One permanent can be sacrificed to bounce Kaya, Geist Hunter and double the current token multiplier.
One Whale produces many Krakens, for an exponentiation.
One Fish produces many Whales, for a tetration.
One Reef Worm produces many Fishes, for a pentation.
Exiling Reef Worm to Dollhouse of Horrors produces many of them, for a hexation.
When the original Reef Worm is sacrificed, it produces two Fishes; after those are fully consumed, the token multiplier is >2^^^4. Then Dollhouse of Horrors makes >2^^^4 copies of Reef Worm, for an end result of >2^^^^(2^^^4). Hold on to one of the Reef Worm tokens, and after the last Whale is sacrificed, start using the first ability of Kaya, Geist Hunter to put +1/+1 counters on it, so that it can attack for >2^^^^(2^^^4) damage.
Unfortunately, this can't be made into a 7-card solution by adding Black Lotus, Channel, and Lich's Mirror, because the token-doubling effects persist through Lich's Mirror's effect.
1st LF copy gives 6 life and one Golem is destroyed. Activate CA targeting PG.
2nd LF copy gives 8 life and one Golem is destroyed. Activate CA targeting PG. 1 mana left over.
3rd LF copy gives 10 life and one Golem is destroyed. Activate CA targeting PG. 4 mana left over.
4th LF copy gives 12 life and one Golem is destroyed. Activate CA targeting PG twice. 2 mana left over.
5th LF copy gives 17 life and one Golem is destroyed. Activate CA targeting PG twice. 5 mana left over.
Second PG trigger copies LF 21 times.
etc.
For the final card, like before, we want an instant/sorcery that can target a Golem and draw a card (or otherwise obtain Lantern Flare from the library to be recast), but that's not all this time. The Golems are being destroyed because of Lantern Flare's damage, which normally makes the final spell fizzle. We want to choose a card that can avoid fizzling.
Oblivion's Hunger would work if not for the +1/+1-counter condition.
Shelter would work if it were a different color from Lantern Flare. (Harsh Sustenance in place of Lantern Flare would solve this problem, but its mana requirements make it impossible to get going.)
Survive the Night would work, except that it goes infinite by saving a Clue to redraw it after it resolves, or by using Cogwork Assembler to copy a Clue.
Adrix and Nev, Twincasters, Ratadrabik of Urborg, Cabaretti Confluence... but the haste isn't copiable. If it were, we could make 2 copies of A&N, make 8 copies of A&N, and then make 2048 copies of Ratadrabik, and get 20490 Ratadrabik triggers on A&N and a bunch on Ratadrabik, and deal >2^^20492 damage.
Instead, Kindred Charge is the best I could find, dealing >2^^8 damage.
Edit: Using a similar method, I believe the 7-card deck finishes its second Slaughter cast with 7,258,845,553,210 copies of Precursor Golem and 16,937,306,290,822 copies of Salvaging Station:
That would be the numbers after the second Precursor Golem trigger on Slaughter, so before the second Slaughter cast.
We can however do better by using the mana from the last two slaughter deaths to create Precursor Golems, instead of only the mana from the last one. The Salvaging Stations we could create on the second to last death only give us 6 mana back before we resolve the next precursor golem trigger or cast the next spell and in either case getting just one more golem beforehand is worth more than the additional salvaging stations. (Though to optimize we can create Salvaging Stations on the second to last death at the cost of effectively 1 mana, so we can use that to finish with 0 to 6 additional stations 0 mana)
I updated my spreadsheet with that play pattern. We end up with 25 Salvaging Stations, 19 Precursor Golems and 36 Golem Tokens before the second Precursor Golem trigger on Slaughter will resolve.
Ooh, good find. And right, yeah, second trigger.
Running the function for (55,25,0) and making that same adjustment, looks like we get 227,479,118,689,477 (227 trillion) Precursors.
Thinking about the infinite mana versions, I think Inferno of the Star Mounts would work for 1-card. As soon as it hits 20 power, its triggered ability kills either itself or one of the players, so the most you can swing for is 19, for a total of 39.
I've also been thinking more about adding layers on top of Busy Beaver. I'm not entirely sure this works, but if so, it looks like an improvement over the Learn from the Past version:
Lotus, Channel, Nature's Spiral, replay Lotus for UUU. 21 mana.
Jadzi, Replication Technique, copy it, get two Jadzi triggers. Use the first to play Cadric, then resolve the copy for a Jadzi token. 7 mana.
Chain the second Jadzi trigger into Arcbond, Artificial Evolution, and Elixir, activate Elixir. 7 mana.
From the other trigger on Artificial Evolution, chain into Channel, chain one trigger into Black Lotus and sacrifice it, chain the other into Nature's Spiral targeting Lotus, chain into Precursor Golem and Elixir. 5 mana.
Return Lotus to hand, resolve down to Artificial Evolution, then activate Elixir (8 mana). Chain the second Arcbond trigger into Channel, then Artificial Evolution, targeting a golem token and getting a Precursor trigger....
End up with a bunch of life and with Lotus back in hand, then discard it to bounce Jadzi and start again with Journey to the Oracle. From here, start repeatedly casting Replication Technique to make more tokens, copying it with Precursor Golem and adjusting creature types to optimize it. Do another round with Fractured Identity, then Journey again, then Comeuppance.
After resolving Comeuppance, play Journey one more time, and this time start running the computations. Fractured Identity transfers permanents to the opponent, then lets us take them back after the computation. Precursor helps Fractured Identity target large numbers of permanents at once, and also means large numbers of generic tokens created by the computation provide more Jadzi triggers when playing targeting spells, letting us replay Replication Technique more times.
Unfortunately, the fact that we need to keep resolving Replication Technique, Fractured Identity, Artificial Evolution, and Arcbond between computations limits how many layers we can get outside of the computation. We can get four layers from Channel, Nature's Spiral, Journey, and Comeuppance, and I think another two from Arcbond and Precursor-Arcbond, resolving one Precursor trigger between each computation, for a total of six.
After a computation, we can replay the original Cadric from our library and make hasted tokens for each Cadric token we took from the opponent. At the end, we'll use these as the final damage source, since we'll no longer be able to replay the original Jadzi.
I wonder what would be the most efficient way to build a stage into a Busy Beaver deck. It'd have to be a traditional stage, since both Saw and Toralf are out.
Edit: Saw might be usable, but it would have to focus on a creature that doesn't have a creature type to keep Coat of Arms from interfering, with a way to move it between our field and the opponent's and to keep it from getting killed by Arcbond.
Edit: Removed a part of the early strategy that tried to get Jadzi back into the chain without discarding her.
Edit: Jadzi gives the opponent the ability to interrupt a computation since they can discard a card to bounce any copies of her they have. Argh.
Mulligan to 6. Lotus, Channel, Purse. 17 colorless, 4 treasures.
Play Precursor Golem, then Audacious Swap targeting Precursor, getting a Precursor trigger. Sacrifice a golem to Casualty, point the Casualty copy at Purse.
Resolve the Casualty copy, shuffle Purse back and flip Replication Technique. Point it and the Demonstrate copy at Precursor. Resolve the Demonstrate copy to get to two Precursors and three regular golems, then resolve the Precursor copy and the original to get to four Precursors and ten regular golems. Resolve the Precursor trigger on Swap, getting a Swap targeting each of them.
Resolve a Swap on a golem, flip Elixir, activate it. (12 colorless, 3 treasures.) Resolve a second, flip Elixir again. Resolve a third, flip Replication Technique, use the Demonstrate copy to make another Elixir and the original to target a golem, getting four Precursor triggers to double the Precursor count four times to 64. Then use the extra Elixir to reshuffle.
Replay Replication Technique 11 times, each time making a new Elixir and tetrating the Precursors, to reach 2^^14. Then play Mnemonic Deluge, exiling Swap to cast three copies of it. Each one gets 2^^14 Precursor triggers, as well as a Casualty copy, sacrificing two regular golems and the original Precursor.
Using the first Casualty, play Twist Allegiance to move all the creatures to the opponent's field. Then sacrifice the extra Elixir to shuffle Precursor and Replication Technique back into our library, use the second Casualty to replay Precursor, and use the third to replay Replication Technique. Use Demonstrate to replace the extra Elixir and use the main Replication on a golem token, triggering all the Precursors. Replication can only target the golems on our field, but it doubles our own board 2^^14 times, resulting both sides of the board now having about that many golems.
Resolve the first Precursor trigger on Audacious Swap, getting a Swap targeting all the golems on both sides. We can use these to finish building our board, starting by resolving Swaps on regular golems. Play Coat of Arms, Bishop of Wings, Arcbond, Comeuppance, and Dralnu's Crusade, then copy and hack our cards as needed with Replication Technique and Artificial Evolution. Dralnu's Crusade is less efficient to copy and hack, but when we do it, we'll want one of them set to turn our fresh Bishops into Golems for more efficient Precursor copy chains.
Once we run low on Swaps on the stack, replay Twist Allegiance to rotate the board again, then use an Elixir token to reshuffle it. At this point, we'll have one Precursor on our side after eliminating the other creatures from that batch, while the opponent will have all the creatures we made over the course of that round, three of which are Prcursors with Swaps still pointed at them. Use one Swap to play Soulblast, sacrificing our Precursor to run a computation, which we'll arrange to generate a large number of golem tokens, and to keep the two remaining Swap targets alive.
After the computation, resolve the next Swap for Twist Allegiance again, moving all those creatures back to our board. Resolve the last one for Replication Technique, getting it copied many times for all of our golems. Then resolve the next Precursor-Swap trigger, copying Swap for each of our new golems. The ones created during the computation are owned by the opponent, so we can't make use of targeting them with Swap, but the ones created by copying those tokens with Replication Technique are owned by us. Before the next computation, we'll need to use Twist Allegiance to move the creatures back to the opponent's board, then replay one of our creatures from our library to get something back to our side to sacrifice to Soulblast.
At this point we can repeat the process for each Precursor-Swap trigger, for over 2^^14 iterations. When we finish, take all the creatures with Twist Allegiance, give them haste, and swing.
Mnemonic Deluge feels a bit underwhelming, but a lot of alternatives would allow going infinite, like running a second copy of Audacious Swap.
For infinite mana strategies, with the Reef Worm one at 4, would 2-3 cards be the same as the current 5-6?
Edit: I think Possibility Storm works as a better method for getting multiple plays of Audacious Swap while keeping things sufficiently controlled.
Possibility Storm, Elixir. Possibility Storm puts Elixir on the bottom of the library and flips Precursor Golem.
Play Artificial Evolution, turn it into Audacious Swap. Use Casualty to play Replication Technique, go to 4 Precursors and 10 regular Golems. Resolve Precursor trigger, get 14 Swaps on the stack (counting the original). Use the first two to play Elixir (keeping it the second time), then the other 12 to play Replication Technique. The first Replication targets a golem, getting four Precursor triggers (making 60 new Precursors), and uses Demonstrate to copy Elixir. Replications 2-10 each target Possibility Storm and Elixir, then 11-12 target Possibility Storm twice, ending with 13 copies for a total of 14.
Play Arcbond, getting 14 Possibility Storm triggers (and putting Arcbond back into our library). Each time, flip into Audacious Swap for a new batch of Precursor triggers. Use those to build our board and start running computations.
Replication Technique makes 3 TYS tokens, Second Harvest gets copied 8 times to reach 1,536 TYS tokens, Emergency Powers gets copied 4,611 times, for 2^^4611. Because there's no way for this deck to have creatures on both players' fields at the same time, we can use Chandra's Triumph to start the computation, partway through resolving an even number of Twist Allegiances.
This does allow us to cast Artificial Evolution on Bishop during the Emergency Powers chain, before playing the final Artificial Evolution, but I don't think there's a way to go infinite from that.
Lotus, Channel, Purse, Precursor, Audacious Swap. Swap triggers Precursor and shuffles back Purse for Replication Technique, go to 4 Precursors and 10 regular. Resolve Precursor trigger for 14 Swaps, play Elixir, Elixir, Possibility Storm, and 11 Replication Techniques, making 11 more Possibility Storms for a total of 12, as well as making 10 Elixirs and doing one round of Precursor cloning.
Artificial Evolution, trigger 12 Possibility Storms. Use them to replay Audacious Swap > Replication Technique: for each Possibility Storm trigger, for each Precursor trigger on Swap, for each golem targeted by Swap, for each Precursor trigger on Replication Technique, for each golem targeted by Replication Technique - 5 layers?
I think that gets us to 2^^^^^14 golem tokens by the time we resolve the last Precursor trigger on Swap, letting us replay Replication Technique that many times for 2^^^^^14 copies of Possibility Storm to trigger on the last instant in our hand.
Alternate 15-card can break computation by copying Bishop before playing Emergency Powers, allowing it to hack multiple Bishops before drawing Artificial Evolution the last time.
Audacious Swap, Casualty on Innkeeper into Replication Technique, go to 4 Precursors, 10 regular. Resolve Precursor trigger, 14 Swaps.
Flip Innkeeper, Epitaph Golem, and Possibility Storm. Use Epitaph to put Replication back and copy Storm and golems, gaining enough life to keep activating Epitaph Golem. Use Replication 10 more times, reach 22 Possibility Storms.
I think the right number of layers is three, so the sixth card gets us to 2^^^24 Possibility Storms and then that becomes our input for the final card. So I think we end up with BB{2}(2^^^24)?
Edit: Wait, there's four other instants in our deck. Two of them are the ones we'll play to trigger Possibility Storm, but the other two will get in the way of replaying Audacious Swap after we put it on the bottom, blocking two of the triggers. One of them is Soulblast, but that's fine, we can choose not to play it and let it go back on the bottom. So BB{2}(2^^^22), then.
Lotus, Channel, Healing Technique getting both back. Replay Lotus for RRR. 21 mana.
Precursor, Audacious Swap. Use the golem tokens for Casualty to play Replication Technique, reaching 4 Precursors and 8 vanillas. Resolve the Precursor trigger on Swap, 12 Swaps on the stack.
Flip Soldevi Digger, put Replication Technique back (10 mana). Flip Possibility Storm and Replication, copying the golems (4 triggers, making 60 Precursors) and PS. Put back Lotus and Replication (6 mana). Flip Lotus, then hold onto the original while putting back and replaying Replication 8 times. Copy Lotus 5 times and PS 11 times, then put back Replication again, finishing with 3 mana and 13 PS.
Play Channel, getting 13 PS triggers. The first one hits Twist Allegiance, play it to get it out of our library. (This moves the creatures to the opponent's board but we can fix that later.) The second hits Replication, the third hits Channel, the others play Replication 10 more times. We'll need to put back Replication 11 more times, as well as putting back Audacious Swap, and Twist Allegiance + Channel which are on top of it, so we'll copy Lotus 10 times and PS 12 times. We'll put back Swap/Twist/Channel right before the last PS trigger, so Swap ends up as the top sorcery in our deck.
Finish this round with 25 PS, letting us play Artificial Evolution for 2^^^27, then Arcbond for BB{2}(2^^^27).
Edit: Jack-o'-Lantern looks even better. Sacrifice it to draw a card at the start, then we have to exile it to get our red mana, so we can't draw cards later on when it would cause problems. This lets us get a third instant in hand after setting up, so we can reach 2^^^2^^^X. (This version can go back to Epitaph Golem, although it's less effective here than with Innkeeper, since we have to keep copying Black Lotus.)
I think it plays out like the first step of the Healing Technique version, getting 13 Possibility Storms after the first round. One whiffs, so that looks like BB{2}(2^^^2^^^14).
Edit (10/17):
Vessel of Endless Rest is interesting. It allows the blue/red mana generation card to also be the one that returns spells to the library, but actually using it tends to be inconvenient, since its ETB trigger needs to happen after the resolution of the spell it wants to tuck. So making Vessels with a copy of a spell won't let us loop the same spell, and we want to be able to have the original castings of copy spells get Precursor triggers. That means needing either a second copy spell, or something like Astral Dragon / Phyrexian Metamorph.
For those reasons, I don't see a way to use this for 14-card computations (at least, ones that use Dralnu's Crusade) and I haven't found anything it can do for 16-card that would be better than Jack-o'-Lantern drawing an additional card.
However, I believe it can get 15-card to 8 computations. Oddly enough, the best support I could find was a second copy of Replication Technique, alternating between the two.
Precursor, Replication Technique, use Demonstrate to make another Vessel. 2 Precursors, 6 regulars, 10 mana.
Replication Technique, use Demonstrate to make another Vessel and put back Technique #1. 4>8 Precursors, 15>38 regulars, 6 mana.
Audacious Swap, get 8 Precursor triggers, use Casualty to tuck Vessel and flip Technique. Use Demonstrate on a Precursor, then resolve 8 Precursor triggers to reach 2,304 Precursors and 28,418 regulars.
Resolve the first Precursor-Swap trigger for 30,721 Swaps. Flip Vessel, tucking Technique. Flip Bishop of Wings, Dralnu's Crusade, and Artificial Evolution, hacking Crusade from Zombie>Golem. Flip Technique, making two Vessels to tuck Technique and Artificial Evolution. Flip Artificial Evolution, hacking Crusade from Goblin>Human, so all clean Bishops are now Golems. Then start flipping Technique, putting Vessel back each time and exponentiating golems, including the Bishops.
Eventually, make more copies of Dralnu's Crusade, and flip Coat of Arms, Artificial Evolution, Arcbond, and Comeuppance to finish building the board. (Each replay of Artificial Evolution also requires burning a Replication Technique to put it back.) At the end, and after tucking the original Precursor, play Twist Allegiance to rotate the board, flip the original Precursor, and play Soulblast to run the computation. Save three Swap triggers for afterward, to replay Technique to put Twist Allegiance back, replay Twist Allegiance, and replay Technique to build a fresh set of golems we own based on the computation. Then go to the remaining seven Precursor-Swap triggers to do it seven more times.
The first Replication Technique can make a second Precursor and use the main Vessel to tuck itself back, then playing Audacious Swap gets two Precursor-Swap triggers. Using Casualty to tuck Vessel lets us replay Technique to get to 12 Precursors and 45 regular golems, so the first Precursor-Swap trigger gives 56 Swaps.
Building a board like this is pretty clunky, especially making new Dralnu's Crusades (copying one and editing it twice costs 5 Swaps), but using the methods here, I think it's enough to set up and run a computation that outputs three or four Knuth arrows. (It helps that having multiple Precursors means we can cast Artificial Evolution once to hack all of our creatures multiple times.) From there, we can replay Scrambleverse to retake all but one of the creatures, then replay Soulblast to kill all those creatures and get a bunch of Bishop triggers that fill our board with a corresponding number of golems we own. Then we can resolve the second Precursor-Swap trigger for a that many Swaps and use them to build a proper computation, and finish with Soulblast.
15-card can add Possibility Storm, which I think gets it to about BB{2}(2^^12). 16-card could add Sakashima's Will to get many computations per Precursor-Swap trigger and reach BB{3}, but honestly, if these actually work, I'm not sure the writeup even needs a 16-card strategy. BB{2}(2^^12) is already a big step over the Eiganjo Uprising 16-card deck.
Edit: I miscounted - it's 12 Precursors, 42 regular golems, 53 Swaps from the trigger. That should still be enough for three arrows, but I think it ends up just below the number we'd need for four.
Those are some great ideas to get the computations!
For 14 cards using a very small computations to get the numbers needed for the big computation should be doable.
I think skipping the small computation and casting Replication Technique 21 times instead would get us to ~2^^24 golems, so that would be our lower bound of what we can reach. IIRC that is a bit below the input sizes where we know how to implement a universal turing machine. So a computation that gets more layers would be preferred.
The special creature type dying will also create a creature for the heartbeat clock, delaying the heartbeat one turn,
For the heartbeat clock, I think we can also use one creature type,
We can't have both of those, or the heartbeat will accumulate more and more creatures.
The constructions are also probably not optimized for the situation we actually get here, where we can create and hack Bishops fairly easily but the Dralnu's crusades are very expensive. I plan to look into those small computations a bit more.
Those are some great ideas to get the computations!
For 14 cards using a very small computations to get the numbers needed for the big computation should be doable.
I think skipping the small computation and casting Replication Technique 21 times instead would get us to ~2^^24 golems, so that would be our lower bound of what we can reach. IIRC that is a bit below the input sizes where we know how to implement a universal turing machine. So a computation that gets more layers would be preferred.
The special creature type dying will also create a creature for the heartbeat clock, delaying the heartbeat one turn,
For the heartbeat clock, I think we can also use one creature type,
We can't have both of those, or the heartbeat will accumulate more and more creatures.
The constructions are also probably not optimized for the situation we actually get here, where we can create and hack Bishops fairly easily but the Dralnu's crusades are very expensive. I plan to look into those small computations a bit more.
The problem is that we can't actually use Replication Technique for tetrations during the Precursor-Swap triggers. The main spell always needs to target Vessel to recur Technique, so we can't get Precursor triggers from it. (The only exception is when we cast it off Casualty, since Vessel is back in our library.) But hopefully we can do better anyway.
Regarding the heartbeat clocks, would adding Dralnu's Crusades to them fix the issue? That would get even more expensive but I think it would at least be enough for two arrows. I think we'd have enough spare Swaps in that case to run the small computation twice, hacking the output creatures from the first one into the input type, to reach 2^^2^^N and beat that target anyway. But yeah, hopefully there's an alternative that leans less on Crusade and more on Bishops.
The problem is that we can't actually use Replication Technique for tetrations during the Precursor-Swap triggers. The main spell always needs to target Vessel to recur Technique, so we can't get Precursor triggers from it. (The only exception is when we cast it off Casualty, since Vessel is back in our library.) But hopefully we can do better anyway.
Ugh, right, I didn't realize that.
So each Bishop copy costs 1 cast, each Dralnu's Crusade costs 1 cast + 2 per hack, so usually 5, but we can save a bit by making use of goblin/zombies. We need additional casts to hack all creatures (3?), get Coat of Arms (1), Arcbond (1?), Comeuppance (1), Soulblast (1), and Scrambleverse to give the setup to the opponent and get the output back (1+2). And all from only 53 casts (and I think there's a card in hand we could use?). That's pretty tight.
Regarding the heartbeat clocks, would adding Dralnu's Crusades to them fix the issue? That would get even more expensive but I think it would at least be enough for two arrows. I think we'd have enough spare Swaps in that case to run the small computation twice, hacking the output creatures from the first one into the input type, to reach 2^^2^^N and beat that target anyway. But yeah, hopefully there's an alternative that leans less on Crusade and more on Bishops.
Adding a Dralnu's Crusade for the Heartbeat should work at least for the exponentiation part. I don't fully understand the tetration and additional layer parts yet.
Starting the computation again with bigger vanilla input might be helpful. After those small computations we then get a lot more casts. Hopefully we can manage to set up the universal turing machine with a fixed number of those casts and have the majority left over. Then we can use the remaining casts to get repeated Busy Beaver computation with the same trick, and wouldn't need to go to 15 cards for that ^^'
The problem is that we can't actually use Replication Technique for tetrations during the Precursor-Swap triggers. The main spell always needs to target Vessel to recur Technique, so we can't get Precursor triggers from it. (The only exception is when we cast it off Casualty, since Vessel is back in our library.) But hopefully we can do better anyway.
Ugh, right, I didn't realize that.
So each Bishop copy costs 1 cast, each Dralnu's Crusade costs 1 cast + 2 per hack, so usually 5, but we can save a bit by making use of goblin/zombies. We need additional casts to hack all creatures (3?), get Coat of Arms (1), Arcbond (1?), Comeuppance (1), Soulblast (1), and Scrambleverse to give the setup to the opponent and get the output back (1+2). And all from only 53 casts (and I think there's a card in hand we could use?). That's pretty tight.
Regarding the heartbeat clocks, would adding Dralnu's Crusades to them fix the issue? That would get even more expensive but I think it would at least be enough for two arrows. I think we'd have enough spare Swaps in that case to run the small computation twice, hacking the output creatures from the first one into the input type, to reach 2^^2^^N and beat that target anyway. But yeah, hopefully there's an alternative that leans less on Crusade and more on Bishops.
Adding a Dralnu's Crusade for the Heartbeat should work at least for the exponentiation part. I don't fully understand the tetration and additional layer parts yet.
Starting the computation again with bigger vanilla input might be helpful. After those small computations we then get a lot more casts. Hopefully we can manage to set up the universal turing machine with a fixed number of those casts and have the majority left over. Then we can use the remaining casts to get repeated Busy Beaver computation with the same trick, and wouldn't need to go to 15 cards for that ^^'
One card in hand, yeah. And in addition to replaying Scrambleverse after the computation, we also need to replay Soulblast to kill the tokens the opponent made and replace them with tokens we make, in order for Audacious Swap to see us as the owner.
If we set a Crusade to Human>Golem in addition to the Goblin>Zombie one, that lets us use a single Artificial Evolution to hack all the creatures at once. If we want to run the computation twice on the first Precursor-Swap trigger, that takes another 7 Swaps: 1 to make another Bishop (which we'll keep on our side and hack to make a token off its own death, to feed the following Soulblast) and 6 for additional plays of Arcbond/Soulblast/Evolution.
So if we do two computations and two Crusades, we need:
5 to play Vessel/Bishop/Crusade/Coat/Comeuppance
3 to play Arcbond/Scrambleverse/Soulblast for the first computation
7 for the second computation
4 to collect the output
At least 3 for additional cloning/hacking
We start with 53+1 casts. With only one Crusade, that should leave 32 for making additional Bishops, and any Precursors to use as input. With two Crusades, that's 29 left. (If the second early computation isn't worth it, we can add another 7.)
I'm extremely intrigued by all of the work that's been done on this and the related 60-card thread. Still trying to get my head wrapped around the math, but CaptainMarcia's writeup of 1-16 card decks has helped a ton for me.
I had a question about how the layering interacts when creature power scales to the same degree as creature count. In particular I was curious if using a card like Craterhoof Behemoth is efficient at all and what happens to the layering when all of the creature tokens in some of these strategies compound their damage dealing capabilities.
In particular, I was looking at the Precursor Golem/Smiting Helix/Slaughter 7-card deck. If each of the 2^^^^(2^^(227 trillion)) Golem tokens have 2^^^^(2^^(227 trillion)) power what is the resulting damage output?
At first I thought this would add a layer for an extra card since each golem is doing as much damage as the number of golems. Still not coming close to the 8-card deck in the writeup but I'm just trying to understand how the math works. Looking at it further, though, it seems like the damage would just be (2^^^^(2^^(227 trillion)))^2. What is required to actually add a layer? How do the stages have to interact?
I may be using a bunch of these terms incorrectly compared to how you all understand them, so please correct me if that's the case. Thanks for any help you can provide in my understanding.
~
As a side note, I found a more mana-efficient option to reach the same result in the 4-card deck. Timbermare is another 5-power creature with Haste. However, with only 4 CMC, it demonstrates that perhaps there is design room for a future card with more power and Haste at 5 CMC with 1 green (one can only hope).
I'm extremely intrigued by all of the work that's been done on this and the related 60-card thread. Still trying to get my head wrapped around the math, but CaptainMarcia's writeup of 1-16 card decks has helped a ton for me.
I had a question about how the layering interacts when creature power scales to the same degree as creature count. In particular I was curious if using a card like Craterhoof Behemoth is efficient at all and what happens to the layering when all of the creature tokens in some of these strategies compound their damage dealing capabilities.
In particular, I was looking at the Precursor Golem/Smiting Helix/Slaughter 7-card deck. If each of the 2^^^^(2^^(227 trillion)) Golem tokens have 2^^^^(2^^(227 trillion)) power what is the resulting damage output?
At first I thought this would add a layer for an extra card since each golem is doing as much damage as the number of golems. Still not coming close to the 8-card deck in the writeup but I'm just trying to understand how the math works. Looking at it further, though, it seems like the damage would just be (2^^^^(2^^(227 trillion)))^2. What is required to actually add a layer? How do the stages have to interact?
I may be using a bunch of these terms incorrectly compared to how you all understand them, so please correct me if that's the case. Thanks for any help you can provide in my understanding.
~
As a side note, I found a more mana-efficient option to reach the same result in the 4-card deck. Timbermare is another 5-power creature with Haste. However, with only 4 CMC, it demonstrates that perhaps there is design room for a future card with more power and Haste at 5 CMC with 1 green (one can only hope).
In that case, (2^^^^(2^^(227 trillion)))^2 is technically closer to accurate, but we'd tend to just round it down to 2^^^^(2^^(227 trillion)) since we're already rounding down even larger operations. For example, the difference between 2^^(227 trillion) and 2^^(227,000,000,000,001) is one tetration, which is much bigger than squaring the output, and that in turn feeds into the number of hexations from the 2^^^^N step. And 227 trillion is rounding down from 227,479,118,689,477, so the actual rounding is even bigger than that.
Honestly, I've also found it difficult at times to keep track of the exact number of layers, but the general idea is that adding a layer requires repeating everything before it an increasing number of times, and being able to go all the way down to the bottom before going back up.
Like, in the Saw deck, saw you have two 33/33 Astral Dragons. You split one into 17/17 Astral Dragons, and then one of those into 9/9s, and then one of those into 5/5s, and then you go through the rest of the 9/9s, and each time, your output is increasing. Then repeat all that with the rest of the 17/17s, and you can finish all of that before doing back to the other 33/33.
That's part of a stage, where the layers are just the same thing with different numbers attached, but the same logic applies to layers made of different things. You take take something from a higher layer, trade it for a large number of things in a lower layer, then trade each of those things for increasing numbers of things in the layer below that, and the number of layers is how many of those you can stack together.
Honestly, I've also found it difficult at times to keep track of the exact number of layers, but the general idea is that adding a layer requires repeating everything before it an increasing number of times, and being able to go all the way down to the bottom before going back up.
Like, in the Saw deck, saw you have two 33/33 Astral Dragons. You split one into 17/17 Astral Dragons, and then one of those into 9/9s, and then one of those into 5/5s, and then you go through the rest of the 9/9s, and each time, your output is increasing. Then repeat all that with the rest of the 17/17s, and you can finish all of that before doing back to the other 33/33.
That's part of a stage, where the layers are just the same thing with different numbers attached, but the same logic applies to layers made of different things. You take take something from a higher layer, trade it for a large number of things in a lower layer, then trade each of those things for increasing numbers of things in the layer below that, and the number of layers is how many of those you can stack together.
Thanks again for clarifying; that makes a bit more sense. This early post from Deedlit11 also helped me understand more about how the layers work.
So even the 5-card deck is using the same principal. We could describe it using one layer:
The amount of power gained by Devilish Valet for each of 38 Sparkcaster castings is accelerating. Therefore, Devilish Valet's power grows by exponentiation.
Using the same language for the 6-card deck:
The delta of the token multiplier gained by the set of Doubling Season tokens is accelerating. Therefore, the multiplier grows by exponentiation.
The number of Doubling Season tokens created for each of the 32 Astral Dragons is accelerating. Therefore, the number of Doubling Seasons grows by tetration.
The intralayer resource exchange is: We trade 1 Astral Dragon triggered ability for more than one token multiplier event.
...and for the 7-card deck, I would assume it is something like:
The number of Smiting Helix copies gained by using Cogwork Assembler on Precursor Golem is accelerating. Therefore, the Smiting Helix copies grow by exponentiation.
The number of Slaughter castings gained by the life gain of Smiting Helix is accelerating. Therefore, the Slaughter castings grow by tetration.
The number of Slaughter copies gained by using Cogwork Assembler on Precursor Golem is accelerating. Therefore, the Slaughter copies grow by pentation.
The amount of mana gained by using Salvaging Stations is accelerating. Therefore, the amount of mana grows by hexation.
The intralayer resource exchanges are: We trade 7 mana for another Precursor Golem which fuels more than one Smiting helix copy event, and more than one Slaughter copy event.
However, what I don't understand is what causes the buildup of layers for some of these effects.
It seems to me like the resources found in each layer don't always increase by its immediately senior layer like I inferred from Deedlit11's explanation of "Also, you have to ability to decrease X2 by 1 (or a fixed amount) to increase X1 by more; the ability to decrease X3 by 1 to increase X2 by X1 or thereabouts, the ability to decrease X4 by 1 to increase X3 by X1 or thereabouts, and so on up to Xn."
Using Deedlit's language and the list in your writeup we have:
We can decrease X4 by 1 to increase X1 and X3 by a bunch (since we will get a bunch of mana with which to create more Precursor Golems), but I don't see how we get more Slaughter buybacks from a later layer. Certainly we get more Slaughter buybacks from the increase in X1, but my intuition tells me that this is a multiplicative effect (i.e. for each copy of Smiting Helix we get 0.75 Slaughter buybacks).
I don't see where we get more life from a senior layer at the cost of X3 or X4 resources. I'm sure I'm missing something but where does X2 become a layer?
Honestly, I've also found it difficult at times to keep track of the exact number of layers, but the general idea is that adding a layer requires repeating everything before it an increasing number of times, and being able to go all the way down to the bottom before going back up.
Like, in the Saw deck, saw you have two 33/33 Astral Dragons. You split one into 17/17 Astral Dragons, and then one of those into 9/9s, and then one of those into 5/5s, and then you go through the rest of the 9/9s, and each time, your output is increasing. Then repeat all that with the rest of the 17/17s, and you can finish all of that before doing back to the other 33/33.
That's part of a stage, where the layers are just the same thing with different numbers attached, but the same logic applies to layers made of different things. You take take something from a higher layer, trade it for a large number of things in a lower layer, then trade each of those things for increasing numbers of things in the layer below that, and the number of layers is how many of those you can stack together.
Thanks again for clarifying; that makes a bit more sense. This early post from Deedlit11 also helped me understand more about how the layers work.
So even the 5-card deck is using the same principal. We could describe it using one layer:
The amount of power gained by Devilish Valet for each of 38 Sparkcaster castings is accelerating. Therefore, Devilish Valet's power grows by exponentiation.
Using the same language for the 6-card deck:
The delta of the token multiplier gained by the set of Doubling Season tokens is accelerating. Therefore, the multiplier grows by exponentiation.
The number of Doubling Season tokens created for each of the 32 Astral Dragons is accelerating. Therefore, the number of Doubling Seasons grows by tetration.
The intralayer resource exchange is: We trade 1 Astral Dragon triggered ability for more than one token multiplier event.
...and for the 7-card deck, I would assume it is something like:
The number of Smiting Helix copies gained by using Cogwork Assembler on Precursor Golem is accelerating. Therefore, the Smiting Helix copies grow by exponentiation.
The number of Slaughter castings gained by the life gain of Smiting Helix is accelerating. Therefore, the Slaughter castings grow by tetration.
The number of Slaughter copies gained by using Cogwork Assembler on Precursor Golem is accelerating. Therefore, the Slaughter copies grow by pentation.
The amount of mana gained by using Salvaging Stations is accelerating. Therefore, the amount of mana grows by hexation.
The intralayer resource exchanges are: We trade 7 mana for another Precursor Golem which fuels more than one Smiting helix copy event, and more than one Slaughter copy event.
However, what I don't understand is what causes the buildup of layers for some of these effects.
It seems to me like the resources found in each layer don't always increase by its immediately senior layer like I inferred from Deedlit11's explanation of "Also, you have to ability to decrease X2 by 1 (or a fixed amount) to increase X1 by more; the ability to decrease X3 by 1 to increase X2 by X1 or thereabouts, the ability to decrease X4 by 1 to increase X3 by X1 or thereabouts, and so on up to Xn."
Using Deedlit's language and the list in your writeup we have:
We can decrease X4 by 1 to increase X1 and X3 by a bunch (since we will get a bunch of mana with which to create more Precursor Golems), but I don't see how we get more Slaughter buybacks from a later layer. Certainly we get more Slaughter buybacks from the increase in X1, but my intuition tells me that this is a multiplicative effect (i.e. for each copy of Smiting Helix we get 0.75 Slaughter buybacks).
I don't see where we get more life from a senior layer at the cost of X3 or X4 resources. I'm sure I'm missing something but where does X2 become a layer?
I think you might have the order backwards. When we flashback Helix, we get a fixed number of Precursor-Helix triggers. We don't get more of those, but the number of Helix copies we get from cashing in a Precursor-Helix trigger increases each time, as our number of golems increases. (If we could get more life or Helixes from a later layer, we'd go infinite.) Then, each time we spend that life to buyback Slaughter, we get an increasingly large batch of Precursor-Slaughter triggers, and each time we cash in a Precursor-Slaughter trigger we get an increasing number of Slaughter copies. Then after looping that as many times as we have the life to pay for, that output determines how many Helix copies we get from the next Precursor-Helix trigger.
The result is that the four layers are Precursor-Helix triggers, Helix copies, Precursor-Slaughter triggers, and Slaughter copies.
I think you might have the order backwards. When we flashback Helix, we get a fixed number of Precursor-Helix triggers. We don't get more of those, but the number of Helix copies we get from cashing in a Precursor-Helix trigger increases each time, as our number of golems increases. (If we could get more life or Helixes from a later layer, we'd go infinite.) Then, each time we spend that life to buyback Slaughter, we get an increasingly large batch of Precursor-Slaughter triggers, and each time we cash in a Precursor-Slaughter trigger we get an increasing number of Slaughter copies. Then after looping that as many times as we have the life to pay for, that output determines how many Helix copies we get from the next Precursor-Helix trigger.
The result is that the four layers are Precursor-Helix triggers, Helix copies, Precursor-Slaughter triggers, and Slaughter copies.
Oh I see. I think I misunderstood the meaning of your numbered list in the writeup:
For each Precursor when we play Helix, get a Precursor-Helix trigger
For each golem when we resolve a Precursor-Helix trigger, gain life that determines the number of times we cast Slaughter
For each Precursor when we play Slaughter, get a Precursor-Slaughter trigger
For each golem when we resolve a Precursor-Slaughter trigger, get a round of Salvaging Station untaps
I thought this was describing the layers, but it was just describing the procedure. Those layers make more sense.
I think you might have the order backwards. When we flashback Helix, we get a fixed number of Precursor-Helix triggers. We don't get more of those, but the number of Helix copies we get from cashing in a Precursor-Helix trigger increases each time, as our number of golems increases. (If we could get more life or Helixes from a later layer, we'd go infinite.) Then, each time we spend that life to buyback Slaughter, we get an increasingly large batch of Precursor-Slaughter triggers, and each time we cash in a Precursor-Slaughter trigger we get an increasing number of Slaughter copies. Then after looping that as many times as we have the life to pay for, that output determines how many Helix copies we get from the next Precursor-Helix trigger.
The result is that the four layers are Precursor-Helix triggers, Helix copies, Precursor-Slaughter triggers, and Slaughter copies.
Oh I see. I think I misunderstood the meaning of your numbered list in the writeup:
For each Precursor when we play Helix, get a Precursor-Helix trigger
For each golem when we resolve a Precursor-Helix trigger, gain life that determines the number of times we cast Slaughter
For each Precursor when we play Slaughter, get a Precursor-Slaughter trigger
For each golem when we resolve a Precursor-Slaughter trigger, get a round of Salvaging Station untaps
I thought this was describing the layers, but it was just describing the procedure. Those layers make more sense.
Thanks for helping me understand.
Those four steps are the layers: Precursor-Helix triggers, Helix copies, Precursor-Slaughter triggers, Slaughter copies. (Helix copies translate at a fixed rate to Slaughter casts, but the number of triggers varies.)
Those four steps are the layers: Precursor-Helix triggers, Helix copies, Precursor-Slaughter triggers, Slaughter copies. (Helix copies translate at a fixed rate to Slaughter casts, but the number of triggers varies.)
Right. The life gain in step 2 is from the Helix copies, and the untaps in step 4 are from the Slaughter copies. I failed to notice that the triggers and spell copies are different layers since they aren't one to one, but compound based on the growing number of golem tokens.
5 cards: Black Lotus, Channel, Dryad's Revival retrieving Black Lotus, use it for green mana, flashback Dryad's Revival retrieving Black Lotus, use it for red mana, Nylea's Colossus, Rionya, Fire Dancer. Go to combat and make 4 hasty copies of Nylea's Colossus, producing 20 of its triggers, all targeting one of the copies. Attack for 6*2^20+18=6291474 damage.
For 2 cards, Devilish Valet and Displacer Beast is better. All three default dungeons have unavoidable limiting effects: "Each player loses 1 life" in Tomb of Annihilation, and card draw at the end of Lost Mine of Phandelver and Dungeon of the Mad Mage.
Start by casting Displacer Beast four times to venture through Lost Mine of Phandelver, getting a Goblin token at Goblin Lair, proceeding from there into Storeroom, and drawing Devilish Valet at the end. Cast Devilish Valet. Then go through Tomb of Annihilation 19 times. At Oubliette, sacrifice the Goblin token the first time, and sacrifice The Atropal on later times. This triggers Devilish Valet 19*4=76 times. Finally, venture six steps into Dungeon of the Mad Mage, going through Muiral's Graveyard, for 8 more triggers of DV, making its power 2^84, and attack.
it would be a lot better for that combo if the deck was 58 wastes too.
I guess for 5 cards, Acererak the Archlich beats out Sparkcaster in that case.
mulligan to 5
lotus, show and tell, omniscience, valet, acerak -> complete lost mine 55 times, then 6 steps into Dungeon of the Mad Mage, for 2^(55*4+55+6+2)= 2^283 (plus some more from the storeroom's +1/+1 counters.)
That can be improved slightly by replacing Omniscience with Aluren, which allows casting at instant speed. This means we can venture through Lost Mine of Phandelver one extra time at the beginning, drawing Devilish Valet, while leaving the token-creating trigger from Goblin Lair on the stack, for one extra trigger of Devilish Valet. It also means we can leave Devilish Valet's triggers on the stack and resolve all the Storeroom triggers first. This increases the final power to 56*2^284.
Another benefit of Aluren is that it costs only 4 mana, but I couldn't find any way to get a big advantage from that; the best I have is starting with Black Lotus and Sunscorched Desert for 1 extra damage. (Crystal Dragon looked promising, but there's no way to get the required mana to cast it from exile without giving up multiple Goblin tokens.)
One other possible point of variation is defining exactly what it means to be limited to N cards. That could mean starting with a deck of only N cards and having an otherwise-empty library, or assuming that there's other cards filling space there and we just can't use them. This affects the power of cards like Ardent Dustspeaker and Junktroller that put cards on the bottom of the library.
(emphasis added)
Private Mod Note
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Unfortunately, this can't be made into a 7-card solution by adding Black Lotus, Channel, and Lich's Mirror, because the token-doubling effects persist through Lich's Mirror's effect.
Black Lotus, Channel, Precursor Golem, Cogwork Assembler, activate once targeting Precursor Golem, Academy Manufactor, Confront the Unknown... but that goes infinite by using Cogwork Assembler to copy Clues/Foods/Treasures.
Adrix and Nev, Twincasters, Ratadrabik of Urborg, Cabaretti Confluence... but the haste isn't copiable. If it were, we could make 2 copies of A&N, make 8 copies of A&N, and then make 2048 copies of Ratadrabik, and get 20490 Ratadrabik triggers on A&N and a bunch on Ratadrabik, and deal >2^^20492 damage.
Instead, Kindred Charge is the best I could find, dealing >2^^8 damage.
Running the function for (55,25,0) and making that same adjustment, looks like we get 227,479,118,689,477 (227 trillion) Precursors.
I've also been thinking more about adding layers on top of Busy Beaver. I'm not entirely sure this works, but if so, it looks like an improvement over the Learn from the Past version:
2 Channel
3 Nature's Spiral
4 Jadzi, Oracle of Arcavios
5 Cadric, Soul Kindler
6 Elixir of Immortality
7 Precursor Golem
8 Replication Technique
9 Fractured Identity
11 Bishop of Wings
12 Artificial Evolution
13 Arcbond
14 Comeuppance
15 Dralnu's Crusade
16 Goblin Boom Keg
17 Kaervek's Spite
Lotus, Channel, Nature's Spiral, replay Lotus for UUU. 21 mana.
Jadzi, Replication Technique, copy it, get two Jadzi triggers. Use the first to play Cadric, then resolve the copy for a Jadzi token. 7 mana.
Chain the second Jadzi trigger into Arcbond, Artificial Evolution, and Elixir, activate Elixir. 7 mana.
From the other trigger on Artificial Evolution, chain into Channel, chain one trigger into Black Lotus and sacrifice it, chain the other into Nature's Spiral targeting Lotus, chain into Precursor Golem and Elixir. 5 mana.
Return Lotus to hand, resolve down to Artificial Evolution, then activate Elixir (8 mana). Chain the second Arcbond trigger into Channel, then Artificial Evolution, targeting a golem token and getting a Precursor trigger....
End up with a bunch of life and with Lotus back in hand, then discard it to bounce Jadzi and start again with Journey to the Oracle. From here, start repeatedly casting Replication Technique to make more tokens, copying it with Precursor Golem and adjusting creature types to optimize it. Do another round with Fractured Identity, then Journey again, then Comeuppance.
After resolving Comeuppance, play Journey one more time, and this time start running the computations. Fractured Identity transfers permanents to the opponent, then lets us take them back after the computation. Precursor helps Fractured Identity target large numbers of permanents at once, and also means large numbers of generic tokens created by the computation provide more Jadzi triggers when playing targeting spells, letting us replay Replication Technique more times.
Unfortunately, the fact that we need to keep resolving Replication Technique, Fractured Identity, Artificial Evolution, and Arcbond between computations limits how many layers we can get outside of the computation. We can get four layers from Channel, Nature's Spiral, Journey, and Comeuppance, and I think another two from Arcbond and Precursor-Arcbond, resolving one Precursor trigger between each computation, for a total of six.
After a computation, we can replay the original Cadric from our library and make hasted tokens for each Cadric token we took from the opponent. At the end, we'll use these as the final damage source, since we'll no longer be able to replay the original Jadzi.
Edit: Saw might be usable, but it would have to focus on a creature that doesn't have a creature type to keep Coat of Arms from interfering, with a way to move it between our field and the opponent's and to keep it from getting killed by Arcbond.
Edit: Removed a part of the early strategy that tried to get Jadzi back into the chain without discarding her.
Edit: Jadzi gives the opponent the ability to interrupt a computation since they can discard a card to bounce any copies of her they have. Argh.
2 Channel
3 Bucknard's Everfull Purse
4 Precursor Golem
5 Audacious Swap
6 Mnemonic Deluge
7 Replication Technique
8 Elixir of Immortality
9 Twist Allegiance
11 Bishop of Wings
12 Artificial Evolution
13 Arcbond
14 Comeuppance
15 Dralnu's Crusade
16 Soulblast
Play Precursor Golem, then Audacious Swap targeting Precursor, getting a Precursor trigger. Sacrifice a golem to Casualty, point the Casualty copy at Purse.
Resolve the Casualty copy, shuffle Purse back and flip Replication Technique. Point it and the Demonstrate copy at Precursor. Resolve the Demonstrate copy to get to two Precursors and three regular golems, then resolve the Precursor copy and the original to get to four Precursors and ten regular golems. Resolve the Precursor trigger on Swap, getting a Swap targeting each of them.
Resolve a Swap on a golem, flip Elixir, activate it. (12 colorless, 3 treasures.) Resolve a second, flip Elixir again. Resolve a third, flip Replication Technique, use the Demonstrate copy to make another Elixir and the original to target a golem, getting four Precursor triggers to double the Precursor count four times to 64. Then use the extra Elixir to reshuffle.
Replay Replication Technique 11 times, each time making a new Elixir and tetrating the Precursors, to reach 2^^14. Then play Mnemonic Deluge, exiling Swap to cast three copies of it. Each one gets 2^^14 Precursor triggers, as well as a Casualty copy, sacrificing two regular golems and the original Precursor.
Using the first Casualty, play Twist Allegiance to move all the creatures to the opponent's field. Then sacrifice the extra Elixir to shuffle Precursor and Replication Technique back into our library, use the second Casualty to replay Precursor, and use the third to replay Replication Technique. Use Demonstrate to replace the extra Elixir and use the main Replication on a golem token, triggering all the Precursors. Replication can only target the golems on our field, but it doubles our own board 2^^14 times, resulting both sides of the board now having about that many golems.
Resolve the first Precursor trigger on Audacious Swap, getting a Swap targeting all the golems on both sides. We can use these to finish building our board, starting by resolving Swaps on regular golems. Play Coat of Arms, Bishop of Wings, Arcbond, Comeuppance, and Dralnu's Crusade, then copy and hack our cards as needed with Replication Technique and Artificial Evolution. Dralnu's Crusade is less efficient to copy and hack, but when we do it, we'll want one of them set to turn our fresh Bishops into Golems for more efficient Precursor copy chains.
Once we run low on Swaps on the stack, replay Twist Allegiance to rotate the board again, then use an Elixir token to reshuffle it. At this point, we'll have one Precursor on our side after eliminating the other creatures from that batch, while the opponent will have all the creatures we made over the course of that round, three of which are Prcursors with Swaps still pointed at them. Use one Swap to play Soulblast, sacrificing our Precursor to run a computation, which we'll arrange to generate a large number of golem tokens, and to keep the two remaining Swap targets alive.
After the computation, resolve the next Swap for Twist Allegiance again, moving all those creatures back to our board. Resolve the last one for Replication Technique, getting it copied many times for all of our golems. Then resolve the next Precursor-Swap trigger, copying Swap for each of our new golems. The ones created during the computation are owned by the opponent, so we can't make use of targeting them with Swap, but the ones created by copying those tokens with Replication Technique are owned by us. Before the next computation, we'll need to use Twist Allegiance to move the creatures back to the opponent's board, then replay one of our creatures from our library to get something back to our side to sacrifice to Soulblast.
At this point we can repeat the process for each Precursor-Swap trigger, for over 2^^14 iterations. When we finish, take all the creatures with Twist Allegiance, give them haste, and swing.
For infinite mana strategies, with the Reef Worm one at 4, would 2-3 cards be the same as the current 5-6?
Edit: I think Possibility Storm works as a better method for getting multiple plays of Audacious Swap while keeping things sufficiently controlled.
2 Channel
3 Bucknard's Everfull Purse
4 Precursor Golem
5 Audacious Swap
6 Possibility Storm
7 Replication Technique
8 Elixir of Immortality
9 Twist Allegiance
11 Bishop of Wings
12 Artificial Evolution
13 Arcbond
14 Comeuppance
15 Dralnu's Crusade
16 Soulblast
Lotus, Channel, Purse. 17 colorless, 4 treasures.
Possibility Storm, Elixir. Possibility Storm puts Elixir on the bottom of the library and flips Precursor Golem.
Play Artificial Evolution, turn it into Audacious Swap. Use Casualty to play Replication Technique, go to 4 Precursors and 10 regular Golems. Resolve Precursor trigger, get 14 Swaps on the stack (counting the original). Use the first two to play Elixir (keeping it the second time), then the other 12 to play Replication Technique. The first Replication targets a golem, getting four Precursor triggers (making 60 new Precursors), and uses Demonstrate to copy Elixir. Replications 2-10 each target Possibility Storm and Elixir, then 11-12 target Possibility Storm twice, ending with 13 copies for a total of 14.
Play Arcbond, getting 14 Possibility Storm triggers (and putting Arcbond back into our library). Each time, flip into Audacious Swap for a new batch of Precursor triggers. Use those to build our board and start running computations.
2 Show and Tell
3 Omniscience
4 Thousand-Year Storm
5 Replication Technique
6 Second Harvest
7 Emergency Powers
8 Twist Allegiance
10 Bishop of Wings
11 Artificial Evolution
12 Arcbond
13 Comeuppance
14 Dralnu's Crusade
15 Chandra's Triumph
Replication Technique makes 3 TYS tokens, Second Harvest gets copied 8 times to reach 1,536 TYS tokens, Emergency Powers gets copied 4,611 times, for 2^^4611. Because there's no way for this deck to have creatures on both players' fields at the same time, we can use Chandra's Triumph to start the computation, partway through resolving an even number of Twist Allegiances.
This does allow us to cast Artificial Evolution on Bishop during the Emergency Powers chain, before playing the final Artificial Evolution, but I don't think there's a way to go infinite from that.
Edit: Improved start for the Possibility Storm deck:
Artificial Evolution, trigger 12 Possibility Storms. Use them to replay Audacious Swap > Replication Technique:
for each Possibility Storm trigger, for each Precursor trigger on Swap, for each golem targeted by Swap, for each Precursor trigger on Replication Technique, for each golem targeted by Replication Technique - 5 layers?
I think that gets us to 2^^^^^14 golem tokens by the time we resolve the last Precursor trigger on Swap, letting us replay Replication Technique that many times for 2^^^^^14 copies of Possibility Storm to trigger on the last instant in our hand.
Edit: I think we can just port the Audacious Swap strategy over to 15-card by dropping Possibility Storm, for BB(2^^^6).Edit: Wait, I think I'm miscounting layers.
Improved start for the Possibility Storm deck:
2 Channel
3 Prosperous Innkeeper
4 Precursor Golem
5 Audacious Swap
6 Replication Technique
7 Epitaph Golem
8 Possibility Storm
9 Twist Allegiance
11 Bishop of Wings
12 Artificial Evolution
13 Arcbond
14 Comeuppance
15 Dralnu's Crusade
16 Soulblast
Audacious Swap, Casualty on Innkeeper into Replication Technique, go to 4 Precursors, 10 regular. Resolve Precursor trigger, 14 Swaps.
Flip Innkeeper, Epitaph Golem, and Possibility Storm. Use Epitaph to put Replication back and copy Storm and golems, gaining enough life to keep activating Epitaph Golem. Use Replication 10 more times, reach 22 Possibility Storms.
I think the right number of layers is three, so the sixth card gets us to 2^^^24 Possibility Storms and then that becomes our input for the final card. So I think we end up with BB{2}(2^^^24)?
Edit: Wait, there's four other instants in our deck. Two of them are the ones we'll play to trigger Possibility Storm, but the other two will get in the way of replaying Audacious Swap after we put it on the bottom, blocking two of the triggers. One of them is Soulblast, but that's fine, we can choose not to play it and let it go back on the bottom. So BB{2}(2^^^22), then.
2 Channel
3 Healing Technique
4 Precursor Golem
5 Audacious Swap
6 Replication Technique
7 Soldevi Digger
8 Possibility Storm
9 Twist Allegiance
11 Bishop of Wings
12 Artificial Evolution
13 Arcbond
14 Comeuppance
15 Dralnu's Crusade
16 Soulblast
Precursor, Audacious Swap. Use the golem tokens for Casualty to play Replication Technique, reaching 4 Precursors and 8 vanillas. Resolve the Precursor trigger on Swap, 12 Swaps on the stack.
Flip Soldevi Digger, put Replication Technique back (10 mana). Flip Possibility Storm and Replication, copying the golems (4 triggers, making 60 Precursors) and PS. Put back Lotus and Replication (6 mana). Flip Lotus, then hold onto the original while putting back and replaying Replication 8 times. Copy Lotus 5 times and PS 11 times, then put back Replication again, finishing with 3 mana and 13 PS.
Play Channel, getting 13 PS triggers. The first one hits Twist Allegiance, play it to get it out of our library. (This moves the creatures to the opponent's board but we can fix that later.) The second hits Replication, the third hits Channel, the others play Replication 10 more times. We'll need to put back Replication 11 more times, as well as putting back Audacious Swap, and Twist Allegiance + Channel which are on top of it, so we'll copy Lotus 10 times and PS 12 times. We'll put back Swap/Twist/Channel right before the last PS trigger, so Swap ends up as the top sorcery in our deck.
Finish this round with 25 PS, letting us play Artificial Evolution for 2^^^27, then Arcbond for BB{2}(2^^^27).
I think it plays out like the first step of the Healing Technique version, getting 13 Possibility Storms after the first round. One whiffs, so that looks like BB{2}(2^^^2^^^14).
Edit (10/17):
Vessel of Endless Rest is interesting. It allows the blue/red mana generation card to also be the one that returns spells to the library, but actually using it tends to be inconvenient, since its ETB trigger needs to happen after the resolution of the spell it wants to tuck. So making Vessels with a copy of a spell won't let us loop the same spell, and we want to be able to have the original castings of copy spells get Precursor triggers. That means needing either a second copy spell, or something like Astral Dragon / Phyrexian Metamorph.
For those reasons, I don't see a way to use this for 14-card computations (at least, ones that use Dralnu's Crusade) and I haven't found anything it can do for 16-card that would be better than Jack-o'-Lantern drawing an additional card.
However, I believe it can get 15-card to 8 computations. Oddly enough, the best support I could find was a second copy of Replication Technique, alternating between the two.
2 Channel
3 Vessel of Endless Rest
4 Precursor Golem
5 Replication Technique
6 Replication Technique
7 Audacious Swap
8 Twist Allegiance
10 Bishop of Wings
11 Artificial Evolution
12 Arcbond
13 Comeuppance
14 Dralnu's Crusade
15 Soulblast
Lotus, Channel, Vessel, tap for U. 18 mana.
Precursor, Replication Technique, use Demonstrate to make another Vessel. 2 Precursors, 6 regulars, 10 mana.
Replication Technique, use Demonstrate to make another Vessel and put back Technique #1. 4>8 Precursors, 15>38 regulars, 6 mana.
Audacious Swap, get 8 Precursor triggers, use Casualty to tuck Vessel and flip Technique. Use Demonstrate on a Precursor, then resolve 8 Precursor triggers to reach 2,304 Precursors and 28,418 regulars.
Resolve the first Precursor-Swap trigger for 30,721 Swaps. Flip Vessel, tucking Technique. Flip Bishop of Wings, Dralnu's Crusade, and Artificial Evolution, hacking Crusade from Zombie>Golem. Flip Technique, making two Vessels to tuck Technique and Artificial Evolution. Flip Artificial Evolution, hacking Crusade from Goblin>Human, so all clean Bishops are now Golems. Then start flipping Technique, putting Vessel back each time and exponentiating golems, including the Bishops.
Eventually, make more copies of Dralnu's Crusade, and flip Coat of Arms, Artificial Evolution, Arcbond, and Comeuppance to finish building the board. (Each replay of Artificial Evolution also requires burning a Replication Technique to put it back.) At the end, and after tucking the original Precursor, play Twist Allegiance to rotate the board, flip the original Precursor, and play Soulblast to run the computation. Save three Swap triggers for afterward, to replay Technique to put Twist Allegiance back, replay Twist Allegiance, and replay Technique to build a fresh set of golems we own based on the computation. Then go to the remaining seven Precursor-Swap triggers to do it seven more times.
I think this reaches at least BB^8(2^^30,000).
I think it actually works to use Vessel of Endless Rest to loop a single Replication Technique, I underestimated how much we can get out of it. In addition, being able to replay Soulblast for damage makes the haste from Twist Allegiance unnecessary, allowing the more flexible Scrambleverse instead.
2 Channel
3 Vessel of Endless Rest
4 Precursor Golem
5 Replication Technique
6 Audacious Swap
7 Scrambleverse
9 Bishop of Wings
10 Artificial Evolution
11 Arcbond
12 Comeuppance
13 Dralnu's Crusade
14 Soulblast
The first Replication Technique can make a second Precursor and use the main Vessel to tuck itself back, then playing Audacious Swap gets two Precursor-Swap triggers. Using Casualty to tuck Vessel lets us replay Technique to get to 12 Precursors and 45 regular golems, so the first Precursor-Swap trigger gives 56 Swaps.
Building a board like this is pretty clunky, especially making new Dralnu's Crusades (copying one and editing it twice costs 5 Swaps), but using the methods here, I think it's enough to set up and run a computation that outputs three or four Knuth arrows. (It helps that having multiple Precursors means we can cast Artificial Evolution once to hack all of our creatures multiple times.) From there, we can replay Scrambleverse to retake all but one of the creatures, then replay Soulblast to kill all those creatures and get a bunch of Bishop triggers that fill our board with a corresponding number of golems we own. Then we can resolve the second Precursor-Swap trigger for a that many Swaps and use them to build a proper computation, and finish with Soulblast.
15-card can add Possibility Storm, which I think gets it to about BB{2}(2^^12). 16-card could add Sakashima's Will to get many computations per Precursor-Swap trigger and reach BB{3}, but honestly, if these actually work, I'm not sure the writeup even needs a 16-card strategy. BB{2}(2^^12) is already a big step over the Eiganjo Uprising 16-card deck.
Edit: I miscounted - it's 12 Precursors, 42 regular golems, 53 Swaps from the trigger. That should still be enough for three arrows, but I think it ends up just below the number we'd need for four.
For 14 cards using a very small computations to get the numbers needed for the big computation should be doable.
I think skipping the small computation and casting Replication Technique 21 times instead would get us to ~2^^24 golems, so that would be our lower bound of what we can reach. IIRC that is a bit below the input sizes where we know how to implement a universal turing machine. So a computation that gets more layers would be preferred.
I'm not sure the constructions Deedlit11 gave work out completely, specifically:
We can't have both of those, or the heartbeat will accumulate more and more creatures.
The constructions are also probably not optimized for the situation we actually get here, where we can create and hack Bishops fairly easily but the Dralnu's crusades are very expensive. I plan to look into those small computations a bit more.
Regarding the heartbeat clocks, would adding Dralnu's Crusades to them fix the issue? That would get even more expensive but I think it would at least be enough for two arrows. I think we'd have enough spare Swaps in that case to run the small computation twice, hacking the output creatures from the first one into the input type, to reach 2^^2^^N and beat that target anyway. But yeah, hopefully there's an alternative that leans less on Crusade and more on Bishops.
So each Bishop copy costs 1 cast, each Dralnu's Crusade costs 1 cast + 2 per hack, so usually 5, but we can save a bit by making use of goblin/zombies. We need additional casts to hack all creatures (3?), get Coat of Arms (1), Arcbond (1?), Comeuppance (1), Soulblast (1), and Scrambleverse to give the setup to the opponent and get the output back (1+2). And all from only 53 casts (and I think there's a card in hand we could use?). That's pretty tight.
Adding a Dralnu's Crusade for the Heartbeat should work at least for the exponentiation part. I don't fully understand the tetration and additional layer parts yet.
Starting the computation again with bigger vanilla input might be helpful. After those small computations we then get a lot more casts. Hopefully we can manage to set up the universal turing machine with a fixed number of those casts and have the majority left over. Then we can use the remaining casts to get repeated Busy Beaver computation with the same trick, and wouldn't need to go to 15 cards for that ^^'
If we set a Crusade to Human>Golem in addition to the Goblin>Zombie one, that lets us use a single Artificial Evolution to hack all the creatures at once. If we want to run the computation twice on the first Precursor-Swap trigger, that takes another 7 Swaps: 1 to make another Bishop (which we'll keep on our side and hack to make a token off its own death, to feed the following Soulblast) and 6 for additional plays of Arcbond/Soulblast/Evolution.
So if we do two computations and two Crusades, we need:
5 to play Vessel/Bishop/Crusade/Coat/Comeuppance
3 to play Arcbond/Scrambleverse/Soulblast for the first computation
7 for the second computation
4 to collect the output
At least 3 for additional cloning/hacking
We start with 53+1 casts. With only one Crusade, that should leave 32 for making additional Bishops, and any Precursors to use as input. With two Crusades, that's 29 left. (If the second early computation isn't worth it, we can add another 7.)
I'm extremely intrigued by all of the work that's been done on this and the related 60-card thread. Still trying to get my head wrapped around the math, but CaptainMarcia's writeup of 1-16 card decks has helped a ton for me.
I had a question about how the layering interacts when creature power scales to the same degree as creature count. In particular I was curious if using a card like Craterhoof Behemoth is efficient at all and what happens to the layering when all of the creature tokens in some of these strategies compound their damage dealing capabilities.
In particular, I was looking at the Precursor Golem/Smiting Helix/Slaughter 7-card deck. If each of the 2^^^^(2^^(227 trillion)) Golem tokens have 2^^^^(2^^(227 trillion)) power what is the resulting damage output?
At first I thought this would add a layer for an extra card since each golem is doing as much damage as the number of golems. Still not coming close to the 8-card deck in the writeup but I'm just trying to understand how the math works. Looking at it further, though, it seems like the damage would just be (2^^^^(2^^(227 trillion)))^2. What is required to actually add a layer? How do the stages have to interact?
I may be using a bunch of these terms incorrectly compared to how you all understand them, so please correct me if that's the case. Thanks for any help you can provide in my understanding.
~
As a side note, I found a more mana-efficient option to reach the same result in the 4-card deck. Timbermare is another 5-power creature with Haste. However, with only 4 CMC, it demonstrates that perhaps there is design room for a future card with more power and Haste at 5 CMC with 1 green (one can only hope).
Honestly, I've also found it difficult at times to keep track of the exact number of layers, but the general idea is that adding a layer requires repeating everything before it an increasing number of times, and being able to go all the way down to the bottom before going back up.
Like, in the Saw deck, saw you have two 33/33 Astral Dragons. You split one into 17/17 Astral Dragons, and then one of those into 9/9s, and then one of those into 5/5s, and then you go through the rest of the 9/9s, and each time, your output is increasing. Then repeat all that with the rest of the 17/17s, and you can finish all of that before doing back to the other 33/33.
That's part of a stage, where the layers are just the same thing with different numbers attached, but the same logic applies to layers made of different things. You take take something from a higher layer, trade it for a large number of things in a lower layer, then trade each of those things for increasing numbers of things in the layer below that, and the number of layers is how many of those you can stack together.
Does that help?
Good find on Timbermare. Based on a search, it looks like Fleetwheel Cruiser is another equivalent.
Thanks again for clarifying; that makes a bit more sense. This early post from Deedlit11 also helped me understand more about how the layers work.
So even the 5-card deck is using the same principal. We could describe it using one layer:
Using the same language for the 6-card deck:
...and for the 7-card deck, I would assume it is something like:
However, what I don't understand is what causes the buildup of layers for some of these effects.
It seems to me like the resources found in each layer don't always increase by its immediately senior layer like I inferred from Deedlit11's explanation of "Also, you have to ability to decrease X2 by 1 (or a fixed amount) to increase X1 by more; the ability to decrease X3 by 1 to increase X2 by X1 or thereabouts, the ability to decrease X4 by 1 to increase X3 by X1 or thereabouts, and so on up to Xn."
Using Deedlit's language and the list in your writeup we have:
X1 = Smiting Helix copies
X2 = Slaughter buybacks
X3 = Slaughter copies
X4 = Salvaging Station untaps
We can decrease X4 by 1 to increase X1 and X3 by a bunch (since we will get a bunch of mana with which to create more Precursor Golems), but I don't see how we get more Slaughter buybacks from a later layer. Certainly we get more Slaughter buybacks from the increase in X1, but my intuition tells me that this is a multiplicative effect (i.e. for each copy of Smiting Helix we get 0.75 Slaughter buybacks).
I don't see where we get more life from a senior layer at the cost of X3 or X4 resources. I'm sure I'm missing something but where does X2 become a layer?
The result is that the four layers are Precursor-Helix triggers, Helix copies, Precursor-Slaughter triggers, and Slaughter copies.
Oh I see. I think I misunderstood the meaning of your numbered list in the writeup:
I thought this was describing the layers, but it was just describing the procedure. Those layers make more sense.
Thanks for helping me understand.
Right. The life gain in step 2 is from the Helix copies, and the untaps in step 4 are from the Slaughter copies. I failed to notice that the triggers and spell copies are different layers since they aren't one to one, but compound based on the growing number of golem tokens.
4 cards: Black Lotus, Channel, Skitterbeam Battalion, Tribal Unity with X=8 attains 36 damage.
5 cards: Black Lotus, Channel, Dryad's Revival retrieving Black Lotus, use it for green mana, flashback Dryad's Revival retrieving Black Lotus, use it for red mana, Nylea's Colossus, Rionya, Fire Dancer. Go to combat and make 4 hasty copies of Nylea's Colossus, producing 20 of its triggers, all targeting one of the copies. Attack for 6*2^20+18=6291474 damage.
For 3 cards, Parallel Lives, Astral Dragon, Mirage Phalanx is still the best I know of.
For 2 cards, Devilish Valet and Displacer Beast is better. All three default dungeons have unavoidable limiting effects: "Each player loses 1 life" in Tomb of Annihilation, and card draw at the end of Lost Mine of Phandelver and Dungeon of the Mad Mage.
Start by casting Displacer Beast four times to venture through Lost Mine of Phandelver, getting a Goblin token at Goblin Lair, proceeding from there into Storeroom, and drawing Devilish Valet at the end. Cast Devilish Valet. Then go through Tomb of Annihilation 19 times. At Oubliette, sacrifice the Goblin token the first time, and sacrifice The Atropal on later times. This triggers Devilish Valet 19*4=76 times. Finally, venture six steps into Dungeon of the Mad Mage, going through Muiral's Graveyard, for 8 more triggers of DV, making its power 2^84, and attack.
I guess for 5 cards, Acererak the Archlich beats out Sparkcaster in that case.
mulligan to 5
lotus, show and tell, omniscience, valet, acerak -> complete lost mine 55 times, then 6 steps into Dungeon of the Mad Mage, for 2^(55*4+55+6+2)= 2^283 (plus some more from the storeroom's +1/+1 counters.)
Another benefit of Aluren is that it costs only 4 mana, but I couldn't find any way to get a big advantage from that; the best I have is starting with Black Lotus and Sunscorched Desert for 1 extra damage. (Crystal Dragon looked promising, but there's no way to get the required mana to cast it from exile without giving up multiple Goblin tokens.)
I didn't think the extra cards could be assumed to be Wastes?
(emphasis added)