Hello, I’m Eph289, and today I’m going to talk about last-hitting under tower. You know that frustrating feeling when the tower is blasting minions and stealing all the gold? I know that feeling. I hate that feeling. When I was first learning to play this game, somebody told me that “Let the tower hit caster minions once, then auto-attack, and then let the tower hit the melee minion twice before you auto-attack it.” Well, as I frustratingly found out, that little aphorism is not universally true. So today, I’m going to discuss the actual math behind last-hitting under tower.

Towers

For the purposes of this article, I’m referring to last-hitting under outer turrets during the laning phase—that’s the most likely case. Inner turrets (aka second-tier turrets) and inhibitor turrets have higher base AD and scale differently, so if you lose your outer turret quickly, none of these results will help.

Outer turrets have a base AD of 152 that increases by four every minute at the ten second mark. Confused? Basically, at 1:10, the outer turret has 156 AD. At 2:10, that AD increases to 160, and so on, up until a cap of 184 AD. Turrets also have an innate 30% armor penetration.

Minions

Caster minions have 280 health, increasing by 15 every three minutes, along with zero base armor that increases by 1.25 armor every three minutes. Melee minions have 445 health, increasing by 20 every three minutes, along with zero base armor that increases by 2 armor every 3 minutes. Siege minions have 700 health, increasing by 27 every three minutes, along with 15 base armor that increases by three every three minutes. Siege minions also innately only take 70% damage from towers.

Note that minions have different life amounts, but their health bars are scaled equally. This makes last-hitting based on looking at the missing life-bar percentage deceptive given that while the bar might look the same for a melee and a caster minion, the actual HP values are different.

Knowing what we do about the turrets and minions, we can calculate how many turret shots each type of minion can take without dying using a bunch of math.

Caster minions are relatively fragile and will only survive one turret hit until 17 minutes, at which point they can tank two hits.

Melee minions are sturdier and can survive two outer turret hits until 14 minutes, at which point they can tank three hits.

Siege minions can tank six turret hits until 11 minutes, at which point they take seven. At 20 minutes, they can tank eight hits, which increases to nine at 26 minutes.

The amount of HP left after the minions have taken enough turret hits so that the next one kills them is what I’m going to call the survivability threshold.

Note that there’s a huge drop-off at 17 and 14 minutes for caster and melee minions respectively, as that point they are able to tank one more turret hit than previously, resetting their “survivability threshold.”

Base Attack Damage

Not all champions are created equally—some have significantly lower base attack damage/attack damage per level than others, with a range of up to ten. Furthermore, AD-based champions such as AD carries . . . er, marksmen (whatever!) or AD-based fighters, generally bring additional AD from runes and masteries to help with last-hitting, while AP-based fighters and mages get no such benefit.

I looked at four representative cases:

Low Base AD, no AD boosts: The champion has 47 base AD and 3 AD/level, with no additional boosts in damage (Spellsword will give something along the lines of 1-2 damage at early levels). These values are representative for champions such as Vladimir, Twisted Fate, Cassiopeia, Elise, Diana, Anivia, and Lissandra within a few percentage points. It’s hard to categorize these champions, but they tend to include mostly ranged mages.

High Base AD, no AD boosts: The champion has 54 base AD and 3 AD/level, with no additional boosts in damage. These values are representative for champions such as Ziggs, Katarina, Kayle, Fizz, Zac, Morgana, Ryze, and Xerath within a few percentage points. It’s hard to categorize all of these, but in general, melee mages and tanks tend to have higher base AD.

Low Base AD, AD Boosts: I took my 47 base/3 per level AD champion and gave them some runes and masteries to help out: +6 armor penetration, +3 AD, +0.66 AD/level, +4 bonus minion damage on basic attacks, 8% armor penetration, and another 8.5 AD from AD marks. This is representative for champions such as Varus, Ashe, Miss Fortune, Tristana, Jayce, Caitlyn, and Ezreal, e.g. most ranged AD carries.

High Base AD, AD Boosts: I gave the same runes and masteries to the 54 base/3AD/level champion. This is representative for champions such as Xin Zhao, Garen, Renekton, Nasus, Gangplank, Nocturne, Wukong, Rengar, Trundle, and Aatrox. Most physical damage fighters fall into this category.

Results

I have some graphs of the results for my four cases, but they’re a little big to place in an article and they’re easy to misinterpret. The important information from those four cases isn’t “how many auto-attacks after the turret hits do I need to finish each type of minion,” it’s the general trends that those four cases illustrate. There’s too much variance in AD and AD/level from champion to champion to just use those cases quantitatively, but qualitatively, we can glean the following results:

For the first ten minutes, siege minions will generally need two auto-attacks after they absorb the equivalent damage of six turret hits.

Last-hitting under tower is the hardest from 11 to 17 minutes depending on what kind of minion you’re trying to farm. This is because the turret’s AD has stopped scaling upward while the minions are continuing to grow stronger, up until the point where they can tank another shot.

Last-hitting under tower is easiest from 14-23 minutes for melee minions and 17 to 26 minutes for caster minions—only one-auto-attack is needed.

Champions who take AD boosts will never have to spend three auto-attacks to last-hit a minion under tower before 30 minutes.

A high base AD/AD per level will help with last hitting at early levels, but unlike someone who’s buying/runing AD, you’ll still struggle to last-hit under tower using basic attacks from the 14-17 minute mark. The base AD is much less than the AD you obtain from runes, masteries, and item choices in most cases.

Virtually everyone needs at least two auto-attacks after the turret hits to clear any type of minion before 14/17 minutes, unless they have a lot of extra bonus AD from items early game (think 50+).

Champions without AD boosts will often need three auto-attacks to finish minions under tower.

Why is three auto-attacks such a big deal? For one, it requires a lot more counting and precision, particularly since your minions are also likely screwing things up. With two auto-attacks, you do something called priming, where you attack the minion once, let the turret hit it once or twice, and then get the easy last-hit with the final auto-attack. Having to use THREE auto-attacks makes it much harder to prime the minion.

Conclusion

If you want to be sure of your ability to last-hit minions under tower with two or fewer attacks after the turret hits for the first 15 minutes of the game, make sure that your auto-attacks do at least 75-80 damage to minions. For most AD champions, this is as easy as getting to level 3ish.

For AP champions, your options are less awesome. Spellsword isn’t going to give you much damage early game (three bonus damage at 60 AP) and without a high base AD/AD level, last-hitting under tower is going to become more of an exercise in clearing with spells if the wave is shoved to your tower, because you may need three auto-attacks after the 1-2 turret hits land. Of course, minion damage can screw all of those niceties up as well. Your best bet is to try and shove back, because even with high base AD and AD/level, the additional stats that AD champions get from their runes and masteries helps appreciably with last-hitting under tower.

Here’s a table to encapsulate the practicalities of last-hitting under tower:

Important Note: The tables assume you are not buying a lot of bonus AD. Buying any BF Sword item, Sunfire Cape + some bonus AD, or Ravenous Hydra first will effectively reduce your required auto-attacks by roughly one, up to a floor of one.

I hope this article helps you better understand last-hitting under tower; it certainly broadened my understanding. Best of luck to you next time the minion wave is shoved to your outer tower.