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...
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...)
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)
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.
Get Inert Glop or Magical Explosion with probability depending on the magical charge.
0-1 → 100% chance of Inert Glop
2 → 50% chance of Inert Glop
3 → neither, skip to next step
4 → 50% chance of Magical Explosion
5+ → 100% chance of Magical Explosion
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).
If didn’t get that either, randomly get one of the potions the ingredients are eligible for, if any.
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
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...
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!
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...)
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.
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!
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)
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.
Get Inert Glop or Magical Explosion with probability depending on the magical charge.
0-1 → 100% chance of Inert Glop
2 → 50% chance of Inert Glop
3 → neither, skip to next step
4 → 50% chance of Magical Explosion
5+ → 100% chance of Magical Explosion
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).
If didn’t get that either, randomly get one of the potions the ingredients are eligible for, if any.
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