Carissa Sevar: "- sure." She's pretty sure that all guesses are equally good if you know nothing, which seems right, but maybe he's making a deeper point. 

Keltham: Right then.  Here is the coin, Asmodia, this isn't even a loan you're just holding my coin.  Let's start.

Keltham writes down some numbers, not particularly showing them to anyone per se, just yet.

Ione Sala: Ione writes down some numbers!  They're all 25s.

Pilar : Pilar is actually going to try to guess those coin-spins because, like, also oracle?  25 or 75 depending on which she expects to happen next.  She's not doing too well so far but maybe she'll get the hang of it.

Carissa Sevar: 25, 25, 25, 25. 

Project Lawful: Pilar's curse says no.

Pilar : He didn't say not to cheat!  What manner of defective Chaotic Good oracular curse is this?

Iarwain: Time for everybody to add up their scores!  Double-Abrogail came up twice in the 12 rounds.

Ione Sala: So, round 1, +25 points, round 2, +25 points, round 3, which is the first double-Queen, -25 points, it's obviously going to work out to +200 points at the end and also she's an idiot and is embarrassed to reveal what she wrote, maybe nobody will ask her.

Pilar : She guessed neither of the two double-Queens, and also got -25 points the three times she falsely foretold them, so 5*-25 + 7*+25 = +50 points.  She probably just needs practice.

Carissa Sevar: On the last round Carissa had an idea, and wrote a 0. 50 points if she's right, and -50 if she's wrong, but she's right more often than not.

Which is 225, on the whole, because the last one isn't an Abrogail. 

Asmodia: So everybody just wrote down all 0s, right?

Keltham: Keltham did, yes.  +50 * 10 + -50 * 2 = +400 points.

Keltham is very good at this game!  Everyone should clearly use his numbers, which say that a coin comes up Abrogail twice in two spins roughly 0 out of 100 times.

lintamande: (It took Meritxell two rounds to catch on. +350.)

Carissa Sevar: "...okay, I don't know how to fix that. Bigger penalties for being wrong? But then you lose - it being better to participate than not -"

Keltham: "Actually, we're not going to end up with that particular property, necessarily?  When you're just - guessing things, trying to know things, it's not like something comes in and takes your real-life gold pieces when you do the thing that loses you game points.  Like, the game can just say, every time you try to guess, that means you might turn out to be wrong and lose points.  But that's fine, because just guessing and just being wrong doesn't hurt your bank storage unless you actually made a bet."

"Who, besides Asmodia, would like to now try stating a fragment of Law that scoring rules ought to obey?"

lintamande: "It ought to be a better idea to give your true guess," says Meritxell.

Keltham: "Can you state that more precisely?"

lintamande: "Say it actually has a one in four chance of happening. Guessing one in four ought to score more points than guessing anything else."

Keltham: "So if double-Abrogail has a 1 in 4 chance of happening, then whoever guesses 25 on that round should be rewarded the most, regardless of whether double-Abrogail actually happened or not?"

lintamande: "....not on that round. But in the long run."

Keltham: "So, if we're just playing one round of something, there's no way it could have any kind of Lawful scoring rule."

lintamande: "I think it'd be something like - escalating gains from being right when you pick an extreme number and from being wrong when you pick an extreme number, so it's only worth being that extreme if you're sure - and there's going to be some rule this implies but I don't know it, that balances it exactly right -"

Carissa Sevar: "When you pick ninety-nine, the penalty for being wrong has to cancel out ninety-nine of being right, so it's only worth doing it if that's exactly how sure you are."

Keltham: "So, considering all the numbers P we could guess between 0 and 100, if the truth is that something happens F out of 100 times, and doesn't happen 100 minus F out of 100 times, we want..."

This room's wall does now function as a whiteboard for people who, like Keltham, can cast Prestidigitation.

argmax P of F*Yes(P) + (100 - F)*No(P) = F

"...which is to say that it seems like a Lawful scoring rule must surely have this property: for every F between 0 and 100, the answer P that maximizes the sum of F of the yes-value you get from P, plus 100 minus F of the no-value you get from P, is F."

"In other words, if something happens 25 out of 100 times, then out of every possible answer between '0' and '100', '25' should do best, when it comes to adding 25 yes-values of the answer to 75 no-values of the answer."

"Putting somebody into a situation like this is what makes their answer mean, 'How often do you actually think this happens?'  There's a lot of ways to put people in weird situations where their answer could mean something else instead, because the most rewarding answer they could give isn't the one that matches reality!  Civilization tries to avoid weird situations like that, so that our words and more importantly numbers go on meaning things."

Carissa Sevar: Carissa's not back to full health yet. She can feel it, even aside from the waking up panicked. But -

- it's like she can feel the thing she's reaching for, just barely out of her reach -

The most rewarding answer is the one that matches reality. That simple, and Hell - might genuinely not have stumbled on it - no, surely they have, at the highest reaches -

Ione Sala: Everyone in this classroom except for her is going to end up executed.  And then she'll also get executed, but she's not going to Hell.

Asmodia: Go master dath ilani thought.  Right.  Sevar had better actually have the pull to avoid them all getting tortured for heresy which, to be fair, it kind of seems she might.

Pilar : ...so on the one hand she is pretty sure this is not how Asmodeanism works, and, on the other hand, it seems pretty persuasive that this is how Lawfulness works, and, on the original hand, she is very sure that Asmodeus is Lawful.  Is she allowed to just think that she'll ask Aspexia Rugatonn about this later or is that cheating?

Project Lawful: Pilar's curse knows the answer to this one, actually.  Is it time for Pilar to hear?

Pilar : Pilar is absolutely not taking theological advice on this subject from Cayden Cailean, especially before Pilar knows the actually correct answer the Grand High Priestess will give.

Project Lawful: No problem!  Chaotic Good tries not to force answers on people that they don't want!

(It's how Asmodeanism can work for someone if they'd turn down Elysium because they actually wanted to go to Hell.)

lintamande: Meritxell isn't sure there's a contradiction here. 

There might be, but - you're not supposed to reason about everything the same way. Obviously you're not supposed to use numbers-reasoning to consider matters in which the Church has instructed you; it's for matters where you can't otherwise figure out what you're supposed to believe.

At least she hopes that's it.

Keltham: Keltham gives those words some time to settle; he's kind of guessing that nobody in Golarion has ever before considered the notion of incentive conditions under which talking becomes communication.

"So now, of course, we have the problem of finding any scoring rule which has this lovely and desirable property."

"Well, and we'd also like it to have the property that, when something happens, the higher the number you assign to it, the higher your score; or, when it doesn't happen, lower numbers get higher scores."

"So long as we're making up a wishlist, we'd like the method to still work if somebody says '17.3 out of 100'."

"We can take advantage of a symmetry which is that, if something happens 47 out of 100 times, that means it doesn't happen 53 out of 100 times.  So the yes-score of 47 should equal the no-score of 53.  Or to put it another way, the no-score of 30 is just the yes-score of 70; we don't need separate yes-score and no-score rules."

"And somebody named an important final condition earlier, does anybody happen to remember it?"

lintamande: "If a prediction breaks down into two separate parts, the points you get for the whole prediction being right should be the points you get for both parts being right," says Gregoria, who said it originally.

Keltham: "Hm hm, what sort of scoring rule could have that property?"

lintamande: Wizards aren't trained in this math at all. They blink at it frustratedly.

Keltham: If you literally do not need to know about logarithms to be a wizard, and gods can't tell Golarion about math on the order of the conjunction rule of probability, that substantially increases the chance that, in fact, the trick to synthesize your own spells is something on the order of 'invert a matrix so you can solve for start state given end state'.

Regardless.  He can work with this.

"If you can't solve the very abstract problem, make up a very specific problem and consider what the scoring rule would have to look like for that," Keltham suggests.

Carissa Sevar: Wizards aren't trained in this math at all but she's lots smarter now, and pure math is one of the things headbands are really good for. 

Gregoria's condition is trickier than it looks. Because what they want is for the scoring rule if it would award you 5 points for a guess a and 5 points for a guess b, to award you ten points if a and b are both true; but the chance of a and b both being true is their individual chances multiplied together, like how two coin flips is 1/4. There isn't anything that has that property - correction, there isn't anything she previously knew about that had that property. What could you possibly do to numbers -

"I think we need to invent a really weird thing to do with numbers to satisfy Gregoria's property," she says. "I'm imagining, uh, defining some property of numbers that scales up a steady amount when they grow by multiplication."

Keltham: ...okay, not bad.  You usually have to prompt a dath ilani five-year-old more simply and more extensively than that before they invent the concept of a logarithm, and they've been hanging around adults talking bits and decibels already.

"Can you give me an example of a few numbers and their weird-property-values such that the weird-property-values obey that rule?" says Keltham.

Carissa Sevar: "...how many powers of 2 fit in them?"

Keltham: "Okay, I give up, how the ass do wizards end up knowing about powers of 2 but not about the function for how many powers of 2 something is?"

Carissa Sevar: "Sometimes the spellsilver cost of an item grows by powers of 2. I've never seen one that grows by the function for how many powers of 2 something is."

Keltham: "Alllllll righty ighty then.  Though one observes that if there's a function from 'how powerful is this magic item' to 'how much money does this cost me', there is generally some reverse function that goes from 'how much money do I have available to spend' to 'how powerful of a magic item can I get'."

"Anyways, I propose that what you want is a Mysterious Function with the following property:"

Whiteboarding:  \x y.  MF(x*y) = MF(x) + MF(y)

"And, again, can you make up some particular xs and ys and MF-values that obey this rule?"

Ione Sala: "x is 3 and has a score of 1, y is 4 and has a score of 2, x times y is 12 and has a score of 3."

Carissa Sevar: "Doesn't work if x is 3 and y is 3."

Ione Sala: "Okay, x is 3, score 1, y is 4, score 3, x times y is 12, score 4... you're going to say that 81 should also have score 4, then.  Okay, I'm not really seeing how to do it if it's not just powers of 2."

Carissa Sevar: "I think we want to count part-powers-of-two somehow except I don't actually know how."

Asmodia: "Prediction," Asmodia says.

Message to Keltham:  If 2 is 1, 9 should be a little bit more than 3, since 8 is 3, so 3 should be a little more than 1-and-a-half.  Should I tell them that?

Keltham: "You're good to repeat that."

Asmodia: Asmodia repeats it.

Keltham: Keltham writes it down:

score(2) = 1score(8) = 3score(9) = 3 + a tadscore(3) = 1/2 * score(9)

Carissa Sevar: So what they need is a rule for the leftovers after you take out the powers of 2 which behaves the same way as the bigger 'take out the powers of 2' rule. Can you....take out powers of something smaller? No, that doesn't feel like it'd work - it treats the places where numbers are whole as different, the real answer won't do that... Can you...define how close, in a multiplying way, the bit leftover is to being another power of 2?

Keltham: Keltham will write some more questions!  Asmodia, give them two minutes and then you're allowed to start telling them.

score(2) = 1score(1) = ?score(1/2) = ?score(1/4) = ?score(2/3) ≈ ?score(99/100) ≈ ?score(0) = ?

Asmodia: Asmodia will take out her own spell-timer pocketwatch and deliberately start looking at it.

Carissa Sevar: Yes, the new puzzles on the board do precisely crystalize the question, they just don't suggest how you answer it.

(But if Asmodia figured it out already then it can't be that hard.)

What does it mean, to find the one halfth power of 2. ...well, presumably, it's the number that multiplied by itself makes 2. ...what does it mean, to find the 99/100th power of 2. 

"Okay, I think it just works to have fractional powers of 2," she says, "so you can use the powers of 2 rule all the way through."

Ione Sala: Ione has that score(1) is 0 and score(1/2) is -1.  Meritxell yells that score(1/4) is -2 before Ione can finish her next sentence.

Asmodia: Time's up!

Score(2/3) is half of Score(4/9), which will be a bit less than Score(1/2), so a bit more - uh, a bit less than negative 1/2.  Score(2/3) = a bit less than -1/2.

Score(99/100) is a bit less than 0 and she's not sure about score(0), she keeps wanting to think score(0) = 0 but score(1) is already 0.

lintamande: Most of the rest of the class is not going to get logarithms with three minutes of discussion and is writing down the answers while somewhat lost!!

Keltham: "Okay, so in dath ilan, everybody in this class would have been sorted here out of thousands of candidates, based on really fine-grained predictions that caused everybody to finish getting the problem within roughly the same minute.  In broken Cheliax schools for people with average Intelligence 10, everybody who falls slightly behind is... left to sit in total confusion for the rest of the class while effort gets focused on the people who are ahead?  Am I missing something there that is more clever than it sounds?  If it's not always the same people who are ahead, we're going to end up with a class no one member of which has all of the pieces.  Fully thirty seconds of thinking about this on my own part has so far failed to yield a brilliant solution."

lintamande: "...mostly it's the 'left to sit in total confusion for the rest of class' thing," says Pela. 

Keltham: "Yeah, so, what about not that."

"How many Rat's Cunnings do we have available to tap people with?"

lintamande: "We ...mostly didn't prepare spells this morning," says Gregoria. "We got introduced to the High Priestess and then went to breakfast and then, uh, watched Carissa prepare fourth circle spells."

Carissa Sevar: Sorry not sorry.

Keltham: 12 minus Carissa Asmodia Meritxell Ione equals eight.  "Security, I don't suppose there's eight Rat's Cunning spells going spare around here?"

lintamande: Security consults each other via raised eyebrows. "We have three of Fox's Cunning," one of them says after a pause.

Keltham: "All right.  This is not going to be the last time we run into this issue while we wait on intelligence headbands for the class, and even then, if I've got this right, they'll just be +2 headbands, so here's my baseline policy proposal to" meliorize "improve from there:"

"This time, I spend extra time tutoring everyone who isn't Carissa Asmodia Meritxel and Ione on 'logarithmic' functions.  I see how much further we can get based on that, then we take a longer lunch break than usual so everyone can prep spells and in particular prep the crap out of Fox's Cunning.  If there's second-circle spells you'd usually want to have, and won't be getting because of this policy, and they're spells a cleric can cast for you, maybe tell me and I'll start praying for those.  Security, I'm not sure what kind of collective resources all the wizards at this installation have in the way of second-circle spells, but I request at least sixteen Fox's Cunnings available for us to allocate on future days.  Thirty-two would be better."

dath ilan: Civilization sometimes sorts people having difficulties to easier classes, which will, of course exist and be optimized for that purpose to the limit of what very smart people can manage.

It doesn't leave them sitting confused.

There's a saying about cryopreservation which is "Civilization doesn't leave anyone behind."  It's sort of a Good saying, so Keltham's not saying it in Cheliax, but the thought did run through his mind.

lintamande: "I'll report that and we can see what we can do, but we only get personal use spells once all the emergency response needs of the installation are met and I don't know if it'll add up to thirty-two." Obviously implicit, to a Chelish listener: and you're commandeering all our personal use spells, you know.

Keltham: This particular subtext is also very legible to Abadar clerics, who are not famous for expecting free services that nobody has to pay for.

"I could be wrong, having not tried it either way let alone both ways, but being able to brute-force bottlenecks like that and keep the class unified, seems like the sort of thing that could easily correspond to a factor of 1.5 speed difference in our work.  The work that is, in fact, the reason this installation exists in the first place."

"I submit a request for - temporarily, until Intelligence headbands arrive - stationing additional wizards here, second-circle or higher, collectively able to supply sixteen Fox's Cunning per day, or to make up deficits in emergency response capabilities produced by reallocating the second-circle spells of those wizards who possess adequate security clearance to be in direct contact with us.  Thirty-two is better.  If that can't happen by tomorrow, I request at least twelve Fox's Cunnings that day collectively among available Security wizards."

"If I had an actual budget I would be asking how much it cost inside that budget to compensate you for any time or inconvenience, or hire those additional wizards; but having an actual budget with line items is not a way that Governance seems to currently be trying to relate to me, and so I can only ask Governance for stuff."

lintamande: "I'll submit that to the site manager." He heads out. 

Carissa Sevar: "Should the four of us go to the library and talk among each other while you cover the rest of the class, or should we stay."

Keltham: "Outside view on the way similar events have previously played out for us predicts that I'll say fifteen different things I wish you'd been present to hear."

Carissa Sevar: Indeed. She sits at her desk and puzzles over what the score for zero is.

Keltham: Dath ilani kids, before they run into logarithms, have prior experience with seeing numbers as bags of prime factors.  Maybe running over that for a few minutes will help with priming this pump?

15 is a bag of a 3 and a 5.4 is a bag of two 2s.15*4 = 60, so 60 is a bag of two 2s, a 3, and a 5.If you multiply 2 and 3, you get 6.So if you divide 60 by 6, you should get a bag of one 2 and a 5.2 times 5 is 10.  Checks out, right?

Now make up your own bags with numbers and play with that to see if your reasoning by bags-of-factors gets you the right answer.

Well, sure, you can use 4s as factors and see what happens?  But if you want to turn numbers into unique bags of numbers, each number in the bag has to not be made up of any numbers smaller than itself.

lintamande: This the students can follow along with. - the application to creating a scoring rule is not clear.

Keltham: Does it help if he mentions that a bag of 1.58496 2s is almost exactly 3?

lintamande: ....how would you possibly figure that out if you didn't know it, though.

Keltham: Did anybody happen to learn calculus since Keltham mentioned they should do that?

lintamande: Yep!! It was one of the major things they did while Keltham was at the Imperial Palace, along with triage on the library.

....they didn't get very far, since that was only one day and they didn't have textbooks or anything.

Keltham: That's weird, he'd expect significant progress on calculus if you were spending a significant part of a day on it.

How were they studying calculus at all without textbooks?  Tutoring from somebody?

lintamande: Some of Security knew some and were willing to trade favors once they were back alive. 

Keltham: Those favors need to be charged to the project budget somehow.

Onward then!  They're going to need calculus anyways, to get all the way through proving that the logarithmic scoring rule works correctly, and the calculus you need for that exact thing shouldn't be hard to teach in a few minutes even if Keltham has to do it from scratch.  But let's keep the focus on logarithms for now.

So first of all, remember that Asmodia had already worked out that since 9 is a bit more than 8, there should be slightly over three 2s inside a bag of two 3s.  So 1.58496 2s inside a bag of one 3 shouldn't be surprising.

And is that one fact Asmodia found, going to be the only fact like that which exists?  Three 3s is 27, and two 5s is 25, so there should be slightly less 2s in a bag of two 5s than in a bag of three 3s.  Say there's a thrice-bit-more than 4.5 2s in a bag of three 3s, then a little fewer 2s in a bag of two 5s, so there ought to maybe be 4.5 2s in a bag of two 5s and 2.25 2s in a bag of one 5.  The actual number is 2.32193 or so, which is, as one would expect, a tad more 2s than are in a 4.

You could also notice that a bag of seven 2s is 128, and a bag of three 5s is 125, so you'd expect a tad less than 7/3 2s in one 5, which would give you an estimate of 2.333... 2s per 5.  Not far off at all, right?

Yes, Keltham is writing this down on the whiteboard:

3*3*3 = 27   <=>   log3(27) = 35*5 = 25      <=>   log5(25) = 22*log2(5) = log2(25)  ≈+  log2(27) = 3*log2(3)3*log2(2) = 3  ≈+  log2(9) = 2*log3(3)log2(3)  +≈  1.5actually log2(3) ≈ 1.584962*log2(5)  ≈  3*1.5 = 4.5log2(5)  ≈  2.25log2(125) = 3*log2(5)  ≈+  log2(128) = 7log2(5) ≈ 7/3 = 2.333...actually log2(5) ≈ 2.32193

Now there's cleverer ways to compute this once you actually get calculus.  But it so happens that 3^12 = 531,441, and that 2^19 = 524,288.  There's slightly more than nineteen 2s in a bag of twelve 3s.  So you'd expect log2(3) to even more precisely be a tad more than 19/12, which will be 1/12 more than 1.5, so 1.58333, which is nicely closer to the true 1.58496 than the previous estimate of 1.5.

Problem time!  If you happened to have memorized the figure of 1.58496 2s per 3, you could derive that log2(8/9) ≈ -0.08496*2, for purposes of scoring a prediction of 8/9 on something that actually happened.  So score(8/9) is about -0.17 'bits', to borrow the Baseline term.  Does anybody see how that figure gets derived?

lintamande: There's a lot of silent scribbling. 

Well, says Gregoria after a bit, log2(8/9) is log2(8) + log2(1/9) - that's the entire desirable scoring property that got them on this horrible tangent in the first place.

And log2(8) is 3.

And log2(1/9) is going to be negative, fractions always are. log2(1/2) was -1. log2(1/4) was -2. log2(1/8) is going to be -3, and log2(1/9) is going to be - log2(9).

She doesn't actually know why this works but she can see that 3 - (1.58496)*2 is about -.17. 

Keltham: Sure.  It's just saying that you have to take around 0.17 2s out of a 9 in order to get an 8.  9 × 8/9 = 8.  3.17 2s minus 0.17 2s equals three 2s so an 8.  8/9 just literally means the number you multiply 9 by in order to get 8, so it's the number you multiply by to take 0.17 2s out of the bag.

If it's a probability of something happening 8 times out of 9 it's the same number and will score the same way, according to the scoring rule that counts 2s in things.  Which is the scoring rule that gives you the same cumulative score whether you assign 1/4 to two events, or 1/16 to their product event.

lintamande: - nods.

Carissa Sevar: Message: If you don't actually understand it don't act like you do.

lintamande: "I don't understand why taking .17 of a 2 out of a bag of twos is a thing you're allowed to do," says Pela.

Keltham: "Well, look at it this way.  A 16 is a bag of two 4s.  What happens if you take half a 4 out of the bag?"

lintamande: "....you take a two out, and now you've got a bag that multiplies up to eight."

Keltham: "A million is a bag of two thousands.  What happens if you take a third of a thousand out of the bag?"

lintamande: "I don't know what a third of a thousand is. ...I mean I know what it is when it's 333. But it's not, here."

Keltham: "A thousand is a bag of three tens.  What happens if you take a third of three tens out of a bag of twice three tens?"

lintamande: "You have five tens left in the bag. So - a hundred thousand."

Keltham: "Yep."  His smile goes away after a moment; it's impossible to have any sense of how well this is going when everybody is supposed to learn this at age five or six and they're adults.

"Well, if you can take half of a four out of a bag of fours, and a third of a thousand out of a bag of thousands, why not take 17 100ths of a 2 out of a bag of twos?"

lintamande: "....I guess."

Keltham: "I mean, there's the problem of figuring out that taking out 0.17 twos from a bag works out to multiplying the contents by roughly 8/9, but you can get that fairly precisely off nineteen twos being a bit less than twelve threes.  Possible self-study problem: rederive that yourself, convince yourself of it, prove it, without looking back at the whiteboard."

lintamande: "Now, or after class?"

Keltham: "How many people in this group think that's now so obvious that there's no point in proving it themselves?  Because if the answer is no, then yeah, maybe everybody pauses and tries to rederive the logic."

lintamande: Most of them don't in fact find it so obvious there's no point in proving it!

They get to work on that. 

Keltham: This would otherwise be a good time to Message Carissa to ask how he's doing teaching-wise, but apparently Asmodia, Ione, Meritxell, and Carissa also think that rederiving this claim is a good exercise for them to do.

Possibly Keltham is overcorrecting for how many fewer exercises ought to be required to grok logarithms if you first encounter them as an adult rather than a five-year-old.

Carissa Sevar: When Carissa was a five year old she required one on one tutoring from her mother to have enough attention for anything at all complicated, constantly forgot things that ought to be in working memory and needed reminding of them, and had about ten minutes' attention span for actual thinking. Trying to teach her math in a group would have been a disaster. 

Anyway, what is 8 9ths as a bag of twos. 

dath ilan: Dath ilan didn't say it was easy to teach it to five-year-olds.  Civilization is staring at that problem and optimizing it roughly as hard as Civilization ever optimizes anything.

Figuring out how to have logarithms be fun to learn about starting one month earlier on maturation timelines is a perfectly respectable accomplishment for a +4sd researcher's entire life's work.  If any single individual made a discovery like that singlehandedly, it would get them well into the 'more money than one person can reasonably spend on themselves' category of rich.

lintamande: Meritxell scribbles until satisfied that the log of (8/9) is going to be the difference between the log of 8 and the log of 9, and then until satisfied that that difference is the difference between 3 and 2*log(3), and then looks around for someone who looks stuck and helpfully helps! Paxti, are you stuck?

Iarwain: Paxti has worked out that 8/9 is 0.888.  She's worked out that nineteen 2s is 524,288 (by multiplying by 8 repeatedly to get to 18 2s, and then doubling the final result), which got her an answer that could've maybe been on the whiteboard she can't look at.

Paxti is currently working on computing twelve 3s via assembling a bag of six 9s.  After that she's going to divide out 19/12 the long way.  Maybe if she computes all the numbers Keltham said to compute, it'll be obvious once she's computed them how to put them together.

lintamande: Message: hey, free hint, all of that's stupid. Further hints available for sale.

Iarwain: Message:  Keltham wandered over to look at what I was doing and nodded approvingly.  Fuck off.

Iarwain: A bit later on Paxti has managed to get 9^6 = 531603 (close enough), and 19/12 = 1.58something.  Now she just has to figure out how that all fits together with 8/9, or three 2s and two 3s.

Nineteen 2s equals twelve 3s.  You need more 2s than 3s to make up something, so that makes sense.  19/12 is the number of 2s in a 3.  There'll be 2*19/12 2s in a 9, so it's 2*19/12 = 19/6.  Subtract 3 2s for the 8, and... 19/6 - 3 = 19/6 - 18/6 = ...

Message Keltham:  I get that there's exactly 1/6th of a 2 in an 8/9, does that make any sense?

Keltham: Keltham will come over and check how she arrived at that conclusion, but will soon approvingly inform Paxti that if, as is not actually the case, 2^19 exactly equalled 3^12, then yes, there'd be exactly -1/6 2s in 8/9.  Please observe that -1/6 is -0.1666... or about -0.17.

To try to see it a glance, consider that if there's 19/12 of a 2, in one 3, that's 1/12 more of a 2 than 1-and-a-half 2s:  19/12 = 18/12 + 1/12.

So in a 9, there should be 2/12 more 2s than in 8.  Though it's actually a bit more, because 3^12 is greater than 2^19, so there's a bit more than 19/12 2s in a 3.

Carissa Sevar: It might be good to give everyone another similar problem, Carissa tells Keltham. To check if they really get it.

Keltham: Three 5s is a tad less than seven 2s; tell me how to score a prediction of 2/5 on something that actually happens.  Sanity check, 2/5 is a noticeable-chunk less than 1/2, so your score should be a noticeable-chunk less than -1 2s.

Ione Sala: If you're a worshipper of Lord Nethys, you can by this point work out in your head that the answer is -4/3 and then write it down on the paper without any visible work accompanying it.

Carissa Sevar: 2/5 is log2(2) - log2(5), so that'll be 1 - something a little greater than 2. If there were 7 2s in 3 5s, then there'd be 7/3rds of a 2 in a 5, so it'd be 1 - 7/3rds. 

lintamande: Everyone is mostly keeping up now, though at obviously varying speeds.

Keltham: Well obviously this learning experience is Done, then!  At least Keltham figures that's how it should work if you don't need to just spend a bunch of time playing around waiting for your brain to mature slightly further.

They should probably all stand up and walk around and eat a tiny snack though.

Pilar : Pilar has some appropriately tiny snacks in her bag.  Cayden Cailean apparently has nothing better to do with the power He received from the Starstone and all His worshippers.

lintamande: Sure, snacks! Even if they're from Cayden Cailean and therefore kind of weird.

Keltham: "This may be a dumb question, but is there any connection, however distant, between the snacks and the Elysium thing?"

Keltham previously knew 2 facts about Pilar, her trip and her fetish, and he was already suspecting there would ultimately prove to be some connection between them (trope-wise, not causally).  Now he has noticed a third fact repeated twice:  Pilar has candy.

Carissa Sevar: Can Pilar be placed in my Telepathic Bond, please, and this fact concealed. 

Yes, almost definitely; they're both the product of some intervention by Cayden Cailean which we think is in support of the project but we're not entirely sure honestly.

Pilar : "Almost definitely yes?  Um, according to what was found out yesterday, the snacks are a product of an intervention by Cayden Cailean, which people seem to think is in support of the project, and Cayden Cailean is Chaotic Good, which is the alignment on Elysium, but nobody's sure about anything."

Keltham: "Wait, so a god intervened to support our project with tiny snacks?"

Carissa Sevar: That would be very stupid, which is why Cheliax's very smart people are confused. Some possibilities: it's actually stupid! Cailean is called the Drunk God etc etc Pilar can fill that in with true stuff. Alternatively, it's meant as a form of communicating that Chaotic Good is backing Asmodeus here. Alternatively, it'll turn out to actually be important for some reason, say if the project is besieged, or if everyone is nutrient deficient on something that the snacks contain. Alternatively, there's some kind of preexisting god agreement which happened to cash out like this.

Pilar : "Repeating back some things that have been said to me:  That would, in fact, be incredibly stupid, the phrase fucking Chaotic fucking Good does come to mind, Cayden Cailean is called the Drunk God because he did the Starstone on a dare while drunk.  But it could also be to show that Chaotic Good is backing Asmodeus on this, or it could be important for some reason like we get beseiged and have to survive on my snacks or everyone ends up deficient in something the snacks contain or there could be a pre-existing god agreement that happens to imply this -"

Keltham: "Could you maybe have mentioned this earlier?  Could you maybe have mentioned this before handing out the snacks?"

Pilar : Seems to demand a fast response.  "There literally hasn't been time since I saw you at breakfast... this probably seems even weirder if you're not from Golarion, doesn't it."

Keltham: "It actually does!  And I would in fact like to be told about all divine interventions on my project as soon as they become known to Governance moving forwards!  Are there any more?"

Pilar : "I just know about Asmodeus on the project twice, Nethys on Ione, and Cayden Cailean on me... uh, sorry?  I think you sort of expect me to know what you've already been told and I don't actually?"

Keltham: "Is everything on Golarion this poorly organized from a management perspective?"