D&D.Sci Alchemy: Archmage Anachronos and the Supply Chain Issues

[aphyer]

This is an entry in the ‘Dungeons & Data Science’ series, a set of puzzles where players are given a dataset to analyze and an objective to pursue using information from that dataset.

After talking with abstractapplic, I’ve stolen the June 7th scenario slot from him. I hope that this scenario should be relatively simple: not as easy as abstractapplic’s recent introductory scenario, but I still think this would be a fairly approachable starting point if you’re new to D&D.Sci.

STORY

Archmage Anachronos stares at you from beneath his bushy eyebrows. “I have called you to beseech your aid as the greatest living practitioner of the Ancient and Forbidden Art of...Data Science.”

“What, my eyes? No, they’ve always been like that.” (Image generated using OpenArt)

(You’ve tried many times to explain to Archmage Anachronos that Data Science is neither Ancient nor Forbidden, and in fact that the Calantha Institute of Technology and Thaumaturgy has regular classes in Data Science that he could just attend. It hasn’t seemed to work. Ever since you used Data Science to help him locate the lair of the Loathsome Lich, he’s decided that it must be a mighty power indeed, and that you must be a great wizard of some sort to be able to use it. Or maybe he just enjoys being dramatic.)

You wait to hear what the problem is. Is the world being swallowed by darkness? Have the Elemental Lords reawoken and begun subjugating nations? What dread occasion has led him to seek the aid of one who wields so perilous an art^[1]?

He tells you that he needs your help brewing some Barkskin Potion.

(One side effect of Archmage Anachronos’s personality—a love of ridiculous drama, a penchant for overcomplicated schemes, and a strong tendency to frequently disappear to conduct secretive ‘archmage business’ - is that it is hard to tell whether this is actually as unimportant a matter as it sounds, or whether there is some vitally important objective he needs this potion for.)

With his great experience in alchemy (he proudly informs you), he brews a successful potion more than half the time, and remembers every potion he’s ever tried to make. (He presents you with some recently-written records of his brewing, the ink still wet). But he doesn’t always succeed in making a successful potion, or in making the type of potion he was hoping to get.

Additionally, he’s encountered some difficulties sourcing his inputs. Adventuring turned out to be a low-interest-rate phenomenon, and now that famous funders like Rakshasa Global Management, World Tree Capital and Andreesen Heroics have stopped pouring speculative investment into low-level adventurers, it’s much harder for him to get his hands on various alchemically-interesting dragon parts^[2].

In fact, he’s run entirely out of about half of his ingredients, and is low on the rest. And he’s got an urgent need for a cauldron of Barkskin Potion, and only enough time and remaining ingredients to do one brew. He’s turned to you for advice: what ingredients should he include to have the best odds of brewing this Barkskin Potion?

DATA & OBJECTIVES

• Archmage Anachronos tells you he is trying to brew Barkskin Potion.

• Some of his ingredients are not available—he has no access to:

  • Angel Feather

  • Beholder Eye

  • Crushed Ruby

  • Crushed Sapphire

  • Dragon Scale

  • Dragon Spleen

  • Dragon Tongue

  • Dragon’s Blood

  • Ectoplasm

  • Eye of Newt

  • Faerie Tears

  • Powdered Silver

• But he does have access to:

  • Badger Skull

  • Beech Bark

  • Crushed Diamond

  • Crushed Onyx

  • Demon Claw

  • Giant’s Toe

  • Ground Bone

  • Oaken Twigs

  • Quicksilver

  • Redwood Sap

  • Troll Blood

  • Vampire Fang

• He’s asked you which of these ingredients he should use to maximize his odds of brewing a Barkskin Potion successfully.

• He can use any number of different ingredients in a potion, but cannot use any ingredient more than once: first, this would apparently “trigger a harmonic overload” and ruin the potion, and second, he only has one of most of these things.

  • So you could tell him to use Badger Skull and Beech Bark.

  • Or you could tell him to use Crushed Diamond, Crushed Onyx, a Demon Claw, and a Giant’s Toe.

  • Or you could tell him to use all of the available ingredients except for the Vampire Fang.

  • But you could not tell him to use two Troll Blood, or three Redwood Sap.

• To help you with this, you have a dataset of past potions he’s attempted to brew. Each row lists which ingredients he included in the potion, and what the result was (either what potion was successfully brewed, or what went wrong.)

I’ll aim to post the ruleset and results on June 17th (giving one week and both weekends for players). If you find yourself wanting extra time, or find this scenario late and want a chance to attempt it yourself, comment below and I can push this deadline back.

As usual, working together is allowed, but for the sake of anyone who wants to work alone, please spoiler parts of your answers that contain information or questions about the dataset. To spoiler answers on a PC, type a ‘>’ followed by a ‘!’ at the start of a line to open a spoiler block—to spoiler answers on mobile, type a ‘:::spoiler’ at the start of a line and then a ‘:::’ at the end to spoiler the line.

1. ^

   Repetitive Strain Injury is no joke!

2. ^

   He assures you that he has only ever used ethically-sourced dragon parts—that is, ones gotten by killing dragons whose scales weren’t all shiny.

[aphyer]

I’m very proud of this scenario. (Even if you’re confident you aren’t going to play it, I think you could read the wrapup doc and in particular the section on ‘Bonus Objective’ so you can see what it involved).

It accomplished a few things I think are generally good in these scenarios:

• There was underlying structure that players could uncover, which created emergent complexity in the output but made sense with the theme once the underlying ruleset was revealed/​discovered.

• Human thought about e.g. the theme and what patterns would be reasonable to observe was valuable, the puzzle was not optimally-solved just by feeding the data into a model and calling it a day.

• Multiple levels of solution were possible, from a decent solution with little effort up to a more-involved solution that went further and dug into the underlying structure.

And also it managed to trick many players with a surprising-yet-thematic twist :P

[abstractapplic]

I continue to think that this is plausibly the best ever installment of that genre I invented, and I continue to think it expands said genre in interesting ways. I also continue to think said genre is a valuable addition to LW, because it provides (limitedly) messy and (tolerably) complicated inferential problems with definitive answers; in other words, it gives us a chance to fail and know we failed.

(This challenge successfully fooled me, and I love it for that. Maybe you’ll do better, dear reader?)

[simon]

Observations so far:

Each potion has two essential ingredients (necessary, but not sufficient).

Barkskin Potion: Crushed Onyx and Ground Bone

Farsight Potion: Beholder Eye and Eye of Newt

Fire Breathing Potion: Dragon Spleen and Dragon’s Blood

Fire Resist Potion: Crushed Ruby and Dragon Scale

Glibness Potion: Dragon Tongue and Powdered Silver

Growth Potion: Giant’s Toe and Redwood Sap

Invisibility Potion: Crushed Diamond and Ectoplasm

Necromantic Power Potion: Beech Bark and Oaken Twigs

Rage Potion: Badger Skull and Demon Claw

Regeneration Potion: Troll Blood and Vampire Fang

Most of these make sense. Except...

I strongly suspect that Archmage Anachronos is trying to trick me into getting him to brew a Necromantic Power Potion, and has swapped around the “Barkskin Potion” and “Necromantic Power Potion” output reports. In character, I should obtain information from other sources to confirm this. For the purpose of this problem, I will both try to solve the stated problem but also how to make a “Necromantic Power Potion” (i.e. an actual Barkskin Potion) to troll Anachronos.

Barksin and Necromantic Power also seem to be the two toughest potions to make, never succeeding with just a third ingredient added.

An attempt that includes essential ingredients for multiple potions does not necessarily fail, some ingredient combinations can produce multiple potion types.

There are four failure modes:

Acidic Slurry never happens if the essential ingredients of any potion are included, but not all attempts that lack essential ingredients are Acidic Slurry. So, I’m guessing Acidic Slurry is a residual if it doesn’t have the required ingredients for any potion and doesn’t hit any of the other failure modes.

Inert Glop tends to happen with low numbers of ingredients, initial guess is it happens if “not magical enough” in some sense, and guessing Magical Explosion is the opposite, dunno about Mutagenic Ooze yet.

Back to Barkskin and Necromantic Power:

Either one has lots of options to make reliably with just one non-available ingredient, but not (in the stats provided) avoiding all the unavailable ingredients. Each has some available options that produce the potion type most of the time, but with Inert Glop some of the time. There’s one ingredient combination that has produced a “Barkskin Potion” the only time it has been tried, but it has the essential ingredients for a Growth Potion so would likely not reliably produce a “Barkskin Potion”.

Many ingredient combos especially towards higher ingredient numbers haven’t been tried yet, so plenty of room to find an actually reliable solution if i can figure out more about the mechanics. The research will continue...

[abstractapplic]

I think your theory about

him switching the Barkskin and Necromantic Power potions

is completely correct and I feel dumb for not thinking of it; ditto your proposed reaction. On reflection, I suspect that this is because

he’s actually the Loathsome Lich in disguise

so your right-ness is a lot more important than it might seem at first glance. Good catch!

[simon]

Maybe...

a love of ridiculous drama, a penchant for overcomplicated schemes, and a strong tendency to frequently disappear to conduct secretive ‘archmage business’

Lying in order to craft a Necromantic Power Potion is certainly a bad sign, but still compatible with him being some other dark wizard type rather than the “Loathsome Lich” in particular.

Regarding your proposal in second comment: even if he is undead he might not need to drink it to tell what it is. Still, could be worth a shot! (now three different potion types to figure out how to make...)

[abstractapplic]

Wait . . . actually, if we’re in the mood for galaxy-brained moves, we could go one better and try to

con the lich into brewing & drinking a regen potion.

[Lorxus]

Potentially? I’d be worried that that would be too obvious of us and he’d notice immediately. I think I weakly prefer giving him

actual Barkskin using the two woods and three magically charged ingredients coding for no potion

instead—no use complaining about getting what you asked for!

[simon]

Followup and actual ingredients to use:

Mutagenic Ooze is a failure mode that can happen if there are essential ingredients for multiple potions (can also get either potion or Inert Glop or Magical Explosion if eligible).

There are 12 “magical” ingredients. An ingredient is magical iff it is a product of a magical creature (i.e.: Angel Feather, Beholder Eye, Demon Claw, Dragon Scale, Dragon Spleen, Dragon Tongue, Dragon’s Blood, Ectoplasm, Faerie Tears, Giant’s Toe, Troll Blood, Vampire Fang).

Inert Glop is a possible outcome if there are 2 or fewer magical ingredients, and is guaranteed for 1 or fewer.

Magical Explosion is a possible outcome if there are 4 or more magical ingredients, and is guaranteed if there are 5 or more.

(Barksin and necromantic power seem “harder” since their essential ingredients are both nonmagical, requiring more additional ingredients to get the magicness up.)

Therefore: success should be guaranteed if you select the 2 essential ingredients for the desired potion, plus enough other ingredients to have exactly 3 magical ingredients in total, while avoiding selecting both essential ingredients for any other potion. For the ingredients available:

To get “Necromantic Power Potion” (actual Barkskin):

Beech Bark + Oaken Twigs + Demon Claw + Giant’s Toe + either Troll Blood or Vampire Fang

To get “Barkskin Potion” (actual Necromantic Power):

Crushed Onyx + Ground Bone + Demon Claw + Giant’s Toe + either Troll Blood or Vampire Fang

To get Regeneration Potion:

Troll Blood + Vampire Fang + either Demon Claw or Giant’s Toe

I expect I’m late to the party here on the solution… (edit: yes, see abstractapplic’s very succinct, yet sufficient-to prove-knowledge comment, and Lorxus’s much, much more detailed one)

[simon]

Post-solution extra details:

Quantitative hypothesis for how the result is calculated:

“Magical charge”: number of ingredients that are in the specific list in the parent comment. I’m copying the “magically charged” terminology from Lorxus.

“Eligible” for a potion: Having the specific pair of ingredients for the potion listed in the grandparent comment, or at the top of Lorxus’ comment.

1. Get Inert Glop or Magical Explosion with probability depending on the magical charge.

   1. 0-1 → 100% chance of Inert Glop

   2. 2 → 50% chance of Inert Glop

   3. 3 → neither, skip to next step

   4. 4 → 50% chance of Magical Explosion

   5. 5+ → 100% chance of Magical Explosion

2. If didn’t get either of those, get Mutagenic Ooze at ¹⁄₂ chance if eligible for two potions or ²⁄₃ chance if eligible for 3 potions. (presumably would be n/​(n+1) chance for higher n).

3. If didn’t get that either, randomly get one of the potions the ingredients are eligible for, if any.

4. If not eligible for any potions, get Acidic Slurry.

todo (will fill in below when I get results): figure out what’s up with ingredient selection.

edit after aphyer already posted the solution:

I didn’t write up what I had found before aphyer posted the result, but I did notice the following:

• hard 3-8 range in total ingredients

• pairs of ingredients within selections being biased towards pairs that make potions

• ingredient selections with 3 magical ingredients being much more common than ones with 2 or 4, and in turn more common than ones with 0-1 or 5+

  • and, this is robust when restricting to particular ingredients regardless of whether they are magical or not, though obviously with some bias as to how common 2 and 4 are

• the order of commonness of ingredients holding actual magicalness constant is relatively similar restricted to 2 and 4 magic ingredient selections, though obviously whether is actually magical is a big influence here

• I checked the distributions of total times a selection was chosen for different possible selections of ingredients, specifically for: each combination of total number of nonmagical ingredients and 0, 1 or 2 magical ingredients

  • I didn’t get around to 3 and more magical ingredients, because I noticed that while for 0 and 1 magical ingredients the distributions looked Poisson-like (i.e. as would be expected if it were random, though in fact it wasn’t entirely random), it definitely wasn’t Poisson for the 2 ingredient case, and got sidetracked by trying to decompose into a Poisson distribution + extra distribution (and eventually by other “real life” stuff)

    • I did notice that this looked possibly like a randomish “explore” distribution which presumably worked the same as for the 0 and 1 ingredient case along with a non-random, or subset-restricted “exploit” distribution, though I didn’t really verify this

[Lorxus]

Giving this a try. I’ll document my progress towards a solution.

(2024-6-7) [Poking at the foundations?]

Looks like you only get magical explosions and mutagenic ooze with >=4 ingredients, and also like potions are determined by a specific pair of ingredients. List is the next spoiler:

Farsight: Beholder Eye and Eye of Newt
Firebreathing: Dragon Spleen and Dragon’s Blood
Fire Resistance: Crushed Ruby and Dragon Scale
Glibness: Dragon Tongue and Powdered Silver
Growth: Giant Toe and Redwood Sap
Invisibility: Crushed Diamond and Ectoplasm
Necromantic Power: Beech Bark and Oaken Twigs
Rage: Badger Skull and Demon Claw
Regeneration: Troll Blood and Vampire Fang

Barkskin: Crushed Onyx and Ground Bone—both of which we have!

So… why aren’t we just done? Why don’t we just tell Anachronos to add Crushed Onyx and Ground Bone and have done with it?

---

(2024-6-8) [Early analysis, first solution proposal]

Well: I still don’t know yet what makes a brew succeed or fail.

My working hypothesis is that a major thing determining what happens when you try to brew a potion is what pairs of ingredients are found in it.

I suspect it’s even deterministic—i.e., the same subset of potion ingredients results in the same outcome every time—but I haven’t yet actually checked that—I’m doing all this in Google Sheets and my data-manipulation-fu is pathetically weak. (Probably someone could check this easily and I look dumb; I’ll poke at looking for identical ingredient-sets resulting in different outcomes later.) Lorxus from like an hour later sez: “Nope, decidedly nondeterministic, given the filter-based checking I did later. Still gotta figure out what causes it and why.”

For instance, I suspect that Inert Glop is the default outcome in some sense, i.e., you get Inert Glop IF no pair of ingredients in the ingredient-set codes for any potion AND no more interesting failure condition is met. Acidic Slurry would be the next-to-default, where nothing more interesting happened BUT something was different such that the result of the brew was AS and not IG.

I suspect this specifically because AS and IG are the two simplest results—each show up in the result table with only 3 ingredients.

Poking a little more with some better use of filters, it looks like my earlier theory was maybe incomplete. It’s true that e.g. Barkskin Potion requires Crushed Onyx and Ground Bone, but that’s not actually sufficient.

Sometimes you add those to a brew and you get nothing. An earlier working hypothesis I had that maybe Magical Explosions happen to a brew precisely when it codes for two or more potions got falsified—there’s plenty of brews with e.g. Crushed Onyx and Ground Bone and also Dragon Spleen and Dragon’s Blood. Some of those make Barkskin Potions. Some of them make Firebreathing potions. Sometimes you get a Mutagenic Ooze or a Magical Explosion.

So maybe that falsified theory was less falsified than I thought, and there’s something about a failure rate (and likely also type) dependent on how many of the valid potions your brew codes for? Maybe something like “if a brew codes for at least two valid potions, then there’s a 25% chance of a ME, a 25% chance of a MO, and equal chances of every other valid potion your brew codes for.”

I didn’t note this above but notably, four ingredients don’t code for any of the potions: Angel Feather, Crushed Sapphire, Faerie Tears, and Quicksilver. I can’t legibly explain why, but I feel like they’ll still matter—have an effect on failure types or rates, or be required as part of a potion base, or maybe even just very clearly have no effect at all.

As a terminological note you may or may not have picked up on by now: I’m going to use “brew” for some subset of potion ingredients, “potion” for one of the 10 potions we can brew, and say that a brew “codes for a potion” if a subset of the brew is a pair that looks to me to determine which of the 10 potions get brewed. The “outcomes of a brew” is the set of things that have observably resulted from a given brew.

Did a quick observational-experiment:

Among brews consisting of Crushed Onyx, Dragon Spleen, Dragon’s Blood, Ground Bone, and Powdered Silver, the outcomes consisted of 4 Barkskin, 9 Firebreathing, 20 Inert Glop, 8 Mutagenic Ooze, and 0 Magical Explosion.

Among brews consisting of Beech Bark, Beholder Eye, Eye of Newt, Troll Blood, and Vampire Fang, the outcomes consisted of 3 Farsight, 4 Regeneration, 0 Inert Glop, 11 Mutagenic Ooze, and 0 Magical Explosion.

(this is the point at which if my data-manipulation-fu were stronger, I’d probably just generate all possible results for all brews of the form [Putative Coding-Pair for Potion 1] + [Putative Coding-Pair for Potion 2] + [Arbitrary Other Ingredient] and look at the spread of results; I would strongly expect that to be something like “sometimes you get potion 1, sometimes you get potion 2, and there’s patterns in the ratios between them and the failure rates; the failure rates, summed up, are always greater than the chance of getting any individual potion and are generally greater than getting any successful potion at all”.)

...actually, that means that I (think that I) understand this problem well enough to make an initial solution-proposal!

OK OK less of a solution-proposal, more of what has turned into another observational-experiment. What happens if we filter by requiring Crushed Onyx and Ground Bone (like we have to) and forbidding the ingredients we don’t have any of anyway?^[1] Basically, let’s just look at the observational-space of brews that already seem kind of like a good idea anyway—what happens in all of those possible-worlds?

We get 201 brews, and we should already spot poor Anachronos’s problem: none of the reliably working 3-ingredient brews that make Barkskin are found among them.

Perhaps worse, of the 4-ingredient brews, 19 make Barkskin and 44 fail as Inert Glop; a lot of these are the specific brew {[Crushed Onyx, Ground Bone], Demon Claw, Vampire Fang}, which is responsible for all of the successes and 10 of the failures for unclear reasons. He was right to call through the void at right angles to try to find Math-Doers, damn the risks!

Another promising-looking brew among the observed ones is the 5-ingredient brew {[Crushed Onyx, Ground Bone], Giant’s Toe, (Troll Blood, Vampire Fang)}, which makes Barkskin 18 times, Regeneration 22 times, and Mutagenic Ooze 35 times. Maybe Anachronos likes those odds and a Regen potion will do in a pinch?

So I guess I’d say if I were under truly tight time constraints I’d present Anachronos with those two options - [a ~2/​3 chance of straight out success and a ~1/​3 chance of harmless failure] vs [a ~1/​4 chance of straight out sucess, a ~1/​4 chance of lesser failure, and a ~1/​2 chance of dangerous failure] and probably push him towards the former of the two - {[Crushed Onyx, Ground Bone], Demon Claw, Vampire Fang}. I have the luxury of existing outside of his time, though, so I’m going to keep trying.

Interestingly—the modified brew {[Crushed Onyx, Ground Bone], Demon Claw, Troll Blood} only gets tested once—a failure. That’s kind of a shame—I’d predict it to be pretty good given the lack of potion-code collision with Regen (which the 5-ingredient brew suffers from). That means there’s something more at work here. {[Crushed Onyx, Ground Bone], Giant’s Toe, Vampire Fang} never even gets tested at all!

Note to future Lorxus—run this same process with the three other potions that we could possibly brew at all—Growth, Rage, and Regen.

Growth:
{[Giant’s Toe, Redwood Sap], Quicksilver, Vampire Fang}: 24 Growth, 29 IG
{[Giant’s Toe, Redwood Sap], Ground Bone, Oaken Twig, Quicksilver, Vampire Fang}: 19 Growth, 16 IG

Rage:
{[Badger Skull, Demon Claw], Quicksilver, Troll Blood}: 21 Rage, 13 IG
(notably, modified brews with only Quicksilver or Troll Blood fail more often, and fail altogether if they have nothing else—this looks to be a pattern that holds up of secondary ingredients? “Require A and B; require at least one of C and D plus something else—maybe one of the four miscellaneous ones?”)

Regen:
{[Troll Blood, Vampire Fang], Crushed Diamond, Giant’s Toe, Oaken Twig}
{[Troll Blood, Vampire Fang], Crushed Diamond, Demon Claw, Ground Bone}
(neither of these brews ever fail???)

---

(2024-6-9) [More analysis, contains a final answer]

Ah, the joys of existing outside of (Anachronos’s) space-time!

I think I’ve squeezed about as much as I can from pure associational observation, and need to actually buckle down and figure out why brews succeed or fail. Here’s my working model:

• There are precisely ten codeable potions: Barkskin, Farsight, Firebreathing, Fire Resistance, Glibness, Growth, Invisibility, Necromantic Power, Rage, and Regeneration.

  • For each of these potions, there’s two ingredients that are absolutely required (the potion’s code, in my terms). I list them off above.

• If a brew codes for two potions, it has a ¹⁄₄ chance of producing potion A, a ¹⁄₄ chance of producing potion B, and a ¹⁄₂ chance of melting down into Mutagenic Ooze.

  • I also hypothesize one of three things:

    • If a brew codes for n>2 potions, it has a 1/​2n chance of producing each of those potions, and a ¹⁄₂ chance of melting down into Mutagenic Ooze.

    • If a brew codes for n>2 potions, it has a 1/​n+2 chance of producing each of those potions, and a 2/​n+2 chance of melting down into Mutagenic Ooze.

    • If a brew codes for three or more potions, then whenever it would turn out as Mutagenic Ooze, it instead turns out as a Magical Explosion. (This one I think is much less likely, given that none of the observed brews that we could replicate result in a ME.)

• A successful brew cannot, for whatever reason, consist of any fewer than 3 ingredients.

  • 4-5 in particular seems to be a sweet spot: 2 ingredients for the code, and another 2-3 for ??reasons?? which also don’t code for any different potion.

• No brew is guaranteed to succeed. Every potion has some base chance of failing as Inert Glop.

  • As a corollary, we should thus reject any brew with a success rate less than that of {[Crushed Onyx, Ground Bone], Demon Claw, Vampire Fang}, that is, ~2/​3.

  • This is likely wrong though! {[Troll Blood, Vampire Fang], Crushed Diamond, Giant’s Toe, Oaken Twig} and {[Troll Blood, Vampire Fang], Crushed Diamond, Demon Claw, Ground Bone} both seem to be no-fail brew for Regen...

• A brew which codes for no potion… gods I wish my data-manipulation-fu were stronger

  • ...is guaranteed to be Acidic Slurry? Maybe?

    • No, that can’t be right - {Ground Bone, Oaken Twigs, Quicksilver, Vampire Fang} doesn’t code for anything and yet we see it in the IG list.

  • ...will turn out as IG if it lacks some factor and as AS if it has that factor?

    • Yeah this isn’t a guess so much as an entire guessing schema.

      • Which still might not even be correct.

So… maybe I want to go looking for relations in or commonalities among those other 2-3 ingredients? Or possibly there’s specific peculiar properties to each ingredient that I need to keep track of? …Actually that sounds a lot more plausible when I think about it that way.

(After a lot more thinking)

Y’know… there’s enough test cases that the proportions I found in the last recipe should really be a lot closer to exact than they are. What’s up with that?

Also, I still have no idea why MEs happen. What’s up with those?

A quick look at the set of observed brews we could go make right now that resulted in MEs:
{Crushed Onyx, Demon Claw, Giant’s Toe, Oaken Twig, Redwood Sap, Troll Blood, Vampire Fang}: 16 MEs, 13 MOs, 4 Growth, 9 Regen
{Demon Claw, Giant’s Toe, Troll Blood, Vampire Fang}: 1 ME/​1 brew (!)

So maybe there’s something special about these four ingredients? You (can) get a ME if you include all four, with the rate of that increasing as the proportion of “magically charged” ingredients in a brew increases?

It’d explain why we probably can’t just tell Anachronos to brew {Crushed Onyx, Ground Bone}: it wouldn’t have sufficient “magical charge”; in this model, a potion goes off (in the sense of “is not IG”) never/​sometimes/​always if it has 0-1/​2/​3+ magically charged ingredients.

The problem is that I’ll need to expand my search to the rest of the dataset (of things we can’t possibly brew now) to figure out whether Crushed Onyx is magically charged, given that it’s not in the codes for Growth or Regen (because codes are nonoverlapping) - though if that theory is correct, the existence of sometimes-working 3-ingredient brews that can make Barkskin or IG suggests that Crushed Onyx probably is magically charged (and if so, good gods, Anachronos, what were you doing brewing the same maximally overcharged potion that many times?). Also, countervailing that is the fact that {[Crushed Onyx, Ground Bone], Demon Claw, Vampire Fang} doesn’t always actually work, which suggests that it’s only the smaller set of four (out of what we have left).

Assuming Crushed Onyx is magically charged: {[Crushed Onyx, Ground Bone], Giant’s Toe, Demon Claw}.
Assuming Crushed Onyx is not magically charged: {[Crushed Onyx, Ground Bone], Giant’s Toe, Demon Claw, Troll Blood} and {[Crushed Onyx, Ground Bone], Giant’s Toe, Demon Claw, Vampire Fang} should both work equally well—all we need to do is avoid code-clash with Regen.

If I had to give a solution right now—given the fact that Troll Blood and Vampire Fang are absent from all of the (4) {[Crushed Onyx, Ground Bone], Giant’s Toe, Demon Claw}-containing brews, and those tend to fail as IG (or MO from code-clash) - I’m pretty sure Crushed Onyx is magically uncharged.

So I’d tell Anachronos to brew {[Crushed Onyx, Ground Bone], Giant’s Toe, Demon Claw, Troll Blood} or {[Crushed Onyx, Ground Bone], Giant’s Toe, Demon Claw, Vampire Fang}, whichever [he feels better about]/​[is cheaper]/​[he likes better]/​[he has better stocks of]/​[he can buy more easily]/​[idk my dude flip a coin]. Either should work with certainty.

https://​​docs.google.com/​​spreadsheets/​​d/​​1kzR25lXiGqhSMg6K_QwDfoE0VnVUnOeK74AbzOSkjmk/​​edit?usp=sharing

(after reading simon’s observation that maybe Barkskin and Necromantic Power are swapped around) If we do in fact doubt Anachronos here, then we can easily put together a 5-ingredient brew that would guarantee getting him actual Barkskin: {[Beech Bark, Oaken Twig], Giant’s Toe, Demon Claw, [Troll Blood XOR Vampire Fang]}

1. ^

   For reference, these are: Angel Feather, Beholder Eye, Crushed Ruby, Crushed Sapphire, all four of the Dragon gibblies, Ectoplasm, Eye of Newt, Faerie Tears, and Powdered Silver.

[abstractapplic]

Thanks for running this!

Unless I made some trivial mistake,

Crushed Onyx, Ground Bone, Demon Claw, Giant’s Toe, Vampire Fang

should work.

Explanation:

First two ingredients specify the potion, remaining three make it juuust impossible enough to guarantee that it will reliably be magical without going boom.

[abstractapplic]

After some consideration (and reading other people’s answers, in particular simon’s) I’ve come to the conclusion that the best answer to give is actually

Vampire Fang, Troll Blood, Ground Bone, Oaken Twigs, Demon Claw

[Unnamed]

With the ingredients he has, he has gotten a successful Barkskin Potion:

1 of the 1 times (100%) he brewed together Crushed Onyx, Giant’s Toe, Ground Bone, Oaken Twigs, Redwood Sap, and Vampire Fang.

19 of the 29 times (66%) he brewed together Crushed Onyx, Demon Claw, Ground Bone, and Vampire Fang.

Only 2 other combinations of the in-stock ingredients have ever produced Barkskin Potion, both at under a 50% rate (4/​10 and ¹⁸⁄₇₅).

The 4-ingredient, 66% success rate potion looks like the best option if we’re just going to copy something that has worked. That’s what I’d recommend if I had to make the decision right now.

Many combinations that used currently-missing ingredients reliably (100%) produced Barkskin Potion many times (up to ¹¹⁸⁄₁₁₈). There may be a variant on one of those, which he has never tried, that could work better than 66% of the time using ingredients that he has. Or there may be information in there about the reliability of the 6-ingredient combination which worked once.

[aphyer]

Are you sure you are reading the dataset correctly? In particular, which row number do you think shows him brewing together Crushed Onyx, Redwood Sap, and Vampire Fang to yield a Barkskin Potion? I suspect you may be looking at row 118078 (or 118079 if you include the header row), in which he did brew a Barkskin Potion and did use those three ingredients—but he also used a Giant’s Toe, Ground Bone, and Oaken Twigs. Are you seeing something different in the dataset?

EDIT: Never mind, looks like you caught this and edited in your comment. Sorry for the bother, just wanted to make sure I hadn’t screwed up the upload in some way.

[Christian Z R]

Okay, I might be late, but to test myself I will post my solution here before checking the right answer:

First I find the subset of recipes that produce a Barkskin potion. By examining this I find that they all need Crushed Onyx and Ground Bone. I make a subset of recipes with only the ingredients at hand AND using both Crushed Onyx and Ground Bone.

We now have two very manageable dataframes, 42 succesful and 152 unsuccessful recipes based on C.O and G.B. Looking at them I quickly realize that all the recipes have some chance of failure. Make a simple script to identify the unique recipes that worked, and figure out how often they failed. At last I am left with the best recipe, which is C.O, G.B, Demon Claw and Vampire Fang. This has produced the right potion 19 times and produced Inert Glob 10 times. So I rate it to have a 65% chance of success.

Then I go here to check my recipe, and whoops, the smarter people all seem to have figured out that I am actually being tricked into a Necromantic Power potion. Well, looks very convincing but too late for me. Better find a safe place to hide...

[aphyer]

MUWAHAHAHAHA! YOU FOOL!

ahem

That is to say, I’m glad to have you playing, I enjoy seeing solutions even after scenarios are finished. (And I think you’re being a bit hard on yourself, I think simon is the only one who actually independently noticed the trick.)

[Yonge]

All potions have between 3 and 8 ingredients. Those with 6 and 7 are most liekly to succeed with 3 being particularly unlikely. In particular Barkskin Potion is never brewed successfully with only 3.
Barkskin Potion, in common with all other potions has two ingredients that must be present, in this case Crushed Onyx and Ground Bone.

The same set of ingredients sometimes produce different potions so there is clearly some randomness involved.

Scanning through the results there are several combinations that always produce the required potion. The most convincing (With 118 entries) is:
Crushed Onyx
Dragon Tongue
Dragons Blood
Eye of Newt
Giants Toe
Ground Bone

Sadly 3 of these ingredients aren’t available however. The same is true of most of the combinationsn that have a 100 percent success rate. There is only one exception but as that is based on a single record using that is highly risky.

The next best with a 65.5 percent chance of success based on 29 records is:
Crushed Onyx
Demon Claw
Ground Bone
Vampire Fang

So I’m going to have to recommend he uses these, and hope he doesn’t end up with Inert Glop instead.

[qwertyasdef]

From looking at every ingredient-result pair and picking the ones which appear with Barkskin Potion more often than would expected if they were completely independent, I’m going to suggest the ingredients
1. Crushed Onyx
2. Demon Claw
3. Ground Bone
4. Quicksilver
5. Troll Blood
I’m very certain about Crushed Onyx and Ground Bone which appear in every successful Barkskin Potion, less so about the rest.

Edit: if I filter to only successful Barkskin Potions without any of the unavailable ingredients, that leaves 42 rows. None of them use Quicksilver, and instead Giant’s Toe and Vampire Fang are looking pretty good. Not sure how much weight to put on this but something to investigate later.