 323,212,233,133.
Retract artifacts to hand floating green again. Replay them all adding 18 to storm count bringing storm up to 323,212,233,151. Cast desire, copy it 8 times, rebuild, recast artifacts, copy 8 more times, rebuild, recast artifacts, copy 8 more times, rebuild, recast artifacts, copy it 8 more times. 323,212,233,151 multiplied by 32 = 10,342,791,460,832 + original desire = 10,342,791,460,833 copies of beacon. 10,342,791,460,833 x 336 = 3,475,177,930,839,888 damage off beacons. Storm count at this point is 10,666,003,694,039.
Cast a praetor's counsel grabbing back 12 retract effects, lotus, channel, 4 remand, 5 regrowth effects, tendrils of agony, show and tell, enter the infinite, and mind's desire. Rinse, repeat several more times what is outlined above. I think it's safe to say that the amount of damage dealt is probably more than 10 times the number of atoms in the known universe seeing as how much storm you're going to generate when you repeat the above iterations 5 more times thanks to yawg's will as well. But I am, again, too lazy to calculate it out beyond this point seeing as going through just the above sequence took a couple of hours or so.
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Using a 60-card vintage-legal deck, with all the cards in the deck stacked, playing against an opponent with an infinite life total who is running a 60 basic lands deck, what is the highest finite amount of damage you can do, without using any infinite combos, by having your deck perform the Ackermann function in a single turn if you were on the play and this was turn one? Note: You are not allowed to use any of your opponent's cards.
In listed form, here's the challenge:
What is the highest amount of damage you can do using the Ackermann function?
Constraints:
-No infinite combos at any time (no infinite mana, turns, combat, etc.)
-Your deck must contain only 60 cards--no more, no less
-Your deck must be vintage legal
-You are on the play
-Your deck is stacked optimally
-Your opponent's deck has nothing but basic lands
-Your opponent's life total is infinite
-You are not allowed to use any of your opponent's cards
Belcher could deal 112 damage. That is the first thing that popped into my head when I read this. Mull to 4. Opening hand: Belcher, Black Lotus, Mox, LED. Last card in Deck is a mountain. No other lands.
I like this challenge... time to start thinking about other decks and how they can work out.
More clarification on what you are looking for would be great.
1 Show and Tell
1 Omniscience
1 Enter the Infinite
1 Mirror Gallery
1 Opalescence
1 Mana Echoes
1 Gemstone Array
1 Xenograft
1 Maelstrom Wanderer
1 March of the Machines
4 Doubling Season
4 Riku of Two Reflections
4 Parallel Lives
4 Copy Enchantment
4 Kiki-Jiki, Mirror Breaker
4 Quicksilver Gargantuan
4 Coat of Arms
4 Fated Infatuation
4 Spitting Image
4 Rite of Replication
4 Cackling Counterpart
4 Heat Shimmer
4 Stolen Identity
1 Craterhoof Behemoth
This is how I'd go about dealing all that damage:
1. Cast Black Lotus, sac for UUU, cast Show and Tell, put Omniscience into play, cast Enter the Infinite.
2. Cast Opalescence. Cast Mana Echoes. Cast Xenograft (naming Pirate of course). Cast Gemstone Array. Cast Mirror Gallery.
3. Cast a Riku. Use the 4 from Mana Echoes to put 2 counters on Gemstone Array. Cast a Doubling Season. Use the counters on Gemstone to copy it with Riku. 3 total Doubling Seasons, 1 Riku. At this point mana becomes a non-issue as you only need 4 for each copy from a Riku and you're adding more than that for each creature that enters the battlefield. However this is not "going infinite" because you can't use the mana to generate tokens without having nontoken creatures enter the battlefield, which you don't do an infinite number of times. It's just an arbitrarily large amount of mana.
4. Cast 2nd Riku, copy him with the first Riku: 3 Doubling Seasons, 10 Rikus.
5. Cast 2nd Doubling Season. Each Riku triggers.
1st: makes 16 tokens (4 doubling seasons in play).
2nd: makes 1046576 tokens (20 doubling seasons in play).
3rd: makes 7.07 x 10^315658. (1048596 doubling seasons in play).
Unfortunately, after this point WolframAlpha wouldn't give me exact numbers so I had to stop calculating there. However, it's pretty easy to see how you'd proceed: Cast all your enchantments, copy everything a million times with Riku, give everything haste with Maelstrom Wanderer, cast Craterhoof Behemoth for extra value and deal more damage than you can count in a lifetime. I'm sure there's better ways to optimize both the deck construction and the line of play itself, but without some serious processing power it'll be hard to figure it out.
Edit: With all the extra mana from the Mana Echoes otherwise going to waste, it's also possible something like an Aggravated Assault would be worth a slot.
Riku can't copy Doubling Season.
Why not?
Opalescence doesn't turn enchantment cards into creature cards, it turns enchantment permanents into creature permanents. Which means that Doubling Season isn't cast as a creature and can't trigger Riku.
A(m,n) is the Ackermann function. A(1,n) = n and represents addition. A(2,n) is iterated addition, otherwise known as multiplication. A(3,n) is iterated multiplication, otherwise known as exponentiation. A(4,n) is iterated exponentiation otherwise known as tetration.
Ackermann Function
Rethinking it, the Spawnwrithe combo is only A(3,2^n). It's left associated iterated exponentiation rather than right associated.
Riku doesn't trigger when you cast creatures, he triggers on nontoken creatures entering the battlefield under your control...
So I plick up the call on Mana Echoes and Xenograft, as this solves my mana problems. Play Mycosynth Lattice so I can spent it everywhere.
1st KJ: makes 16 tokens for a total of 20 doubling seasons in play.
2nd KJ: makes 1046576 tokens for a total of 20 doubling seasons in play.
3rd KJ: makes 2^1046596 tokens.
4th KJ: makes 2^(2^1046596)tokens.
Play Djinn Illuminatus. Play Twiddle and replicate it with the Djinn for as many times as we have Doubling Seasons on the battle field to untap KJ to make more Doubling Seasons.
Play Regrowth and replicate it for as many times as we have Doubling Seasons on the battle field on Twiddle. Let one replication of Twiddle resolve, play Twiddle on KJ and replicate it as in the step above.
Play Leyline of Anticipation
Play Recollect and replicate it for as many times as we have Doubling Seasons on the battle field to get Regrowth back. Let one replication of Recollect resolve, play Regrowthas in the step above.
Do the same with Elven Cache on Recollect.
Play Eternal Witness. Play Momentary Blink on the Witness and replicate it for as many times as we have Doubling Seasons on the battle field to get the Elven Cache back from the step above.
Play Flicker on the Witness, replicate it, and get Momentary Blink back.
Play Ghostly Flicker on the Witness, replicate it, and get Flicker back.
So basically we just need more effects to add new layers to the recursion from above until we will hit the 60 cards in the deck.
You ended up pretty similar to the line of play that I was using. I didn't think of Mycosynth Lattice, it's obviously better than Gemstone Array (like I was using) as you don't lose any mana on converting colourless to coloured. I also didn't think of using stuff like Eternal Witness or possibly Izzet Chronarch to rebuy token making effects.
First card I would want to add to your method would be Riku of Two Reflections since he basicaly fills a similar role to Djinn Illuminatus, but would also be able to make tokens of your Doubling Seasons, Parallel Lives and your recursion in Eternal Witness. I wonder what the optimum number of regrowth effects, token makers, etc. would be for the maximum damage output.
A fairly old but classic answer to this. It gets larger than you can comprehend.
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Pretty awesome to read. It'd be interesting to see an updated version, ie. with an easier method of drawing your library (Enter the Infinite) and ways of casting them (Omniscience), and also since cards like Parallel Lives and Riku of Two Reflections have been printed since it was written.
Yeah, a lot of the opening hand can be substituted out for Show and Tell into Omniscience and then Enter the Infinite. That saves you slots on all the rituals, Wheel and Deal, and Dream Salvage, which probably lets you eke out a few more layers here. I'm not sure Parallel Lives really helps (it's just another Doubling Season, so it's like starting with two instead of one which barely affects the outcome). Ditto Riku (he's basically another Mirari/Mirrorworks)
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Yes, this was the combo that actually spurred the challenge. Apparently, a certain friend of mine boasted that he could come up with a vintage legal combo that could surpass that one. I proposed a combo to him that was of O(n!), but he said the number he had in mind was far larger than that. Wanting to one-up him, I then consulted another friend of mine who knew quite a bit about math and asked him what kind of function would be bigger than O(n!), and he said "Ackermann."
Hence, why there is this challenge. I believe the key to surpassing that combo is being able to replicate the steps of the Ackermann recursion via Magic cards, THEN devising a damage combo out of those steps. Currently, most of these posts I feel are disregarding imitating the Ackermann function and are just generating combos for damage. C'mon people, technique! Technique!
That's true about Parallel Lives if you're going by the original combo. Also you wouldn't want to use Riku going by from the one you linked, since it's largely focused on returning Minion Reflector to your hand with Hanna, Ship's Navigator. But I imagine there's more efficient combos in terms of the number of cards used now, and that might change the nature of the combo such that Riku is something you want. For example, Xenograft and Mana echoes along with Mycosynth Lattice solve all of your mana issues straight away.
1 Black Lotus
4 Gilded Lotus
1 Mana Crypt
4 Pyromancer's Gauntlet
4 Strionic Resonator
4 Voltaic Key
4 Furnace of Rath
1 Leyline of Anticipation
1 Omniscience
4 Pyromancer's Swath
4 Hurkyl's Recall
4 Rebuild
4 Remand
4 Retract
1 Channel
1 Enter the Infinite
1 Call to Mind
4 Regrowth
1 Mind's Desire
4 Praetor's Counsel
1 Tendrils of Agony
1 Show and Tell
1 Yawgmoth's Will
If I had to wager a guess I'd say the above list isn't optimal, but changing things around affects the outcome dramatically in a challenge such as this. Obviously you don't start with leyline in your opener/on the field since it reduces the storm count by 1. Enter the infinite puts back beacon of course for mind's desire to cast thousands of times. Since gauntlet, furnace, and swath are all replacement effects you apply the order in which they apply/you can apply furnace last so that a single copy of grapeshot will deal 272 points of damage (first doubling goes to 34. Double that 3 times more and you get 272.) For beacon a single copy deals 336 points of damage. Tendrils is the weakest card in the deck in comparison as it isn't doubled by furnace as it's loss of life/not damage and it isn't helped out by swath or gauntlet. An idea would be to swap tendrils for remands. Remand draws beacon after remanding desire/grapeshot and then you beacon for free essentially. I'm far too lazy to calculate all the damage out though.
EDIT: Just realized regrowth and call to mind are cards/tendrils is worse than those when they just regrab grapeshot as well as ramp storm up more. -3 Tendrils +3 Regrowth. Another change is -4 venser +4 remand as remand deals free damage via drawing beacon and then casting beacon for free to shuffle back in when all venser did was bounce spells anyways to up storm count and remand works double duty in this way as it provides a storm count as well as beacon providing another storm count and more damage.
Just realized that with just 6 artifacts to bounce really retract is worse than venser at generating storm as venser basically doubles storm when it bounces the storm spell and allows you to recast it effectively doubling storm versus just retracting 6 artifacts and recasting them. On second thought there are more than 6 artifacts to bounce AND strionic resonator quadruples storm essentially. It is correct to go 4 ofs for retract, recall, and rebuild in this case then.
On third thought, since beacon deals more damage than grapeshot per copy essentially via mind's desire grapeshot is inferior to more regrowths and call to mind's (for desire). So -4 Grapeshot +2 Regrowth +2 Call to mind
Final edit: So the number was way too high for me to count in terms of how much damage would be dealt if I went through and did the entire spell chain. Let's just say it's probably more than 10 times the number of atoms in the entire universe or something absurd like that in terms of what I would deal damage wise.
Final edit #2 as dumb as that sounds: Forgot 4 regrowth in the decklist making it 56 cards so I added those as well as changed one call to mind to tendrils.
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I'm imagining the line of play would be something like as follows:
1. Black Lotus, Show and Tell, Omniscience, Enter the Infinite putting back Beacon of Destruction, Leyline of Anticipation, 4 Pyromancer's Swath, 4 Furnace of Rath, Channel. Storm count 14.
2. Chain playing your artifacts and Rebuild, Retract, and Hurkyl's Recall, adding mana from Gilded Lotus and Mana Crypt each time. Storm count 247, 286 mana.
3. Cast Mind's Desire. Wait on copying the storm trigger with Strionic Resonator to maximise damage potential. Let all the copies resolve, then Remand the original, casting the Beacon of Destruction you draw. Repeat for each Remand, then do the same with 4 Regrowth and 2 Call to Mind. ~510,000 storm count and ~170,000,000 damage.
4. Praetor's Counsel and do it all again! This time you get 9 times the storm triggers because of being able to Rebuild your Strionic Resonators between Mind's Desires. 5.1 x 10^16 storm and 1.7 x 10^19 damage after the first counsel, 5.1 x 10^27 storm and 1.7 x 10^30 damage after the second, 5.1 x 10^38 storm and 1.7 x 10^41 damage after the third, and 5.1 x 10^49 storm and 1.7 x 10^52 damage after the fourth. After the final Yawgmoth's Will I'd put the total storm count at about 5.1 x 10^60 and the total damage output at about 1.7 x 10^63. These numbers are pretty approximate but you don't need to be too accurate as long as it's only a couple of factors of 10 when you're talking about numbers this big.
Unfortunately it falls short of number of atoms in the universe by a factor of about 10^15. It's a pretty decent amount of damage but not even close to the possible amount that you could deal. Please let me know if I screwed up my numbers very badly at some point.
The chain goes like this:
Lotus, show and tell in omniscience, enter the infinite put back beacon of destruction.
Lay down all the permanents, channel, storm count is 30. Channel down to 1 life, tap mana crypt, float 12 blue with gilded lotus. Mind's desire copying it 8 times via resonator so 1 colorless floating and 12 U floating. 30x9=270 along with the original copy of desire so you cast beacon 271 times adding 271 to the storm count. 336x271=91,056 damage.
Cast retract and replay all the artifacts adding 18 to the storm count. Storm at this point is 319.
Cast tendrils of agony for 320 copies copying it zero times with resonator to deal 640 damage and gain 640 life/mana via channel/have no mana problems ever.
Call to mind/regrowth desire storm 321.
Cast mind's desire copying it 8 times with resonators. 321x9=2889 copies + original desire. So 2890 copies of beacon to deal 971,040 damage and add 2890 to the storm count, bringing the storm count up to 3,211.
Retract artifacts upping the storm count by 18 floating green off of the lotuses and 2 colorless because why not. Storm is 3,229.
Regrowth desire storm count is 3,230. Cast it copying it 8 times. 3,230x9=29,070 + 1 original desire so add 29071 to the storm count via beacon casting. 29,071x336=9,767,856 damage via beacons. Add 29,071 to the storm count bringing the storm count up to 32,301.
Retract artifacts floating some green and colorless yet again adding another 18 to storm count. Storm count is 32,319.
Regrowth desire storm count is 32,320. Cast it copying it 8 times. 32,320x9=290,880 copies + original. Add 290,881 to the storm count and beacon them that many times. 290,881x336=97,736,016 damage. Storm count is 323,201.
Retract artifacts floating green off lotuses adding 18 to storm. Storm is 323,219.
Regrowth desire storm count is 323,220. Cast desire copying it 8 times. 323,220x9=2,908,980 + original so 2,908,981 copies of desire/beacon. 2,908,981x336=977,417,616 damage. Add 2,908,981 to the storm count bringing it up to 3,232,201.
Retract artifacts and regrowth desire adding 19 to storm count to bring it up to 3,232,220.
Cast desire copy it 8 times storm count is 3,232,221. 3,232,220x9=29,089,980 + original = 29,089,981. Add that much to storm count and beacon deals 9,774,233,616 damage. Storm count is 32,322,202.
Retract artifacts floating green and regrowth desire adding 19 to storm count. Storm count is 32,322,221.
Cast desire and copy it 8 times and then remand the original copy and cast the beacon for 336 damage. Storm count is 32,322,224. 32,322,221x9=290,899,989 copies of beacon to deal 97,742,396,304 damage and add 290,889,989 to storm count bringing storm up to 323,212,213.
Recall all artifacts to hand and float green off of lotuses recast them all upping storm by 18 bringing it to 323,212,231.
Desire, remand original, beacon them upping storm to 323,212,234 and dealing 336 damage. Copy it 8 times via resonators. 323,212,231x9=2,908,910,079 copies of beacon. 2,908,910,079x336=977,393,786,544 damage. Add 2,908,910,079 to storm count bringing storm count to 3,232,122,313.
Recall all artifacts to hand floating green off of lotuses. Recast upping storm by 18. Storm is 3,232,122,331.
Desire, remand, beacon again upping storm to 3,232,122,334 copy it 8 times. 336 damage off beacon. 3,232,122,331x9=29,089,100,979 copies of beacon multiplied by 336 = 9,773,937,928,944 damage. Add 3,232,122,331 to storm count bringing storm count up to 32,321,223,313.
This is where it get's so complicated that it might make your brain explode (you have been warned.) Desire, remand the original, beacon them, copy it 8 times. 32,321,223,313 x 9 = 290,891,009,817 copies of beacon. Add 290,891,009,820 to storm count. 290,891,009,817 x 336 = 97,739,379,298,512 damage. Storm count is 323,212,233,133.
Retract artifacts to hand floating green again. Replay them all adding 18 to storm count bringing storm up to 323,212,233,151. Cast desire, copy it 8 times, rebuild, recast artifacts, copy 8 more times, rebuild, recast artifacts, copy 8 more times, rebuild, recast artifacts, copy it 8 more times. 323,212,233,151 multiplied by 32 = 10,342,791,460,832 + original desire = 10,342,791,460,833 copies of beacon. 10,342,791,460,833 x 336 = 3,475,177,930,839,888 damage off beacons. Storm count at this point is 10,666,003,694,039.
Cast a praetor's counsel grabbing back 12 retract effects, lotus, channel, 4 remand, 5 regrowth effects, tendrils of agony, show and tell, enter the infinite, and mind's desire. Rinse, repeat several more times what is outlined above. I think it's safe to say that the amount of damage dealt is probably more than 10 times the number of atoms in the known universe seeing as how much storm you're going to generate when you repeat the above iterations 5 more times thanks to yawg's will as well. But I am, again, too lazy to calculate it out beyond this point seeing as going through just the above sequence took a couple of hours or so.
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