The Skill Point Formula Has Been Reverse Engineered

Well folks, it's been done. Though I'd been working on this some weeks ago, I never truly nailed the answer until today: the formula for extrapolating the effectiveness of a skill point based on the points invested in it.

Spoiler: THE ACTUAL DISCOVERY

Today we discovered the formula was something no one (presumably) had previously considered; a geometric series.

Spoiler: WHAT WAS KNOWN BEFORE

While there were incredibly accurate attempts prior, such as that of ferricles,
we had never been quite there until today. With hours of time poured into doing the math for this, we were finally able to determine the exact formula, which is the following:


Here's what you need to know about this formula:
- x is the number of skill points in a certain category, and E(x) is the effectiveness out of 100. This means that at x = 150, E(x) = 80.8.
- The round(x,y) function rounds the value of X to the yth digit. For example, round(1.6368, 2) = 1.64, whereas round(1.6368, 3) = 1.637.

Spoiler: THE MATH AND VERIFICATION PROCESS

While my original formula followed the format
,

I had decided after seeing the results of ferricles much more accurate equation that an exponential was the way to go. However, the idea of using a geometric sum actually just occurred to me while I had walked fresh out of my math class. With the thought fresh on my mind, I sat down and attempted to use the new form:

.
Using the built in logarithmic regression feature in Desmos, I was able to calculate that a = -1.00289 and b = .990801; while not the cleanest numbers, they net me an R^2 value of 1, or at least so close to it that Desmos rounded to that. After this, I decided that was good enough for me, and let it go for a while. This was around in May.

Fast forward to today, June 27, 2021, and I'm talking to epic moderator SayaDuck about something or other in the Wynncraft #questions channel. During this conversation I mention my old formula, thinking it's a neat function but nothing more. When Saya asked if mine was closer than ferricles, I went to his friend hppeng, the creator of Wynnbuilder. He suggested to me that it might be more accurate if a = -1. From here, I removed the .000001 on the end of b to get a = -1, b = .9908. When this new formula is rounded to the nearest 10th, you get an equation of perfect accuracy except for one edge case; x < 0, which would return -1% effectiveness instead of 0. However, all values above x = 0 were accurate. With a suggestion from iTechnically, I switched it from


to this

.

To verify it's accuracy, I used a program which would return false if the value output by my formula E(x) was not equal to the real value of the effectiveness at a given point. After writing this algorithm, the equation checked out; we had found it.

This wouldn't have been possible without the help of some lovely folks who I'd like to credit below with their discord names :)

- @ferricles#6344 for his equation, which opened my mind to considering exponentials as an option
- @hppeng#8609 for simplifying some numbers to increase the overall accuracy of the equation
- @iTechnically#4671 for simplifying it to account for x-values less than 0
- @Saya#8503 who pushed me to continue my work after I had given up
- @touhoku#8699 who helped me write this thread

:)

 

first?????

okay i have actually read it now. nice work! was cool to see this project at many steps and glad to see it done so neatly.

 

second???

at least we got the form right an attempt was made

nice job! programming go brr

 

Sometimes I wonder what kind of person would design a game this way, why they would subject us to all this arbitrary strangeness. Could it really be a mere mortal who created such an elaborate maze of contradictions and nonsense?

 

so many numbers???????

there is a joke here about me not understanding it and intelligence thresholds

 

This is a funi bloc game I did not expect college grade mathematics

 

Interesting that a is 0.9908. Intriguing that the team would pick this specific constant.
Still an interesting equation, makes a lot more sense than using a log (to me), as it would be easy to program, just using a for loop.

 

oinuk really is playing ETFA irl

 

I have absolutely 0 idea what any of this means

 

A lot of complicated math that's going to make online building tools like WynnData more accurate in the future

 

:Saltedpog: great work
Here's a Desmos link with the different function forms, and a table for verification https://www.desmos.com/calculator/btjkullbz2
As a sidenote, because this is the sum of a geometric series with n terms, there's a closed-form for it which is really nice to deal with (although it's inapplicable to n<0):

​

 

It's not that complicated.
To explain the formula:


The effectiveness of x skill points is equal to:
The sum of (0.9908 to the power of n), for every whole number between 1 and x.
Rounded to 1 decimal place.

So, the effectiveness of 5 skill points is:
0.9908 + 0.9908^2 (to the power of 2) + 0.9908^3 + 0.9908^4 + 0.9908^5 , all rounded to 1 decimal place.
i.e. 0.9908 + 0.9817 + 0.9727 + 0.9637 + 0.9548
Which equals 4.9 when rounded to 1 decimal place.

 

Regarding the part where x<0 did not work, coding-wise, you could make it work by including something equivalent to an elif command which then takes x and multiplies it by 0 (not that we could ever go below 0 anyway). If you wanted to make it positive, you could apply the absolute value, which would (again, coding-wise) be adding an elif command where x<0 causes x * -1 which would result in a positive number. Anyway, those things aside, this is an amazing find. Good job dude.

 

Great work, it's nice to see this finally solved.

 

welcome to wynncraft

 

And yet, the real mystery remains: Was this an admin overzealously balancing the skill point system, or a deliberate attempts to stop people from discovering it?

 

LFG it's an actual formula that makes sense

Congrats and gj Lennon

 

Your definition of complicated and mine differ greatly when it comes to mathematics. You explained it, and I still don't understand it. That's ok though, in the end I don't need to. I can still recognize the achievement for what it is, and see how it will benefit the community.

 

they did it... they finally did it