It’s forty minutes in, and you’re on edge, playing your favorite AP mid. Neither team has a clear advantage and both sides are down to their bases. Baron’s coming up in one minute. You’ve got your core build, a Void Staff, and a defensive/utility item of your choice, but the enemy AD carry is scaling up to dangerous levels. You need something to insure you take him down instantly. You open the shop and click and stare at the three late game AP burst options:  Deathfire Grasp, Rabadon’s Deathcap, and Lich Bane. What do you pick? I’m Sudunem and today, I’m going to break down the decision between each option.

Late Game Itemization

Why those three items?

First, I assume you have your core. Whether that core is Liandry’s + Rylais, Rod of Ages + Archangel’s, Athene’s + Void Staff, or Morello’s + Deathcap, or whatever you prefer for your core two big items, I assume you have that.

Also, I assume you have one or more defensive items, or else you don’t feel you need them. Such items include (most importantly) Zhonya’s Hourglass, but could also be Abyssal Scepter or Rylai’s Crystal Scepter.

Lastly, I assume you have Void Staff. If you’re familiar with DiffTheEnder and his past work, you’ll know that Void Staff generally adds more damage than Deathcap (or any other option barring Liandry’s Torment) provided the target has over 90 MR and you spend the additional gold saved from buying Void Staff over Deathcap on additional AP.

So basically, you’re looking for a boomstick. You want to make somebody go away quickly, and that means Deathcap, Deathfire Grasp, or Lich Bane.

Deathfire Grasp:

• +120 ability power
• +10% cooldown reduction
• Unique Active: Deals 15% of target champion’s maximum health in magic damage and amplifies all magic damage they take by 20% for 4 seconds. 60 second cooldown (750 range).
• 3100 gold

Rabadon’s Deathcap:

• +120 ability power
• Unique Passive: Increases ability power by 30%.
• 3300 gold

Lich Bane:

• +80 ability power
• +250 mana
• +5% movement speed
• Unique Passive – Spellblade: After using an ability, your next basic attack deals bonus magic damage equal to 50 + 75% of your ability power. 2 second cooldown.
• 3000 gold

Deathcap versus Lich Bane

First up, let’s compare Deathcap versus Lich Bane. Roll up your sleeves, because it’s time for some algebra!

Let R = your combo’s AP ratio, let X = your ability power. This yields the equation:

 R(x+80)+(50+0.75(x+80)=R(x+120)*1.3

Solve for X. This yields the rational equation:

 x=(110-76R)/(0.3R-0.75)

Solving this equation analytically with varying values of R is  not very helpful. So instead, I solved it statistically. I varied your AP and AP ratios for each item, and then subtracted them from each other. The results are presented in tabular form.

If the cell is orange, it means that Lich Bane does more damage to your combination. The negative value represents how much more damage Lich Bane adds to your combination.

If the cell is red, it means that Rabadon’s Deathcap adds more damage to your combination. The value is how much more damage Deathcap adds to your combination.

Note that at an AP ratio of 2.5, the difference is always the same no matter how much AP you have due to an asymptote in the equation.

Now, what happens if you have two Lich Bane procs in your rotation instead of just one?

Lich Bane wins by a lot. Note however, that this is single-target damage. Remember, the premise of this article is about finding an item to help delete somebody from the game. If you’re primarily AoE-focused (e.g. Brand), then Deathcap wins. If you’re hitting two targets, Deathcap beats out two Lich Bane procs for AP ratios above 2.1. For more than two targets, Deathcap is victorious on all fronts.

Also, it’s worth noting that if your champion is never in range to auto-attack (e.g. Xerath), then Lich Bane is of dubious value. Furthermore, Lich Bane does nothing for shields, heals, and any other utility spell that scales off of AP, so utility mages (most of whom are heavy AoE anyway) will definitely prefer the Deathcap.

A quick note on Karthus and Cassiopeia:

Karthus and Cassiopeia do more damage with their Q and E respectively than they would from using Lich Bane. Cassiopeia has 570 + 1.65 AP damage on the three Twin Fangs you can get during the cooldown of a single Lich Bane proc while Karthus hits 240 + 1.2 AP on two Lay Wastes (single-target, or at least twot argets). Lich Bane is an inefficient buy on both champions, particularly since their spells out-range their auto-attacks.

Lich Bane versus Deathfire Grasp

For the next comparison, we’ll compare Lich Bane versus Deathfire Grasp. This equation is more complicated and introduces a couple of new variables into the equation that we’ll have to solve for.

Let R = your combo’s AP ratio, let X = your ability power, b = your combo’s base damage, and h = the target’s max HP. This yields the equation:

R(x+80)+(50+0.75(x+80)+b=1.2(b+(x+120)R)+0.15h

Solving for x, we get the following rational equation:

x=(0.2b+0.15h+64R-110)/(.75-.2R)

Let b (base damage) vary from 600 to 2000, incrementing by 200, and let h vary from 2000 to 3500, incrementing by 500.

It should come as absolutely no surprise that at no point will one Lich Bane proc ever beat out Deathfire Grasp. The equivalence point is over 900 AP at its lowest. Let’s change the equation to measure two Lich Bane procs versus DFG. Some mathematical hand-waving yields:

x=(0.2b+0.15h+64R-220)/(1.5-.2R)

 The numbers across the top row represent varying AP ratios. The numbers in each cell represent the break-even point. Your AP must be higher than the value in each cell for Lich Bane to be better. What does this mean?

If your AP ratios are 1.8 or lower and your base damage is lower than 1000, you’re better off with Lich Bane over Deathfire Grasp as long as you have 350 AP from your core. Who has that low of a ratio? Well, not many people, but Twisted Fate comes to mind.

Let’s also evaluate 3 Lich Bane procs:

x=(0.2b+0.15h+64R-330)/(2.25-.2R)

If you’re hitting 3 Lich Bane procs in your combination, you’re going to get more single-target damage out of Lich Bane than Deathfire Grasp against a squishy target as long as your AP ratios are under 2.5 and base damage of under 1500, with at least 300 AP. To be fair, though, that requires 6 seconds and not the instant-deletion you’re going for. On the other hand, it does give you better sustained damage.

Two other notes:

Lich Bane works quite well against towers. If you’re looking to help your team push instead of deleting a single target, it’s a good buy.

Deathfire Grasp’s debuff also amplifies your team’s magic damage, so if you’re running double AP (such as Ryze or Rumble top lane) or a heavy magic damage, AoE jungler, you can really put the hurt on the target of your DFG burst. Regarding Deathfire Grasp, I did not factor in the 10% CDR on the item, because if you’re buying DFG, you mainly want to delete someone. Nonetheless, it is there, and it is a factor when considering the purchase.

What about both?

Most assassins/burst mages are capable of using both Lich Bane and Deathfire Grasp, which makes for some pretty incredible damage. It’s expensive and makes you very squishy, but if you already have Zhonya’s and Void Staff and don’t need the Abyssal Scepter or any other item, buying both Lich Bane and DFG is a good way to guarantee that your insta-gib potential stays very high. That said, you should buy DFG for higher burst in most cases. Lich Bane is more of a sustained damage item.

Deathcap versus Deathfire Grasp

For our last comparison, we’ll compare Rabadon’s Deathcap with Deathfire Grasp.

Let R = your combo’s AP ratio, let X = your ability power, b = your combo’s base damage, and h = the target’s max HP. This yields the equation:

1.3R(x+120)+b=1.2(b+(x+120)R)+0.15h

Solving for X, we get

x=(0.2b-12R+0.15h)/(0.1R)

The numbers in this chart represent how much Ability Power you would need to make Rabadon’s Deathcap equal the damage output from Deathfire Grasp.  Your AP must be higher than the value in each cell for Deathcap to be better. The simple answer is that you need a lot. All you need to know from this chart is that when it comes to deleting a single target, Deathfire Grasp is far, far superior to Rabadon’s Deathcap.

If you want to consider the use of Rabadon’s Deathcap hitting two or three targets against just DFG, here are those numbers as well.

2 targets: x=(0.2b-48R+0.15h)/(.4R)

3 targets: x = (0.2b – 84R + 0.15h)/(.7R)

The numbers have a few important results. If you’re hitting two targets, DFG will be superior to Deathcap against squishies in scenarios where you have less than a 2.2 AP ratio (unless you somehow manage to get 500 AP without buying Deathcap). If you’re hitting three or more targets, Deathcap wins considerably.

Summary

If you skipped down to here, don’t feel bad. There is a lot of statistics and algebra between the intro and the article here, and you probably don’t care about that. What you care about is which boomstick to buy.

• For long-range poke and 3+ targets: Deathcap >> Lich Bane and Deathfire Grasp.

• For two targets (moderate AP ratios 2.2-): Deathfire Grasp > 2 Lich Bane procs  > Deathcap

• For two targets (high AP ratios 2.5+): Deathcap > Deathfire Grasp > 2 Lich Bane procs

• For non-damage AP scaling (shields/heals): Deathcap > Deathfire Grasp > Lich Bane

• For single target bursting: DFG > 2 Lich Bane procs > Deathcap > 1 Lich Bane proc

• For sustained damage/pushing: Lich Bane > Deathcap > Deathfire Grasp (3 Lich Bane procs is more damage than either other option)

Champions:

Prioritize Deathcap:

Brand, Fiddlesticks, Galio (AP build), Janna (AP Build), Karma, Karthus, Kennen, Kog’Maw (AP build), Lux, Nidalee, Morgana, Nidalee, Orianna, Rumble, Vladimir, Xerath, Ziggs, Zyra

Prioritize DFG:

Ahri, Fizz, LeBlanc, Kassadin, Mordekaiser, AP Poppy, Swain*, Tristana (AP build), Veigar

*Swain is a special case because Deathfire Grasp stacks additively with Torment for a ludicrous 40% damage amplification. Poppy’s kit also allows for this (60% damage amplification).

Prioritize Lich Bane:

**Twisted Fate

**Twisted Fate is a special case because his ratios are low enough that 2 Lich Bane procs are about equal to DFG’s damage on a single squishy target. It also helps him split push.

Situational/Play Style Dependent :

Akali, Anivia, Annie, Cassiopeia, Diana, Elise, Evelynn, Gragas, Lissandra, Katarina, Kayle, Malzahar, Sion, Syndra, Teemo, Viktor, Zilean

Conclusion

A few final thoughts. First, consider your ranges. Lich Bane’s range can vary from 125 (melee) to 425-625 (ranged). Deathfire Grasp’s active has 750 range. If you’re never in range to use Lich Bane or Deathfire Grasp, they’re not good choices. Second, consider your role. Mage! Assassin! Duh! Simple questions of course, but are you playing your mage as more a single-target assassin, or does your team need you to hit multiple targets and/or poke? Some champions are locked into the utility/AOE/poke role, but some can both do that and serve as insta-gibbers.

Second, consider your cooldowns. If you can’t proc it at least three times in a row, then perhaps Lich Bane isn’t the best buy for you, because you won’t be able to maximize its use. On the other hand, if you can proc it constantly and are able to consistently use your sustained damage, three straight procs of Lich Bane is more damage than DFG or Deathcap on a single target for most cases.

And if you ever see an enemy burst mage/AP assassin with both Lich Bane and Deathfire Grasp and the range to use both, particularly paired with a Void Staff, be very afraid. If this person gets to combo your squishy, Guardian Angel, Zhonya’s, or Chronoshift might be the only thing that can save them.

Good luck on the Fields of Justice, and happy insta-gibbing!