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### Optimal Level to Buy Armor Pen

f3ried chicken

Junior Member

What is the optimal level to buy armor penetration? The intuitive answer is sometime during the late game since your opponents have more armor. However, as your opponents buy more armor, each point of armor is worth less in terms of % damage reduction. Therefore, it might actually be optimal to buy armor pen during the early game.

Results:
1. Flat armor pen is best early
2. 35% armor pen (last whisper) ONLY is best at around 125 armor
3. A brutalizer (+10) and last whisper (35%) combination is best at 60 armor

We are going to look at % extra damage you do to your opponent with versus without armor pen. For positive armor, damage reduction is 1-(100/(100+Armor)). So if you have 0 armor, you receive 0 damage reduction but if you have 100 armor, you receive 0.5 or 50% reduction. In order to look at the effectiveness of armor pen, we need to look at the damage reduction with and without armor pen. Namely,
% Extra Damage You Do = [1-(100/(100+Armor))] - [1-(100/ (100+f(Armor)))]
where f(Armor) is your armor after applying % armor pen and armor pen.
Note runes and masteries have been neglected.

I plotted the graph of % Extra Damage You Do versus your Opponent's armor for a brutalizer, a last whisper and both brutalizer and last whisper. A bigger negative percentage means more damage you are doing with armor pen versus without armor pen.

p.s. to find the optimal armor to buy a certain % and flat armor pen combination, plot [1-(100/(100+Armor))] - [1-(100/ (100+f(Armor)))] on Wolfram Alpha and replace f(armor) with (1-percentarmorpen)x - flatarmorpen

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