MTG Salvation

Quote from Deedlit11 »

Ooh, brilliant work yet again! Very tickled to see a natural w^2 construction.

@CaptainMarcia: I would really advise against having multiple bases in your estimates, the average reader is really not going to be able to figure out which is bigger between Knuth expressions with different numbers in the expressions other than at the very end. So they're not going to know that 2^^^ and 3^^^ are pretty much indistinguishable other than at low numbers, so that 2^^^^ and 3^^^^ will differ by at most a small offset. Everyone seems to like using 2 as the base whenever possible, and to use 3 if 2 doesn't work... because the smallest numbers are the best when pursuing large numbers? I don't really agree though, and the thing that I was worried about did appear in discussions of Varhey's 3 card challenge, when the record was about 2^^2^^7; a lot of commenters had a hard time understanding how much that was (understandably), and were unsure how it compared to even numbers like googol or googolplex. I'm not saying they would have necessarily figured that out if it were 10^^10^^4 or 10^^10^^5 (not sure if the latter still works as an upper bound for that particular deck), but it's definitely easier to see with 10 instead of 2, and maybe at least one person in the thread would have figured that those numbers are bigger than a googolplex and could have explained it to the rest.

Of course, replacing "use the smallest number possible as the base" with "use the biggest number possible as the base" doesn't work since you can go very very high, and also using numbers like googolplex or Graham's number as the base might seem a bit gratuitous... so I'm fine with using 10 as the default base, that's the base for scientific notation anyway.

So that's my spiel for using 10 as the base, but really the most important thing is to use a consistent base for all the estimates.

Converting bases is an option, but I'm hesitant to use conversions that would reduce the reported score in situations where it doesn't make it shorter to write. The writeup does make a point of contextualizing the numbers in terms of more familiar ones like googolplex, and I can add examples of how they might translate to other bases, but for this specific deck, the topic of how its score translates to base 3 is particularly relevant due to that being the base used for Graham's Number.

Which doesn't necessarily rule out using a consistent base for the final scoring, but I'm not sure how significant it would be.

While it's true that (3^^^^(3^^^(3^^3^^3^^18))) is larger than (2^^^^(2^^^(2^^2^^3^^18))), the notion that the change between the early 3's and 2's has any significance is far more misleading than just thinking of them as identical. For example, (3^^^^(3^^^(3^^3^^3^^18))) is less than (2^^^^(2^^^(2^^2^^([3^^18)+2])), and is much closer to (2^^^^(2^^^(2^^2^^3^^18))) than tp (2^^^^(2^^^(2^^2^^([3^^18)+2])). So making those early bases higher in this situation, feels fraudulent; if you want a better lower bound, start with the most significant part of the expression, which in case is the power tower 3^^18. While expanding 3^^18 into an exponential tower is kind of a pain, that's the only way you'll get a legitimately good estimate, in the sense that the best lower bound is somewhere between 3^^18 = 3^3^3^...^3, and 3^^19 = 3^3^3^...^27, so there is some number 3^3^3^...^x where the choice of x gives the highest integral value of the power tower that still makes the full expression a lower bound. So figuring out that x is between, say, 16.454 and 16.455, gives you a significantly better estimate. On the other hand, increasing the earlier bases from 2 to 3, or even from 2 to googolplex, will be the same as increasing 3^3^3^...^3 to 3^3^3^...^(3.0000000....), where if we wanted to get to something other than 0 in the 3.00000... we would need more digits than could be stored in with the computational capacity of the observable universe. So REALLY doesn't get you anywhere from 3 to the best value... but if you use your expression with increased bases, people will not know that this is the case, rather they will expect that it makes some kind of difference if you deliberately put in larger bases.

Anyway that's my position. The biggest point for me is that readers will be confused by all the different bases, and how that will affect the size of the estimates. The most important thing to understand is that generally the value of the bases don't make a difference, but probably the best way to handle that is to simply use the same bases.


Edit: Oh and I also argued against having 2 be the preferred base in the previous post. I think 10 is fine, and of course the estimate will be *larger* with 10's than with 3's, if you consider that important.

Yes, the reason I wrote 2^^^^ instead of 3^^^^ was for easier comparisons, as the difference is almost nothing.

I spotted several errors in the writeup:

• (6 section) "which all die, triggring Ratadrabik 2048 times..."
• (7 and 8 sections) You have misspelled "hierarchy" in two places.
• (after 11 section) "for as long as its power remains large enough to survive" -- it's toughness that matters for survival.
• (12 section) You seem to have missed the 2 mana spent to cast Exalted Sunborn warped.

Quote from Deedlit11 »

While it's true that (3^^^^(3^^^(3^^3^^3^^18))) is larger than (2^^^^(2^^^(2^^2^^3^^18))), the notion that the change between the early 3's and 2's has any significance is far more misleading than just thinking of them as identical. For example, (3^^^^(3^^^(3^^3^^3^^18))) is less than (2^^^^(2^^^(2^^2^^([3^^18)+2])), and is much closer to (2^^^^(2^^^(2^^2^^3^^18))) than tp (2^^^^(2^^^(2^^2^^([3^^18)+2])). So making those early bases higher in this situation, feels fraudulent; if you want a better lower bound, start with the most significant part of the expression, which in case is the power tower 3^^18. While expanding 3^^18 into an exponential tower is kind of a pain, that's the only way you'll get a legitimately good estimate, in the sense that the best lower bound is somewhere between 3^^18 = 3^3^3^...^3, and 3^^19 = 3^3^3^...^27, so there is some number 3^3^3^...^x where the choice of x gives the highest integral value of the power tower that still makes the full expression a lower bound. So figuring out that x is between, say, 16.454 and 16.455, gives you a significantly better estimate. On the other hand, increasing the earlier bases from 2 to 3, or even from 2 to googolplex, will be the same as increasing 3^3^3^...^3 to 3^3^3^...^(3.0000000....), where if we wanted to get to something other than 0 in the 3.00000... we would need more digits than could be stored in with the computational capacity of the observable universe. So REALLY doesn't get you anywhere from 3 to the best value... but if you use your expression with increased bases, people will not know that this is the case, rather they will expect that it makes some kind of difference if you deliberately put in larger bases.

Anyway that's my position. The biggest point for me is that readers will be confused by all the different bases, and how that will affect the size of the estimates. The most important thing to understand is that generally the value of the bases don't make a difference, but probably the best way to handle that is to simply use the same bases.


Edit: Oh and I also argued against having 2 be the preferred base in the previous post. I think 10 is fine, and of course the estimate will be *larger* with 10's than with 3's, if you consider that important.

Okay, yeah, from that perspective I can see it. I've edited to base 2, with an explanation of the logic behind it as well as the applicability of base 10 as well.

Quote from plopfill »

Yes, the reason I wrote 2^^^^ instead of 3^^^^ was for easier comparisons, as the difference is almost nothing.

I spotted several errors in the writeup:

• (6 section) "which all die, triggring Ratadrabik 2048 times..."
• (7 and 8 sections) You have misspelled "hierarchy" in two places.
• (after 11 section) "for as long as its power remains large enough to survive" -- it's toughness that matters for survival.
• (12 section) You seem to have missed the 2 mana spent to cast Exalted Sunborn warped.

Thanks, fixed.

I've continued the 12-card writeup to the hyperstage section, but looking back over what I've written - we fuel the hyperstage with -1/1 Exalted Sunborn tokens, but playing Saw in Half on a -1/1 makes more -1/1s. Doesn't that go infinite?

I thought that it rounds -½ to 0, as it says "Round up", and 0 is greater than -1.

Yeah, "divide by two and round up" should take -1 to 0.

Ahh, yeah, you're right. I've added a note about that and finished the 12-card section!

Oh, I'll also point out that Black Lotus + {c]Ball Lightning[/c] was not possible since Alpha, since Ball Lighting came out in The Dark.

Quote from Deedlit11 »

Oh, I'll also point out that Black Lotus + {c]Ball Lightning[/c] was not possible since Alpha, since Ball Lighting came out in The Dark.

Huh. That's been there since the first version years ago, but I guess the error became a lot more obvious once I switched the image to the Masters 25 printing.

Wow, that 12 card hyperstage is cool! And the new writeup is looking great

I think one small correction is necessary: To repeat the final poison layer we cannot use a leftover 0/X, as those don't have a +1/+1 counter to gain the doubling ability. Instead we have to use the last -1/X we get from the stage above and do the grafted skeleton moving again.

The writeup could make it a bit more clear that we keep our high value tokens with massive negative power around as long as possible (without using poison), so they see as many Sunborns as possible when they are eventually sawed. Same for the other tokens from the hyperstage itself.

Quote from Iijil »

Wow, that 12 card hyperstage is cool! And the new writeup is looking great

I think one small correction is necessary: To repeat the final poison layer we cannot use a leftover 0/X, as those don't have a +1/+1 counter to gain the doubling ability. Instead we have to use the last -1/X we get from the stage above and do the grafted skeleton moving again.

The writeup could make it a bit more clear that we keep our high value tokens with massive negative power around as long as possible (without using poison), so they see as many Sunborns as possible when they are eventually sawed. Same for the other tokens from the hyperstage itself.

Thanks, fixed.

Say, do you have the notes on how to get 440 million tokens out of the small computation in the 13-card deck?

Looking through my old files I didn't find a good explanation for 440 million. Not sure how I got to that number. But I found this:
I couldn't get my old programs to run on my new pc, so I didn't check simulation and cost again.

With 12 Precursors from the setup and I think 2 more generated to have the vanillas in this list we could probably get 14 arcbonds on each bishop. That gets us to ~293,601,280 output golems. (The notes seem to assume all the dragons die at the end and we need to convert them to golem then to get ready for the trigger. Not sure if we could use the dragons directly? I guess they would be damaged and might die early)

To maxmize Arcbonds we would cast it from hand sometime between precursor triggers on the second artificial evolution. We need to be careful to not get it on anything that survives, so finish modifying the types of everything that doesn't get arcbond, cast it, then finish modifying the types of everything else. Probably easier to not use human for the computation because of that.

We could get another multiplier if we cast arcbond more often. Using a swap for that might be better than using 2 swaps to get another type in the doubling chain. But I'm not sure if the timing with AE works out if we do that.


Edit: got my programs to run and it says this about the cost:

So it seems to be one swap under what we can afford.

It also reminded we why the extra conversion from dragons to golems is there: dragons are on opponents side, so we need to scrambleverse them to us and sacrifice everything to get golems that we "own" to be targeted by swaps.

Edit 2: To get all the Bishops we want to modify we might need to resolve a few swaps on existing precursor golems before we cast the arcbond. So we probably get only ~10 copies of arcbond per bishop. Please double check any numbers I provided here, mistakes are likely

Edit3: Thought about it some more, and I think we can get full arcbonds by not casting it in between Precursor triggers on AE, but instead casting it before creating the bishops that are not supposed to get it. So the steps become:
- use swaps to create any extra Precursors
- use AE to turn golems that don't get arcbond into dragons, the rest into humans
- create all the bishops that get arcbond
- cast arcbond, everything gets max amount
- create the rest of the bishops
- cast second ae to give everything the creature types we want

I also updated the list to use the 53rd swap and include another improvement: Turns out using a multiplier of 3 is more resource efficient than 2. This lets us get almost a billion golems

Quote from Iijil »

Looking through my old files I didn't find a good explanation for 440 million. Not sure how I got to that number. But I found this:

1x Coat of Arms

--we sacrifice a 1/3 bishop to get 1 damage arcbond ticks.
1x Bishop 1:Ape 2:Ape Control
1x Vanilla Swap

--every tick of arcbond damage the heartbeat angel will die and be replaced by one of the bishops listening for that.
--the second bishop produces a human every tick. With Coat of Arms that keeps the bishops alife as long as the hearbeat keeps going.
1x Vanilla 1:Angel Arcbond Blast <Heartbeat
1x Bishop 4:Angel Arcbond <Program >Heartbeat
1x Bishop 4:Human Arcbond <Output >Program

1x Vanilla 1:Bug <OutputHeartbeat
1x Bishop 1:Dragon 3:Bug 4:Bug <Output >OutputHeartbeat
2x Bishop 1:Dragon 3:Bug 4:Dragon <Output >Output
-- produces 2 dragons for every resolved arcbond
-- keeps the dragons alive forever

--input
5x Vanilla 1:type1 Arcbond <Program
2x Bishop 3:type1 4:type2 Arcbond <Program
2x Bishop 3:type2 4:type3 Arcbond <Program
2x Bishop 3:type3 4:type4 Arcbond <Program
2x Bishop 3:type4 4:type5 Arcbond <Program
2x Bishop 3:type5 4:type6 Arcbond <Program
2x Bishop 3:type6 4:type16 Arcbond <Program
2x Bishop 3:type7 4:type8 Arcbond <Program
2x Bishop 3:type8 4:type9 Arcbond <Program
2x Bishop 3:type9 4:type10 Arcbond <Program
2x Bishop 3:type10 4:type11 Arcbond <Program
2x Bishop 3:type11 4:type12 Arcbond <Program
2x Bishop 3:type12 4:type13 Arcbond <Program
2x Bishop 3:type13 4:type14 Arcbond <Program
2x Bishop 3:type14 4:type15 Arcbond <Program
2x Bishop 3:type15 4:type16 Arcbond <Program
1x Bishop 3:type16 4:Angel Arcbond <Program >Heartstopper
-- Should produce 327680 million Angels in about that many damage ticks
-- There are 33 Bishops with Arcbond around that die now
-- So there are ~n*32*327680 > n*10000000 arcbonds on the stack
-- where n is the number of arcbonds on each bishop

--profit
4x Vanilla 1:Dragon VIP <Output
1x Bishop 1:Dragon 3:Dragon 4:Golem VIP <Output

I couldn't get my old programs to run on my new pc, so I didn't check simulation and cost again.

With 12 Precursors from the setup and I think 2 more generated to have the vanillas in this list we could probably get 14 arcbonds on each bishop. That gets us to ~293,601,280 output golems. (The notes seem to assume all the dragons die at the end and we need to convert them to golem then to get ready for the trigger. Not sure if we could use the dragons directly? I guess they would be damaged and might die early)

To maxmize Arcbonds we would cast it from hand sometime between precursor triggers on the second artificial evolution. We need to be careful to not get it on anything that survives, so finish modifying the types of everything that doesn't get arcbond, cast it, then finish modifying the types of everything else. Probably easier to not use human for the computation because of that.

We could get another multiplier if we cast arcbond more often. Using a swap for that might be better than using 2 swaps to get another type in the doubling chain. But I'm not sure if the timing with AE works out if we do that.


Edit: got my programs to run and it says this about the cost:

|-----------------------------------------------|----------|------------|-----------|
|                                   Description | Quantity | Price/Unit | LineTotal |
|-----------------------------------------------|----------|------------|-----------|
|                Replication Technique recovery |        1 |          1 |         1 |
|-----------------------------------------------|----------|------------|-----------|
|                                  Coat of Arms |        1 |          1 |         1 |
|-----------------------------------------------|----------|------------|-----------|
|                  Free Vanilla (from start up) |        6 |          0 |         0 |
|-----------------------------------------------|----------|------------|-----------|
|                                       Vanilla |        6 |        1/3 |         2 |
|-----------------------------------------------|----------|------------|-----------|
|                               Bishop of Wings |       38 |          1 |        38 |
|-----------------------------------------------|----------|------------|-----------|
|                              Dralnu's Crusade |        0 |          1 |         0 |
|-----------------------------------------------|----------|------------|-----------|
|      Artificial Evolution on Dralnu's Crusade |        0 |          2 |         0 |
|-----------------------------------------------|----------|------------|-----------|
|         Artificial Evolution on all creatures |        1 |          1 |         1 |
|-----------------------------------------------|----------|------------|-----------|
| Extra AE without Human/Cleric->Golem Dralnu's |        1 |          2 |         2 |
|-----------------------------------------------|----------|------------|-----------|
|                           Arcbond (from hand) |        1 |          0 |         0 |
|-----------------------------------------------|----------|------------|-----------|
|                                   Comeuppance |        1 |          1 |         1 |
|-----------------------------------------------|----------|------------|-----------|
|                                 Scrambleverse |        1 |          1 |         1 |
|-----------------------------------------------|----------|------------|-----------|
|                 Soulblast (computation start) |        1 |          1 |         1 |
|-----------------------------------------------|----------|------------|-----------|
|             Scrambleverse (after computation) |        1 |          2 |         2 |
|-----------------------------------------------|----------|------------|-----------|
|                 Soulblast (after computation) |        1 |          2 |         2 |
|-----------------------------------------------|----------|------------|-----------|
|                                               |          |            |           |
|-----------------------------------------------|----------|------------|-----------|
|                                               |          |     Total: |        52 |
|-----------------------------------------------|----------|------------|-----------|

So it seems to be one swap under what we can afford.

It also reminded we why the extra conversion from dragons to golems is there: dragons are on opponents side, so we need to scrambleverse them to us and sacrifice everything to get golems that we "own" to be targeted by swaps.

Edit 2: To get all the Bishops we want to modify we might need to resolve a few swaps on existing precursor golems before we cast the arcbond. So we probably get only ~10 copies of arcbond per bishop. Please double check any numbers I provided here, mistakes are likely

Edit3: Thought about it some more, and I think we can get full arcbonds by not casting it in between Precursor triggers on AE, but instead casting it before creating the bishops that are not supposed to get it. So the steps become:
- use swaps to create any extra Precursors
- use AE to turn golems that don't get arcbond into dragons, the rest into humans
- create all the bishops that get arcbond
- cast arcbond, everything gets max amount
- create the rest of the bishops
- cast second ae to give everything the creature types we want

I also updated the list to use the 53rd swap and include another improvement: Turns out using a multiplier of 3 is more resource efficient than 2. This lets us get almost a billion golems

1x Coat of Arms

--we sacrifice a 1/3 bishop to get 1 damage arcbond ticks.
1x Bishop 1:Ape 2:Ape Control
1x Vanilla Swap

--every tick of arcbond damage the heartbeat angel will die and be replaced by one of the bishops listening for that.
--the second bishop produces a human every tick. With Coat of Arms that keeps the bishops alife as long as the hearbeat keeps going.
1x Vanilla 1:Angel Arcbond Blast <Heartbeat
1x Bishop 4:Angel Arcbond <Program >Heartbeat
1x Bishop 4:Human Arcbond <Output >Program

1x Vanilla 1:Bug <OutputHeartbeat
1x Bishop 1:Dragon 3:Bug 4:Bug <Output >OutputHeartbeat
3x Bishop 1:Dragon 3:Bug 4:Dragon <Output >Output
-- produces 3 dragons for every resolved arcbond
-- keeps the dragons alive forever

--input
8x Vanilla 1:type1 Arcbond <Program
3x Bishop 3:type1 4:type2 Arcbond <Program
3x Bishop 3:type2 4:type3 Arcbond <Program
3x Bishop 3:type3 4:type4 Arcbond <Program
3x Bishop 3:type4 4:type5 Arcbond <Program
3x Bishop 3:type5 4:type6 Arcbond <Program
3x Bishop 3:type6 4:type7 Arcbond <Program
3x Bishop 3:type7 4:type8 Arcbond <Program
3x Bishop 3:type8 4:type9 Arcbond <Program
3x Bishop 3:type9 4:type10 Arcbond <Program
1x Bishop 3:type10 4:Angel Arcbond <Program >Heartstopper
-- Should produce 157464 Angels after a total of >236192 damage ticks
-- There are 30 Bishops with Arcbond around that die now
-- So there were ~15*30*236192 arcbond triggers produced
-- where 15 is the number of arcbonds on each bishop

--profit
4x Vanilla 1:Dragon VIP <Output
3x Bishop 1:Dragon 3:Dragon 4:Golem VIP <Output
-- each dragon we produced turns into 3 golems we own
-- total ~15*30*236192*3*3 = 956577600

Thanks! I think the extra Swap explains the discrepancy - a third output Bishop should multiply the tokens created by 1.5, hitting 440 million.

A minor improvement to the writeup for 2 cards:

Chancellor of the Forge + Blazing Shoal for 8 damage. Idk if is was already discussed here