Thornmail: E̶x̶a̶c̶t̶ Approximate Math to Kill an ADC Without Doing Anything
Last Updated: March 4th 2016



Summary

  • Enemy having Blood Thirster makes it quite difficult for you to survive before the enemy dies to Thornmail
  • Enemy having Blood Thirster makes it very difficult for you to survive before the enemy dies to Thornmail
  • Enemy having Negatron Cloak makes it extremely difficult for you to survive before the enemy dies to Thornmail

Results

  • You must have at least 7456 HP with 290 Armor to make Enemy1 (see below) kill himself 
  • You must have at least 4674 HP with 290 Armor to make Enemy2 (see below) kill himself

Enemy
1 stats: 1800 HP, 43 Magic Resist,  with Blood Thirster + Blood Thirster
Enemy2 stats: 1800 HP, 0 Magic Resist (Result of your full Magic Pen.) with Blood Thirster + Blood Thirster



So we can completely ignore their ADC 
and still kill him? Nice!
WarwickDianaSinged


Well, not if the enemy adc has %armorpen, lifesteal, or magic resist!

But the point is that if they don't have any of those stats and you have full magic pen over them then it is quite practically possible that they can just die to Thornmail alone. Whatever autoattack damage you receive from them, they'll be dealt about the same amount.  Therefore, they'll have no choice but only use abilities to deal damage against you, which essentially has no DPS compared to their autoattacks

Ignite

And for better and quicker results, consider using ignite.  It will seriously kill them, 100% guaranteed.



You might think that this math trick won't always work since it is possible for the other enemy champions (e.g. mages) to hit you instead.  However, that's not the point here.  The fact that whatever autoattack damage you take is dealt right back to the attacker by the same amount is just overpowered.  This means if you kill all the casters, no one would even dare to attack you.

Keep in mind that you will need full magic pen, a lot of armor and HP for this
Thornmail threat to work efficiently.

Enemies that have even a little bit of Magic Resist will severely hinder the effectiveness of Thornmail.



For the mathematical analysis on how I arrived with the results above, please read below.


Ever since the Thornmail damage reflection changed from 30% incoming damage to 15% incoming damage + 25% bonus armor now the math is a little different now.

Thornmail is now more effective against faster attacking champions because the damage of 25% bonus armor scaling is applied on every hit.

This also means that it is no longer possible to generalize how fast Thornmail can kill an ADC as every champion has varying attack speed / on-hit abilities.




We are going to model a problem based on the stats of a common adc with a common build: 

Most ADC these days have around 1800 HP, 43 MR, 1.5 attack speed, 70% crit chance (with IE), 300AD, 45% bonus armor pen,  15% bonus damage, 7% armor pen, and 20% lifesteal at level 17.

  • The average raw damage per autoattack is 707 (300 * 70%*150% * 1.15), thus the 15% reflected damage is 106

And you have 81 base armor + 209 bonus armor for a total of 290 armor. We will now find out how much HP you must have in order for the Thornmail to take out the enemy adc.

  • You lose 45% of 209 bonus armor  and you now end up with 81 + 115 = 196 armor
  • You also lose 7% (Masteries) of 196 armor and now you end up with 182 armor
  • Therefore you still have 64.54% damage reduction against the enemy adc



Let's not forget that the enemy adc also has 20% lifesteal in addition to 1800HP

= (Base HP + Life Steal Amount)
= (1800 + 
L

And we don't know what L is just yet.
Now, your Thornmail damage per hit is

= (15% Reflection + 0.25*Your bonus Armor) * Enemy Magic Resist
= (106 + 0.25*209) * 100/(100+43)
= 158.25 * (0.6993)
= 110.66

Now the enemy ADC's death formula is as follows:

= TotalHP - number of autos * thornmail = 0
= (1800 + 
L)  - number_of_autos * (110.66) = 0
= (1800 + L) -  n(110.66) = 0

Therefore,
n = (1800 + L)/110.66 


That's it for the enemy death formula.
Now we need to figure out the minimum HP required for you to kill the enemy ADC before you die first.

YourHP =
number_of_autos * (enemy raw damage per auto * (1 - damage reduction from armor))
YourHP = n * (707* (1 - 0.6454))
YourHP = n * (707* (1 - 0.6454))
YourHP = n*250.7

Therefore,
n = YourHP/250.7

Aha! So you take 250.7 damage from every auto of the enemy adc while they take 110.66.
So this is pretty obvious that YourHP must be 250.7/110.66 =  2.265 times more than the enemy ADC's total HP.
Anyways, equating the two n's above:

(1800 + L)/110.66  = YourHP/250.7
YourHP = (250.7/110.66) * (1800+L)
YourHP = 2.2655 * (1800 + L)

Wait, what is L? Oh right that's 20% lifesteal!
If the enemy adc kills you the exact same time he dies then he would have lifesteal 20% of YourHP

YourHP = 2.2655 * (1800 + 0.20*YourHP)
YourHP = 4077.9 + 0.4531
*YourHP
0.5469*YourHP = 4077.9
YourHP = 7456

MY GOODNESS!
There is no way in hell I'd have 7456HP and 290 armor for any champion in the game.
Is there a better way to make this Thornmail trick work?
Well first of all, the enemy adc has too much %armor pen and lifesteal but that's the reality, sorry.
However, we can stack magic pen and make the outcome much better.


Say you got 43 magic pen from Runes, Masteries, Sorcerer's Boots, and Haunting Guise.
Now the enemy takes full 158.25 from thornmail.


So doing the math again,

YourHP = (250.7/158.25) * (1800 + 0.20*YourHP)
0.68316*YourHP = 2851.56
YourHP = 4174


Much better eh? 4174 HP, 290 armor, and full magic pen can bring down a late game adc without doing absolutely nothing if he only autoattacks.