Keltham is starting to suspect that Chelish wizards do not routinely memorize 12 factorial (479,001,600) and didn't recognize the number when he said it, which may make this problem harder to mentally chunk.

In which case they couldn't have studied a lot of combinatorics??  Keltham would really have guessed that 'this bit of spell with 12 elements has 479,001,600 possible conformations' would be an important chunk of spellcraft, unless things only work at all when there's only 1 possible conformation.

Maybe you don't get to that part at second circle.

Or maybe - this is a weird thought, but Keltham is starting to feel suspicious of a trend - Cheliax teaches combinatorics in some incredibly narrow way where they've only learned combinatorics for spells and not combinatorics for everyday life??

This probably isn't the most important thing right now, file it with the other 'Why are they so inconsistently X??'

"Correct, but I'm not sure everyone was following along with that, so let's try a smaller scale version.  Suppose I took four of you, lined you up in a randomized order - you can imagine it being visibly randomized, if you like - and gave 8 jellychips to whoever was standing second in line.  On average, how many jellychips should you expect to receive if I run this procedure on you?"

"Two," they chorus cheerfully.

"How could the answer possibly be two?  There's four times three times two times one ways to pick the first person in the line from four people, the second person in the line from the three remaining people, the third person in the line from two remaining people, and one way to tack on the last person in the line.  Four times three times two times one is 24.  You get 8 jellychips at the end, if you get any at all.  So the answer is going to be something divided by 24 different possibilities, maybe 8 divided by 24 or something like that, so the answer should be one-third.  Or something with thirds in it, anyways, because you're dividing by 24, which has a factor of 3 in there."

They stare at him warily.

"You're second in line a quarter of the time," says Tonia. "So it's two." Probably dath ilan does this kind of thing because of it being illegal to light anyone on fire so they have no other outlets.

(Illegal isn't quite the same concept when you don't have threats; but lighting somebody on fire would get you barred from most cities, yes, since most cities contain people who prefer not to be lit on fire.)

"But how... does one obtain... that result?"

"You take the payout, which is eight, and you multiply it by how often you get the payout, which is a quarter of the time, and eight times a quarter is two."

Keltham furthermore suspects that Chelish education may also possibly put more emphasis on guessing the right answer for spell problems than on proving the answer correct.  Which there's obviously a place for!  In fact, if he were to treat them as kids, an old dath ilani rule implies that Keltham needs to find a problem that forces them to use a more rigorous method, rather than complaining that the correct answer was obtained too quickly.  You are not allowed to tell a child 'That answer was correct but I want you to obtain it my way instead of your way,' that is not good for kids.  And it's not actually clear to Keltham if that rule is supposed to hold relative to absolute age or to mathematical maturity.

"If there's twenty-four different ways to stand in line, how does it end up that you're getting a payout one quarter of the time?" Keltham tries instead.  "Shouldn't it be more like 1/24 or something like that?"

"There's not twenty-four different ways to stand in line! There are four places you can be in line and then you don't care what the other three kids are doing."

"I am supposed at this point to find some actual problem which forces you to compute it out the long way, instead of complaining that you got the correct answer but you didn't get it the way I wanted, which I am not supposed to ever do.  But I don't have a workbook full of carefully composed problems like I would if this were a real lesson, unfortunately."

"If we were trying to figure out your marginal contribution to a more complicated economic situation, though, the particular people ahead of you in line might be important -"

"You know, I should just give you a simpler problem that forces you to compute it the long way.  Let's say there are three tokens with numbers that say 2, 3, and 5.  Bringing a group of tokens together gives the group a number of jellychips equal to the product of every number in the group, so if you had the tokens for 2 and 5 together, the group would receive 10 jellychips."

"What does this method say is the fair distribution to the holder of the 5 token, if three token-holders pool 2 and 3 and 5 to get 30 jellychips for the group?"

"So you sum up adding the five to nothing, adding the five to the two, adding the five to the three, and adding the five to the pool with the two and the three," says Meritxell, "and that's everything the five could possibly be worth in every world, and you divide by how many worlds there were."

"Or if you actually bother to do the work, 5 plus 10 plus 15 plus 30 divided by four," says Ione.  "So 15."

Asmodia rolls her eyes.  "Really.  What do the other two tokens get, then?  The 2 and the 3?"

Ione suspects a trap, and tries to rapidly work it out in her head.  For the '2', it's 2 + 6 + 10 + 30, divided by 4, which is... damn it, this is harder to do in her head... 12?  And for the '3', it's 3 + 6 + 15 + 30 = 54, divided by 4 is no it doesn't matter it's not all going to add up to 30.  "Wait, I see my mistake -" Ione begins.

"Mistakes.  Plural.  The divisor is 6, not 4, and you're supposed to sum over the marginal productions rather than the total productions.  If it's ordered 5-3-2, that's a marginal production of 5.  If it's ordered 5-2-3, that's a marginal production of 5.  If it's ordered 2-5-3, the product starts at 2, and goes to 10, which is a marginal production of 8.  3-5-2 goes from 3 to 15, marginal production 12.  2-3-5 and 3-2-5 go from 6 to 30, marginal production 24 repeated twice."  Asmodia has been writing down these numbers, thank you, she is not trying to keep it all in her head without a Fox's Cunning.  "5 + 5 + 8 + 12 + 24 + 24 = 78, divided by 6... 13."

She quickly checks the other two numbers to make sure she's got it right.

2:  2 + 2 + 3 + 5 + 15 + 15 = 42 / 6 = 7
3:  3 + 3 + 4 + 10 + 20 + 20 = 60 / 6 = 10

13 + 7 + 10 = 30.  Okay, she didn't just make a (tiny bit, unimportantly, bigger) fool of herself.

(Is she playing at anything, by being prominently the best at math today?)

The other students are trying as hard as they can at math.  They don't believe themselves to have been instructed by you to diminish their math efforts as such.  Asmodia is just better at this problem.

(Unfortunately.)

"Yeah, the thing I was trying to force you to do with the four students in twenty-four possible orders was sum over the 6 possible ways you could be standing second in line, to make the point about how the sum is defined as being over every permutation.  In retrospect, clearly, I should've started with the case of tokens labeled 2, 3, and 5, but I'm sort of making this up as I go along because it's been a few years and I don't remember some of the exercises let alone their ordering.  Sorry about that.  Anyways -"

"When you're trying to see if there's a way to do what ideal agents would do - or gods, if you think gods are powerful enough to be ideal about that particular case - you want to distinguish the Law that defines what the solution is, and any clever ways you come up with to compute the Lawful solution faster."

"When you've got 12 identical tokens, such that any group of 11 or 12 of them will produce 12 jellychips, there's a symmetry argument which says that each token must get one jellychip.  If you thought there ought to be a coherence constraint on the Law of fairness saying that holders of identical tokens should end up with identical payouts, you could use that to compute the answer even if you had no idea what the actual Law was.  Often when you do see how the Law works, you can go back over a lot of your intuitions, and say, 'Oh, yes, that intuition I had previously was shadowing this coherence of the Law, even though I didn't know how the whole Law worked' and that's a kind of sanity check on whether you're reasoning correctly at all."

"But the Law of fairness that defines the target answer for the '11 tokens of 12' problem is in principle a sum over 479,001,600 marginal productions, of which all but 39,916,800 are zero, and 39,916,800 of which are 12, divided at the end by 479,001,600.  Which means that we can say there's a single ideal fairness formula that governs both the '11 of 12' game, and the '2, 3, 5' game, even if shortcuts or approximations for the particular cases of the formula can be different, in cases where a shortcut exists."

"Which does imply that identical tokens will get identical payouts," says Meritxell. "Right?"

Carissa does not want the kids to be bad at math. Imitating being bad at math seems like another thing where the things Keltham would expect to be correlated won't be and he'll end up suspicious, which is almost definitely happening anyway but at least since it's the product of their real legitimate math education it'll make more sense to him as he learns more. 

Carissa wanted to know whether Asmodia was being impressive on purpose because an Asmodia who is trying to get Keltham's attention, or an Asmodia who is trying to be hard for Cheliax to replace - an Asmodia who has started playing for her survival against the project's interests, more than everyone in Cheliax is doing all the time - is a different problem than an Asmodia who is doing her best but bitter because she had been consoling herself that Cheliax was lying about Good and they turned out not to be. She thinks a disillusioned angry-at-Good Asmodia is probably usable. She is open to learning from someone with more experience with this, though.

This is her being weak and reactive, not strategic.  And she's quite pissed at the Good gods, yes.

(Security doesn't explain why.)

It's not really the kind of thing that requires explanation! The Good gods suck. 

Carissa tries to think what Maillol will think if she tells him that she wants to try to talk Asmodia around. It would be nice if she could predict what Maillol thought about things so she could stop bothering the real one so often, but he still surprises her as often as not, and she isn't sure if he'll think this is Carissa being inexperienced at having a real command and accordingly stupid, or Carissa having weirdly good instincts because Asmodeus dropped Keltham near her for a reason. 

...she should focus on the lesson or she's going to get behind. And then Keltham will think she's kind of stupid, which ....might be good, if it means he thinks she's not a ringleader, but would interfere with attachment to her, she's pretty sure. Lesson it is.

"Yup.  Identical tokens getting identical payouts is one of several coherence properties that this solution has, called 'equal treatment of equals'.  Another example of an obvious coherence property is that the sum over every agent's fair distribution equals the total distribution - we don't have any jellychips left over.  Yet another coherence property is that combining two games into a single game will make the agent's fair reward be the sum of their fair rewards in the component games.  Or another obvious-sounding one, if your marginal production is zero for every permutation, your fair reward is also zero."

"Would you say those four properties sound like properties that any fair formula for a game like this one ought to have?  Again, that's identical agents being treated identically, distributing all of the gains, the reward for playing two games is the sum of the reward for playing the games separately, and agents who contribute nothing receive nothing."

Those seem obviously true but there's still a suspicious pause while they try to think of counterexamples. 

"Did we get a technical definition of a fair formula such that 'split the rewards evenly', which does not have the last of those properties, gets disqualified?"