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

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.

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."

...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.

"...how many powers of 2 fit in them?"

"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?"

"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."

"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?"

"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."

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

"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."

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

"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?

"You're good to repeat that."

Asmodia repeats it.

Keltham writes it down:

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

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

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

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

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 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.

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.

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!!

"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."

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