Emstar
The Shining One
- Location
- Trapped in an extradimensional rift.
- Pronouns
- She/They
@eaglejarl @Velorien @OliWhail
Assuming that Round 2 eliminations didn't get their word halves taken from them. And that that wasn't a typo. (I do this in the edit though!)
Heres the best idea I can come up with that uses some handwaving and some actual fancy math behind the scenes.
The Philosophy of Square-Root Cancellation:
We should expect square root cancellation in the partial sum of any arithmetic function that behaves "randomly".
I would expect about sqrt(600)*ln(600) =~ 150 word-halves left available in totality when we are done.
So we should have SQRT(600)*Log(600) =~ 150 word halves in play. Almost right down the middle.
Where that logarithm is to correct for error margins introduced when I waved my hands and invoked square root cancellation.
So, we have 60.
ISC + everyones partners have 12
(I think three or so of our collective teammates are missing or need to be missing word halves going into Event 5. Proctor said so in Chapter 194
One is Ikemoto. One is Lightning guy. One is sand. Solution: Other two were on a scratch squad on Shika's team. Handwavium activate).
There should be about 80-90 left, distributed throughout the other teams.
How to simulate:
Pick a handful of other teams and start rolling the good old d12 for how many extra they have!
Note: This is off by @Adept_Woodwright 's ORIGINAL by about 75. This is what I expect you would get if you add the additional assumption roughly half of the 150 eliminated ninja also got eliminated with their word halves instead of having it stolen and massage the rest of his calc to fit that. Plus minus some extra noise. I didn't look at his calculations BTW, I was too busy herding ghosts!
Edit: I think Adept is still not being sufficiently pessimistic about how many random factors could cause an elimination (not just ninja v ninja combat) and so is assigning slightly more weight to word halves being stolen and hoarded. But thats fine, people are allowed different assumptions. If everyone wrote the same calcs, life would be boring.
EDIT #2 AND WARNING: Do not apply the above in the case that Round 2 eliminations also lost their word halves. The nature of Event 4 and how it effects word half distribution is that it probably didn't change the distribution at all, really. Then half of the crowd is culled.
Having a second bottleneck event for word halves in play at the Event 4 mark is basically just cutting the available pool in half.
In this case: We'll use N/2 + sqrt(N). With N=~150 (as above)
This gives 75 + 12 = 87 word halves in the wild. Ours, Shikas, the other dudes, and a few leftover.
View this as:
1) Apply the above to get the "In Play" amount as of elimination #1. Then cut it in half (bit more than half of the contestants were eliminated) since the word half pool "in play" was cut down by 1/2 on average. Add sqrt(N) in to account for better teams still having a few extra.
or
2) Apply the sqrt cancellation twice.
Our first N is 300. We know this gives us ~150.
Second N =~150
Double (add the ghosts in), plug in. sqrt(300)*log(300) =~99
These give the same answer roughly, so this seems to be right.
The difference is whether you expect Event 4 to be relatively calm with respect to word half distribution, or whether you think the big players are gonna shake the box again.
or by more naive probabilistic methods
3)Lets pretend we start the exams at 150 genin with all their word halves on their person a few hours after Event 3. Put everyone in a big box.
Shake the box a bit, and apply event 4. I'd expect a bit more than half of the word halves leftover, half of the time.(and a bit less the other half. We likely didn't land here.)
Some of the word halves clearly fell out and into another genin's pocket during the box shaking.
Tsk tsk. Careless.
Assuming that Round 2 eliminations didn't get their word halves taken from them. And that that wasn't a typo. (I do this in the edit though!)
Heres the best idea I can come up with that uses some handwaving and some actual fancy math behind the scenes.
The Philosophy of Square-Root Cancellation:
We should expect square root cancellation in the partial sum of any arithmetic function that behaves "randomly".
I would expect about sqrt(600)*ln(600) =~ 150 word-halves left available in totality when we are done.
The word halves are sort of a give and take process that happens at random.
Some people lose some to proctors, some people trade them, some people get them stolen.
OTOH: Some people spend extra's and the influx of word halves into the "wild" could be heavily cut by round 1 elims.
So "# word halves in the wild" should behave like some sort of slightly excited random walk. It jumps up, then down, then up, but it jumps up slightly more than down.
Slightly more fun analogy:
Alternatively, imagine I'm standing in a center of a (radius 1) circle, and for each ninja I draw an arrow (vector) pointing in some random direction between -90 and 90 degrees, whose tip is on the edge of the circle.
For each ninja, I conjure up an Adversary Ghost Ninja! This represents the chance of fate or fell circumstance or other ninjaic interactions stopping their word halves from remaining in play! Perhaps the AGN chased them through the swamp all night, or made it so their merchant couldn't sell anything and they failed and got eliminated. perhaps it even caused them to say the word out loud like a complete moron.
(Some of the AGN's are very bad at their job, and they only sort of deliver misfortune and pain upon their counterparts. These guys might keep their word halves, but they also might drag their teammates down or something and get eliminated. )
I give each AGN an arrow, this time pointing somwhere between 90 degrees and -90 degrees on the other half of the circle. The angle is randomly chosen.
I do this for everyone!
Now I add all the arrows up. I take the magnitude. This tells me how many word halves are in play.
This normally gives me SQRT(N) where N= large # arrows.
But, I posit that ninja are slightly more likely to cause their word halves to "remain in play" than not. Its should be a bit rarer to get DQ-ed, Eliminated without yours being stolen, spending it, than it is to just get it stolen or to keep it.
Put simply:
Good ninja ain't afraid of no ghosts!!
Thus all of the arrows should be slightly skewed to the right. Correct this by multiplying with a big fat
Log(N), for the wiggle room. These guys are trying really hard to keep the word halves!
Thus, SQRT(600)*Log(600) =~150.
Some people lose some to proctors, some people trade them, some people get them stolen.
OTOH: Some people spend extra's and the influx of word halves into the "wild" could be heavily cut by round 1 elims.
So "# word halves in the wild" should behave like some sort of slightly excited random walk. It jumps up, then down, then up, but it jumps up slightly more than down.
Slightly more fun analogy:
Alternatively, imagine I'm standing in a center of a (radius 1) circle, and for each ninja I draw an arrow (vector) pointing in some random direction between -90 and 90 degrees, whose tip is on the edge of the circle.
For each ninja, I conjure up an Adversary Ghost Ninja! This represents the chance of fate or fell circumstance or other ninjaic interactions stopping their word halves from remaining in play! Perhaps the AGN chased them through the swamp all night, or made it so their merchant couldn't sell anything and they failed and got eliminated. perhaps it even caused them to say the word out loud like a complete moron.
(Some of the AGN's are very bad at their job, and they only sort of deliver misfortune and pain upon their counterparts. These guys might keep their word halves, but they also might drag their teammates down or something and get eliminated. )
I give each AGN an arrow, this time pointing somwhere between 90 degrees and -90 degrees on the other half of the circle. The angle is randomly chosen.
I do this for everyone!
Now I add all the arrows up. I take the magnitude. This tells me how many word halves are in play.
This normally gives me SQRT(N) where N= large # arrows.
But, I posit that ninja are slightly more likely to cause their word halves to "remain in play" than not. Its should be a bit rarer to get DQ-ed, Eliminated without yours being stolen, spending it, than it is to just get it stolen or to keep it.
Put simply:
Good ninja ain't afraid of no ghosts!!
Thus all of the arrows should be slightly skewed to the right. Correct this by multiplying with a big fat
Log(N), for the wiggle room. These guys are trying really hard to keep the word halves!
Thus, SQRT(600)*Log(600) =~150.
So we should have SQRT(600)*Log(600) =~ 150 word halves in play. Almost right down the middle.
Where that logarithm is to correct for error margins introduced when I waved my hands and invoked square root cancellation.
So, we have 60.
ISC + everyones partners have 12
(I think three or so of our collective teammates are missing or need to be missing word halves going into Event 5. Proctor said so in Chapter 194
One is Ikemoto. One is Lightning guy. One is sand. Solution: Other two were on a scratch squad on Shika's team. Handwavium activate).
There should be about 80-90 left, distributed throughout the other teams.
How to simulate:
Pick a handful of other teams and start rolling the good old d12 for how many extra they have!
Note: This is off by @Adept_Woodwright 's ORIGINAL by about 75. This is what I expect you would get if you add the additional assumption roughly half of the 150 eliminated ninja also got eliminated with their word halves instead of having it stolen and massage the rest of his calc to fit that. Plus minus some extra noise. I didn't look at his calculations BTW, I was too busy herding ghosts!
Edit: I think Adept is still not being sufficiently pessimistic about how many random factors could cause an elimination (not just ninja v ninja combat) and so is assigning slightly more weight to word halves being stolen and hoarded. But thats fine, people are allowed different assumptions. If everyone wrote the same calcs, life would be boring.
EDIT #2 AND WARNING: Do not apply the above in the case that Round 2 eliminations also lost their word halves. The nature of Event 4 and how it effects word half distribution is that it probably didn't change the distribution at all, really. Then half of the crowd is culled.
Having a second bottleneck event for word halves in play at the Event 4 mark is basically just cutting the available pool in half.
In this case: We'll use N/2 + sqrt(N). With N=~150 (as above)
This gives 75 + 12 = 87 word halves in the wild. Ours, Shikas, the other dudes, and a few leftover.
View this as:
1) Apply the above to get the "In Play" amount as of elimination #1. Then cut it in half (bit more than half of the contestants were eliminated) since the word half pool "in play" was cut down by 1/2 on average. Add sqrt(N) in to account for better teams still having a few extra.
or
2) Apply the sqrt cancellation twice.
Our first N is 300. We know this gives us ~150.
Second N =~150
Double (add the ghosts in), plug in. sqrt(300)*log(300) =~99
These give the same answer roughly, so this seems to be right.
The difference is whether you expect Event 4 to be relatively calm with respect to word half distribution, or whether you think the big players are gonna shake the box again.
or by more naive probabilistic methods
3)Lets pretend we start the exams at 150 genin with all their word halves on their person a few hours after Event 3. Put everyone in a big box.
Shake the box a bit, and apply event 4. I'd expect a bit more than half of the word halves leftover, half of the time.(and a bit less the other half. We likely didn't land here.)
Some of the word halves clearly fell out and into another genin's pocket during the box shaking.
Tsk tsk. Careless.
Last edited:
