One of the things I have been wondering is just how do AD runes compare to Apen runes? It led me on a journey that ultimately had me solving how much one additional point of AD is worth vs. one additional point of Apen. The answer is

eq. 1---------------

where:
dA is your additional attack damage (ex. from runes)
A is your base attack damage (after addition of dA)
X is your opponent's final armor or MR - this means after reduction effects have taken place, including flat pen.
dX is your additional flat pen.

In other words, the dependent factor is (100+x)/A. If this factor is greater than 1, each additional point of attack damage is worth more than one additional point of flat pen. If this factor is less than 1, the converse is true. When the factor is equal to 1, then a point of attack damage has the same effect as a point of flat pen.

Worried about %pen from last whisper/void staff/masteries?

Just remember that to add multiple sources of %pen, the formula is

eq. 2--------------- total % penetration = 100% - [(100% - % pen from source 1) * (100% - % pen from source 2) * (100% - % pen from source 3) ... etc.]
Thanks to Alex Dude for a more elegant form of the equation.

Reiterate the formula for a 3rd source, etc (plug in total %pen back in as one of the sources)

And your effective flat pen is simply

eq. 3----------------effective flat pen = flat pen * (100% - %pen)

So your dX in eq. 1 is simply your effective flat pen from eq. 3

What about abilities?

Just take your hero in question and substitute the following formula for A and AD into eq. 1:

A = ability base damage + (AD ratio * total bonus AD)
dA = AD ratio * "extra" AD

where "extra" AD is the change in bonus attack damage you are trying to measure; for example, from runes, masteries, buying a new weapon, or a combination of these sources.

If the ability ratio is total attack damage, substitute:

A = ability base damage + (AD ratio * total AD)
dA = AD ratio * "extra" AD

In this way, you can figure out whether AD runes/Apen runes are better for each ability for your hero, and for his/her autoattacks! You can even compare at what level one becomes more effective than the other. It is, however, a case by case basis for you to judge whether autoattacks or abilities are more important for your hero.

Hope that helps anyone interested!

PS. if you want to know how to evaluate eq. 2 and 3 with % reduction / flat reduction, just let me know, and I will get you those equations, too!

PSS. And the beauty of eq. 1 is, Attack Speed, CrC, CrD, Executioner and Havoc do NOT matter. This is because any bonus attack damage you have (in the numerator) has to be multiplied by these stats, and your original attack damage (in the denominator) has to be multiplied by the same values. Therefore, they cancel out.

No offense intended, but this kind of reminds me of my college textbooks: wordy and not particularly helpful when in search of an ultimate conclusion.

Rather than going through all of the equations above, I see armor penetration is a valuable resource that is extremely hard to obtain. In fact, there are ONLY 3 items in the game which will grant it: Brutalizer/Youmuu's, Black Cleaver, and Last Whisper. As a result, most of the time runes should be assigned to armor penetration as the attack damage items ingame have a much wider variety of options.

And even then, you have to remember they all give AD, and Black Cleaver negates Last Whisper/Youmuu's. AD Runes aren't really that useful, all considered.

this is only meant to be helpful

real equation 2:
total % penetration = 1 - [(1 - % pen from source 1) * (1 - % pen from source 2) * (1 - % pen from source 3) ... etc.]

real equation 3:
effective flat penetration = actual flat penetration * (1 - total % penetration)

I'm not sure what equation 1 is measuring since the units are inconsistent, but it can't be right since it doesn't account for % armor reduction (i.e. Jarvan), flat armor reduction (i.e. Corki), % damage taken reduction, flat damage taken reduction, auto-attack damage reduction, etc (I have no idea how these stack)

Black Cleaver does not, in any way, negate LW or Youmuu's. Reductions and penetration goes hand in hand.
It's true that getting LW and BC isn't a great idea in terms of gold investment, but you'll still end up with more armor pierced.

Let's take targets ate 100, 200 and 300 armor as examples. With only LW, those targets drop to 60, 120 and 180 armor. With BC and LW, they drop to 39, 93 and 153. No negation there.

By the time you get Black Cleaver, your targets probably have more than enough armor to not negate ArPen runes.

That said, just do Brutalizer+Ghostblade+LW on every champ and be k.

True story. This on Janna..

I understand your sentiment. However, there are more sources of Apen than just items. You can compare AD runes vs Apen runes, in whatever ratio you prefer. You can also see at what point, if ever, or also how, AD and Apen can outscale each other. This also works for magic pen.

For example, Irelia tends to build only a trinity force, maybe a Frozen Mallet for damage. You can see that AD runes might actually be better for her than Apen runes, because her base attack damage never gets very high. Considering the Crystal Scar aura and masteries, her Apen runes would already be less effective.

It can also be useful in SR, where the line gets fuzzy after a few levels which rune set is better, but that dominant early game is really attractive, and I heard on a stream that a lot of top players are running AD runes on their AD carry. It might be good to know what kind of trade off you are making.

It's not a be all, game-ending equation. Why is it important for players to know that a point of armor/MR is equal to 1% your base health? It's not applicable all the time; it's not even necessary or interesting to 95% of players. But it does allow some calculations about the game, and it's mostly for furthering your understanding of how the game works,

I did mention this is only for people who enjoy math and theorycrafting . It can be a useful tool for those people, but the rest of people will look at this and not understand a thing, or not care or see any value. I tried to highlight the main points, and I skipped completely over the derivation, so it isn't quite like a textbook, at least not one that I am used to. I like to think of it more as a cheat sheet

Hope that clears it up!

Eq. 2 is fine, I just didn't include a third source because there are only one or two instances of a third source of %pen, if any, AFAIK. Although there is the crystal scar aura, so maybe I should have included the potential third source since this is Dominion board. I wanted to write it in an, elegant, concise formula, but I couldn't, so I just figured others would know to reiterate the formula for a third source.

eq 3. has been updated, had the % on the wrong way when I did that oops! Thanks for pointing that out!

The units are fine in Eq. 1. dA and A cancel, leaving units of armor in the numerator which is equal to units of armor from dX.

I highlighted X as current armor because it is meant to be after reduction has been applied, if that hadn't been clear. Essentially, flat pen's value is not dependent on reduction, so reduction of any kind has to be applied first.

% damage and flat damage reduction are extremely rare cases, but are in an interesting point. I believe they would have to be calculated in under X by being converted to an effective armor/MR, but there are only a couple champs in the game who can do this, and that only for a few seconds at a time (Alistar, Maokai ult), and they aren't targets you generally want to focus.

Auto-attack reduction (can only think of Tryndamere's Mocking Shout) is simply applied to base damage, or A in the equation.



LW reduces BC, Youmuu's, and any other source of flat pen. % pen in general does this. See above post or eq. 3.

It could be said that % pen and flat pen don't synergize efficiently, but flat pen has better marginal value per point at lower armor/MR (I believe I could prove the general case, but it is apparent in eq. 1 vs AD), so it is a little risky to make a generalization that they don't fit well together. But for the purposes of eq. 1, an "effective flat pen" is necessary.

Hope this answers the comments satisfactorily!

A rule of thumb is always easiest, so it's good to have one. For the sake of discussion for anyone interested, I'll bring up some things I have learned.

If you consider that it takes several hits to kill an opponent from full to zero, you can average the flat pen of BC. I averaged it to 34 (based on 8 hits on average to kill an opponent), but you might choose a different value, depending on where your build is at and which target you are hitting.

If you average the BC to 34 flat pen, then, assuming a base flat pen of 51 (Ghostblade + 25 from marks/quints), a BC provides superior armor pen to a LW up until the opponent has 136 armor. At that point, LW becomes better. If you include a brutalizer, then a BC will be more useful than a LW up until, say 145 armor or so - AFTER reduction effects.

In other words, if your champion can stack BC reliably, then it will be better than a LW for anyone who does not buy a thornmail or FH. If you have some sort of armor reduction (Jarvan, Kayle, Eve, Kog), then it is likely that BC will be better than LW in almost every case.

Also while they may equal Apen efficiency at opponent 136 armor after reductions, the BC also applies better stats, which is why I think it's better than a LW if the opponent has a chain vest + ninja tabi, for example, or merc treads and a Randuin's only.

It's simpler to speak in a rule of thumb, as you mentioned, and more practical for almost any person. But this is a math and theorycrafting thread , so I am bringing this up.

I also wanted to say that unless you need the CDR, you will probably get more out of a pure damage weapon than a brutalizer. Considering the short Dominion games, and assuming you buy one or two defensive items, the brutalizer is probably not a good choice if you already have ghostblade and LW, unless you know you only have 1337 gold for the rest of the match. In fact, unless you need the ghostblade, your build might be better off without that, too, and just the LW, depending on the hero. That's what I made this thread for, because you can do these comparisons with eq. 1

People always forget that flat out damage runes boost your skills. Most skills scale off BONUS AD, your runes will give you more damage using skills. Late game those AR pen are worthless since its not uncommon for the enemy team to have 200+ armor, going pure damage then LW is the way to go in the best scenario.