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  1. (If anybody knows a better place to put this post, please move it there) Hi all! This is my first post over on the reborn side of things, although to be honest I didn't really know where to put this ... Anyway! Before we go any further, a warning: This post contains maths. If you do not like maths, you will not like this post. But TL;DR, nature important. There's also a summary at the end, so maybe go read that too. ---------------------------------------------------------------------------- Introduction With that warning out of the way, let's get started. During a recent run of mine I caught this beautiful Zangoose called Disaster: I thought she was pretty neat, 26 Atk IVs and +Spd nature, despite only having 12 Spd IVs. However, I showed Disaster to a couple of people and got mixed reactions, with some thinking that she was absolutely trash because of the 12 Spd IVs. It got me thinking "I'm pretty good at maths...let's figure out just how good she is". So that's what we're gonna do here, but a little bit more generally. I'll try and answer the question: Just how good is a good nature? ---------------------------------------------------------------------------- Calculating the IV-Nature equivalent: A tale of floor functions To answer this question, we first need to know how a stat is calculated. Stats other than HP (which is irrelevant here, since we can't change it with nature) can be calculated using: Stat = Floor[ Nature * (Floor[ (2 * Base + IV + Floor[ EV/4 ]) * Level/100) ] + 5) ] Where Nature is 1 for a neutral nature, 0.9 for a hindering nature and 1.1 for a beneficial nature, whilst the floor function is defined such that it picks out the largest integer less than or equal to its argument. e.g: Floor[ 1.2 ] = 1, Floor [ -3.7 ] = -4. Ok great. So what next? To try and quantify how good or bad a nature is, I'll try to convert it into and IV equivalent. So let's suppose something has a neutral nature (Nature = 1) and IV IVs. I'll answer the question "for a different nature, which we will call Nature', what IV value, which we will call IV' , would give the same stat total?". Seems simple right? Let's get started. Well, there are a few problems here. First off, the floor function has no inverse so we can't just "solve for IV' ". But what we can do is use the definition of the floor function to say that: Stat <= Nature' * (Floor[ (2*Base + IV' + Floor[ EV/4 ]) * Level/100) ] + 5) < Stat + 1 Great! That's one of the floor functions gone at least. We can tidy up a little more and get the following inequality: (Stat/Nature') - 5 <= Floor[ (2 * Base + IV' + Floor[ EV/4 ]) * Level/100) ] < ((Stat + 1)/Nature') - 5 Aaaand, we're stumped again. Getting rid of this next floor function isn't very easy (trust me, I've tried...a lot). If anybody knows how to get the most general solution here then please tell me! Since I haven't been able to figure it out though, we'll stick with a more specific case. If we're at Level = 100, we see that everything inside the floor function is an integer, so the floor function goes away nicely. Leaving us with: (Stat/Nature') - 5 <= 2 * Base + IV' + Floor[ EV/4 ] < ((Stat + 1)/Nature') - 5 Which we can solve quite easily for IV': (Stat/Nature') - 5 - (2 * Base) - Floor[ EV/4 ] <= IV' < ((Stat + 1)/Nature') - 5 - (2 * Base) - Floor[ EV/4 ] But we want our new IV stat, IV', to be an integer. So we're going to stick in some floor and ceiling (the opposite of the floor function...great names I know) functions of our own in. Giving: Ceiling[ (Stat/Nature') - 5 - (2 * Base) - Floor[ EV/4 ] ] <= IV' < Floor[ ((Stat + 1)/Nature') - 5 - (2 * Base) - Floor[ EV/4 ] ] For those of you still reading, congratulations! We've found the range of values of IV' that will give the same total stat with a nature Nature' as something with IV and Nature. ---------------------------------------------------------------------------- A worked example A worked example is always nice to put these things into perspective and give you an idea of exactly what I've been trying to do here. Let's suppose we're ready to take on the champion with our trusty Armageddon: He's got perfect attack IVs and EVs, so we're set, right? Well, let's find out just how good Armageddon is using what we've learnt so far. With those stats and that nature, Armageddon has: Stat = 219 Base = 60 EV = 252 Using what we derived so far, this means that if instead of a neutral nature Armageddon had a beneficial nature, Nature' = 1.1, he'd have the same stats with: IV' ~ 12 Yep, that's right! Stated in words: "Armageddon with 31 Atk IVs and a neutral nature is the same as a Torchic with 12 IVs and a beneficial nature." Oops! Looks like Armageddon isn't so perfect at all... Better get back to breeding! ---------------------------------------------------------------------------- Calculating the Base Stat-Nature equivalent: It's easier this time...because we've done it already We could have asked the question above in a different way: "for a different nature, which we will call Nature', what base stat value, which we will call Base', would give the same stat total?". All we're doing here is saying that instead of treating the beneficial nature like having more IVs, we could also picture it like having a higher base stat to start with. Given what we've done so far, this isn't too hard to calculate. We just rearrange what we have above to find: Ceiling[ (1/2) * ((Stat/Nature') - 5 - IV - Floor[ EV/4 ]) ] <= Base' < Floor[ (1/2) * (((Stat + 1)/Nature') - 5 - IV - Floor[ EV/4 ]) ] That wasn't so bad now, was it? So what does this mean for our trusty Armageddon? Well, just like before we can go ahead and plug in the numbers for Armageddon: Stat = 219 IV = 31 EV = 252 And we find using the above: Base' ~ 50 And once more in words, to make it clear: "Armageddon with 31 Atk IVs and a neutral nature is the same as a Torchic with 50 base attack and a beneficial nature." This, coincidentally, is exactly the attack stat of an Oddish. So maybe we should have said it a little differently: "Armageddon with 31 Atk IVs and a neutral nature hits as hard as a very angry Oddish." Wow, we really need to work on Armageddon... ---------------------------------------------------------------------------- So what does this mean for Disaster, our beatiful Zangoose? So just how good is Disaster? After all, that's why I started thinking about this in the first place. Let's calculate the Lv 100 IV and base stat equivalents, just like we did for Armageddon. We'll have to work slightly differently here, since we're starting with a beneficial nature and trying to find the "bad nature" equivalent instead. Unfortunately the floor functions make this hard to do analytically, so I told my computer to do it for me instead. For Disaster's speed stat, we have: Nature = 1.1 Base = 90 IV = 12 EV = 252 So if instead she had a neutral speed nature, what IVs/base stat would she need to be as fast? Our computer tells us that: IV' ~ 38 Base' ~ 103 So our fabulous Disaster (I hope somebody gets this reference) is as good as a Zangoose with neutral nature and 38 Spd IVs, or a base stat of 103. To put that into perspective, that's faster than both Garchomp and even Oddish. ---------------------------------------------------------------------------- Summary If you're reading this coming straight from the start, or you've struggled to follow along, I'll try to summarise nicely: Nature is very important. Probably even more so than IVs and base stats, depending on what you're using to start with. I didn't realise quite how important it was until I started making this post, and I've pleasantly surprised myself even. Adamant nature Oddish hits as hard as Serious Torchic. Yep, it's true. Disaster is beautiful and very fast. So stop being mean to her. That is all. Peace!
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