### Math regarding Zileas' N^y formula

Viro Melchior

Senior Member

THERE IS NO TL;DR. If it's TL, then DR and just hit BACK.

N^2 example used is Annie, even though she's really N^3 (Zileas forgot Magic Penetration).
You are looking at sustained teamfight damage.

If you spend 1000 gold on AP, Annie gains N damage. If you spend 2000 gold on AP, Annie gains 2N damage. Etc.
If you spend 1000 gold on CDR, Annie does her damage more often, resulting in N more damage. If you spend 2000 gold on CDR, Annie reaches the cap, but is doing 2N more damage.
If you spend 1000 gold on AP AND 1000 gold on CDR, you get multiplicative results.

For increasing AP:
Total damage = Base damage * (1 + N). N is typically a fraction relative to base damage (and you could say that since Annie has a total AP ratio of 1.9 on offensive abilities, she requires 1035 damage, or 544 AP to double her damage. Therefore 1000 gold is ~50 AP, or an N value of .09)

For increasing CDR:
Total damage = Base damage * (1 / [1 - CDR]), with cdr expressed as a value from .00 to .40. At 40% CDR, you are doing 66% more damage. Since there are no flat CDR items, I'm going to use Kindlegem for math. For 375g upgrade cost you get a minor amount of "free health", and 10% CDR, so I'm valueing 10% CDR at 375g. 50% CDR would double your base damage, and 1000g is worth 26% CDR. So 1000g would produce an N value of about .52.

Therefore, if you spend 1000g on AP, your damage goes up 9%. Meanwhile, if you spend 1000g on CDR, your damage goes up by 52%. Now, these numbers aren't raw game value, as burst often is more important than sustained damage for a mage, especially given the high damage yield of Annie's AoE potential and long cooldown on Tibbers.
However, the point is that if you spend 1000g on AP *after* getting your CDR, you get more damage out of it than if you didn't have any CDR. This is because they are multiplicative.
1.52 * 1.09 > 1 * 1.09

Now, let's move on to carries:
Level 18 damage is about 100, with attack speed about 1.0. No base armor penetration, typically minimal/no critical chance.
1000 gold gives 25 damage, 40% attack speed (about .3), or 20% critical. Additionally, it scales off Armor Penetration and Critical Damage.
So...
25 damage is a 25% increase.
20% crit is a 20% increase (that caps at 100%)
40% aspd (.3 speed) is a 30% increase.

If you have 6000g to spend, you can get:
150 damage (250% yield)
120% crit (conceptually a 220% yield, but caps at 200%)
240% attack speed (280% yield)

OR you can spend it equally on all 3, and get
1 * 1.5 * 1.4 * 1.6 = 3.36. That's a 336% yield. 50% increase more than the best straight single stat, and 20% more damage than straight attack speed.
THAT is what N^Y is all about. The more stats that factor into your damage that are multiplicative with each other, the higher you can spike your damage.

Champions like Twitch effectively begin with one stat (attack speed) capped out, and thus benefit more from other stats, causing them to scale faster.

PRNmeds

Senior Member

Good explanation, +1.

also, inb4 Rammus "ok".

RaptorSenpai

Senior Member

xavarn

Senior Member

I dont think she can be N3 because spending twice as much on magic pen wont make you do tiwce as much dmg.

Classicola

Senior Member

.TL DR

OP can't inb4

Uccisore

Senior Member

ok

noodle0117

Senior Member

as much as i wished there really was a tl;dr (just because I'm feeling a bit lazy), that was some good math there.

Mages scale linearly, while dps/carries scale quadratic wise because of this.

Viro Melchior

Senior Member

Quote:
Originally Posted by xavarn
I dont think she can be N3 because spending twice as much on magic pen wont make you do tiwce as much dmg.
Many stats have caps.
Many stats are hard to put direct N values on.

Penetration provides a bigger bonus the closer you get them to 0, but after they reach 0, it provides 0 bonus. Reduction does the same, but continues to scale (which is why it is so powerful, and factored in first).
Cooldown reduction caps very easily, while attack speed is (usually) hard to cap.

Directly at your quote, Xavarn. Spending twice as much on Magic Penetration will actually MORE than double the bonus damage you got from the first X gold spent, *unless* you wind up with more penetration than they have resist.

Tinymath for you:
• Typical champion with 30 MR. You are dealing 77% damage to them (you suck and have no runes).
• You buy Sorcerer Boots. You are now dealing 91% damage them (since they only have an effective 10 MR). That's an increase of 18%.
• You now buy Haunting Guise, bringing you to 40 penetration. You are dealing 100% damage them. This is a net increase of 29%, and an increase from just-boots of 10%.
• Now they buy Mercury Treads, bringing them to 55 MR. If you had no penetration, you would deal 64.5% damage. Your boots raise it 74%, and the guise brings it up to 87%. Your damage went up by 14.7% from the boots, and a net 34.9% from both. That's a larger increase for the 2nd item than the first.
So Annie is N^3. It's just that one of them (CDR) caps very quickly, and another scales oddly and caps quickly against targets that build glass cannon. So if you have a somewhat typical 15% runes+ CDR build and get golem, with Sorc Boots as you movement choice, then yes, Annie scales very linearly against glass cannon targets. You probably want a Haunting Guise or Void Staff depending on your target, but the other 4 items are all either damage, defense, or a mix, not an exponential scaling for her damage.

Thanks for the positive feedback everyone. I do hope this clears things up since Zileas decided to post on N^y different threads, instead of taking the time to fully explain his math.

xavarn

Senior Member

Quote:
Originally Posted by Viro Melchior
Many stats have caps.
Many stats are hard to put direct N values on.

Penetration provides a bigger bonus the closer you get them to 0, but after they reach 0, it provides 0 bonus. Reduction does the same, but continues to scale (which is why it is so powerful, and factored in first).
Cooldown reduction caps very easily, while attack speed is (usually) hard to cap.

Directly at your quote, Xavarn. Spending twice as much on Magic Penetration will actually MORE than double the bonus damage you got from the first X gold spent, *unless* you wind up with more penetration than they have resist.

Tinymath for you:
• Typical champion with 30 MR. You are dealing 77% damage to them (you suck and have no runes).
• You buy Sorcerer Boots. You are now dealing 91% damage them (since they only have an effective 10 MR). That's an increase of 18%.
• You now buy Haunting Guise, bringing you to 40 penetration. You are dealing 100% damage them. This is a net increase of 29%, and an increase from just-boots of 10%.
• Now they buy Mercury Treads, bringing them to 55 MR. If you had no penetration, you would deal 64.5% damage. Your boots raise it 74%, and the guise brings it up to 87%. Your damage went up by 14.7% from the boots, and a net 34.9% from both. That's a larger increase for the 2nd item than the first.
Thanks for the positive feedback everyone. I do hope this clears things up since Zileas decided to post on N^y different threads, instead of taking the time to fully explain his math.

Ok, that would be nice, but how bout this, Said champion has 250 MR. then how much of a difference does 20 Mpen actually make?

Viro Melchior

Senior Member

Quote:
Originally Posted by xavarn
Ok, that would be nice, but how bout this, Said champion has 250 MR. then how much of a difference does 20 Mpen actually make?
Upvoted for good discussion even if arguing against it!

That is why I admit that MR scales strangely.
However, if you are Annie and worried about a 250 MR target, you'll want to scale your damage up with a Void Staff, retaining your N^3 scaling ability.

250 MR means you are doing 28.6% damage.
Dropping them to 150 MR with a Void Staff gives you 40% damage, a 40% increase in damage (ignoring the AP it provides).
Since that 40% increase still multiplies any damage values you gain from AP or CDR, it is therefore N^3.

12