"Yeah, so, what about not that."

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

"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."

Sorry not sorry.

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

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

"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."

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.

"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.

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."

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

"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."

"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."

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

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.

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

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

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

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

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.

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?

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

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) = 3
5*5 = 25      <=>   log5(25) = 2

2*log2(5) = log2(25)  ≈+  log2(27) = 3*log2(3)
3*log2(2) = 3  ≈+  log2(9) = 2*log3(3)
log2(3)  +≈  1.5
actually log2(3) ≈ 1.58496
2*log2(5)  ≈  3*1.5 = 4.5
log2(5)  ≈  2.25
log2(125) = 3*log2(5)  ≈+  log2(128) = 7
log2(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?

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. 

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.

- nods.