Hi. I'm a League of Legends player who doesn't actually like playing PvP. My accounts don't even reach Level 100 so let's say I'm basically nobody. What I lack in experience I make up with maths, though! And research! So with this mathematical build I hope it will be good and strong to use for everyone!

Lillia is a champion in League of Legends, one of my favorites. She’s classified as a Skirmisher and is usually used as Jungle or Top. Her gameplan consists of stacking Q passive to move very very fast, burning enemies with Dream-Laden Bough, and utilizing R as her powerful engage/disengage tool. And she's so cute ^w^

Recently, Riot has reworked her gameplay. Her passive damage now scales with AP and heals her for every champion and large monster hit. Her Q passive only has 4 stacks instead of 5 but scales much better with AP. Her Q is more expensive with longer cooldown but hits harder. Her W now deals more damage.

With this change, Lillia has higher sustain in lane and jungle. Her early damage is reduced but she scales much better until late game. With better scaling, you can also deviate from her usual Liandry + Demonic build and try other builds like full AP.

The goal of this build is not maximizing damage like other mathematical builds. Why? Because if you want to count damage, there is a damage counting site called Alphameta.gg. The site has some bugs and miscalculations but it still helps, nonetheless.
So, instead of dealing as much damage as possible, the goal of this build is to make Lillia run as fast as possible without wasting any items or summoner spells when she has reached 4 Prance stacks.

To achieve that, other than using speed from our items, we can also utilize her new Q passive.


Blooming Blows

Old Passive: Lillia gains a Prance stack for 5 seconds whenever she hits at least 1 enemy with her abilities, refreshing up to 5 stacks. Stacks are lost by one every 1 second. Each stack grants Lillia 7 / 8 / 9 / 10 / 11% (+ 1% per 100 AP) bonus movement speed.

New Passive: Lillia gains a Prance stack for 5 seconds whenever she hits at least 1 enemy with her abilities, refreshing up to 4 stacks. Stacks are lost by one every 1 second. Each stack grants Lillia 3 / 4 / 5 / 6 / 7% (+ 3% per 100 AP) bonus movement speed.

Thanks to this mini rework, now you can build AP for speed instead of using the usual speed items, which gives Lillia’s build a new fresh air.

To achieve maximum speed, there are two simple rules:
a. No active items (including Predator) because they all have cooldown
b. No out of combat movement speed, because you have to be constantly in combat to stack Q

Why? Because she's constantly running and fighting. We want her to have consistent movement speed while running in and out the battlefield.

This is the item order for this build. Build should be open-ended according to the game situation (and I'm not so good at this game lol) so I just make a simple guideline.

Starting Item: Doran's Ring, Doran's Shield (sustain), Dark Seal (snowball), Tear of the Goddess (stacks) or Corrupting Potion (healing), but I don't really know how to do start items lol.

1. Mythic Item: Everfrost

Everfrost gives Lillia health, mana, ability haste, a lot of AP, and an active. Mana will scales with Archangel's Staff for even more AP. Even better, the item active can also help you to land your W sweet spot.

2. Boots: Boots of Swiftness

Highest movement speed in combat. Also slow resist. Why not Mobility Boots? Because to stack Prance you have to be constantly in combat.

3. Snowball: Mejai's Soulstealer

100 Health, 10% movement speed, and up to 145 AP. Perfect for your running deer. If your team can't snowball, though, you might want to skip this build to make Archangel + Rabadon instead.

4. Item combo: Archangel's Staff (Seraph's Embrace) and Rabadon's Deathcap

They are best build together and shouldn't be removed in any stages of the game. Quoting League of Legends Wiki, Seraph's Embrace has great synergy with a Rabadon's Deathcap, as it increases ability power by 35%. This means that whenever your mana is increased, the Seraph's Embrace will increase your ability power, and then the Rabadon's Deathcap Rabadon's Deathcap will increase it again due to the 35% increased ability power passive

5. Optional items: Dead Man's Plate, Lich Bane, Cosmic Drive

Dead Man's Plate can give some tankier stats to Lillia while also maintaining her movement speed. Lich Bane gives Lillia a Sheen effect, and she's always in-range to proc Sheen most of the time. Cosmic Drive gives Lillia a lot of Ability Haste and flat movement speed. However, I will not calculate Cosmic Drive's total speed because of the reason I will tell you in Mathematics section. Still, these three items are good.

When calculating movement speed, first, we have to be aware that League of Legends has movement speed soft caps on every champions

When the raw movement speed is greater than 415, there are two soft caps applied:
• The raw speed between 415 and 490 gets multiplied by 80%.
• The raw speed over 490 gets multiplied by 50%.
• As an example, if the calculated raw speed is 600 before soft caps kick in, it is reduced to (600 − 490) × 0.5 + (490 − 415) × 0.8 + 415 = 530, which means an 11.7% reduction of the original movement speed.
These are the simplified formulas:
• If the raw speed is 415 or lower, no cap is applied.
• If the raw speed is between 415 and 490, final speed is soft capped to RawMS × 0.8 + 83.
• If the raw speed is above 490, final speed is soft capped to RawMS × 0.5 + 230.

(https://leagueoflegends.fandom.com/wiki/Movement_speed)


The formula for total movement speed is (Base MS + Flat MS) * (100% + total MS%), then the cap is applied. Most of movement speed multipliers stack additively, including Lillia's Prance.

Related to this topic, there are some things I discovered about movement speed:

a. Both Dead Man’s Plate and Force of Nature stacks also multiply with their own 5% movement speed passive, which means both 100 stacks of Momentum and 5 stacks of Absorb actually grant at least 42 movement speed despite what the text says.

b. The 20 movement speed bonus from Cosmic Drive can’t be increased by %MS, but still applies the movement speed cap (ikr, this item sucks). Which means it only grants 10 Movement speed from this Lillia build.

With our newfound knowledge, let’s re-calculate Lillia’s movement speed! To make it shorter, we only use Lillia’s best builds.
 
The Final Calculation

This is the final calculation of this Lillia build before counting summoner spells and runes.
We are using Level 18 Lillia (1260 mana, 330 base movement speed)
Lillia’s Prance also have 8% MS each stack by itself, which means 32% MS at max level.

We are only using the items with best movement speed, so I won't put Cosmic Drive here, because, as I explained earlier, the Cosmic Drive 20 bonus movement speed doesn't scale with anything. The result will be much smaller than the other builds.

1. Everfrost, Boots of Swiftness, Mejai’s Soulstealer (25 stacks), Seraph’s Embrace, Rabadon’s Deathcap, Dead Man’s Plate (full Shipwreck stack)
Flat movement speed: 330+60+40=430
Base AP = ([80+60] + 145 + 65 + 120) = 470T
Mana = (1260+600+860) * (100 + (5 + 2,5% of 470)) % = 3175.6
AP from Archangel’s Staff= (3175.6-1260) * 5%= 95.78
calculating bonus mana
Total %MS at 4 stacks = (470+95.78) * 135% * 12% + 10% + 5% = 106.65636%
Total Movement Speed + Cap = (430 * (100 + 32 + 106.65636) %) * 0.5 + 230 = 743.111174

2. Everfrost, Boots of Swiftness, Lich Bane, Seraph’s Embrace, Rabadon’s Deathcap, Dead Man’s Plate (full Shipwreck stack)
If your team can’t snowball with Mejai, you might want to replace it with Lich Bane instead.
Flat movement speed: 330+60+40=430
Base AP = ([80+60] + 75 + 65 + 120) = 400
Mana = (1260+600+860) * (100 + (5 + 2,5% of 400)) % = 3128
AP from Archangel’s Staff= (3128-1260) * 5% = 93.4
Total %MS at 4 stacks = (400+93.4) * 135% * 12% + 8% + 5% = 92.9308%
Total Movement Speed + Cap = (430 * (100 + 32 + 92.9308) %) * 0.5 + 230 = 713.60122

3. Meme Build (with Knight’s Vow)
For the best result, I’m replacing either Everfrost, Dead Man’s Plate, or Mejai with Knight’s Vow. Rabadon’s Deathcap and Seraph’s Embrace are must-build.
Knight’s Vow actually counts as ‘active item’ that adds movement speed, but since you can proc the effect by simply moving towards your Worthy target when they’re low on HP, this item still counts.

a. Without Everfrost
Flat movement speed: 330+60+40=430
Base AP = (145 + 65 + 120) = 330
Mana = (1260+860) * (100 + (5 + 2,5% of 330)) % = 2400.9
AP from Archangel’s Staff= (2400.9-1260) * 5%= 57.045
calculating bonus mana
Total %MS at 4 stacks = (330+57.045) * 135% * 12% + 10% + 5% + 35% = 112.70129%
Total Movement Speed + Cap = (430 * (100 + 32 + 112.70129) %) * 0.5 + 230 = 756.1077735

b. Without Dead Man’s Plate
Flat movement speed: 330+60=390
Base AP = ([80+60] + 145 + 65 + 120) = 470
Mana = (1260+600+860) * (100 + (5 + 2,5% of 470)) % = 3175.6
AP from Archangel’s Staff= (3175.6-1260) * 5%= 95.78
calculating bonus mana
Total %MS at 4 stacks = (470+95.78) * 135% * 12% + 10% + 35% = 136.65636%
Total Movement Speed + Cap = (390 * (100 + 32 + 136.65636) %) * 0.5 + 230 = 753.879902

c. Without Mejai’s Soulstealer
Flat movement speed: 330+60+40=430
Base AP = ([80+60] + 65 + 120) = 325
Mana = (1260+600+860) * (100 + (5 + 2,5% of 325)) % = 2398.25
AP from Archangel’s Staff= (2398.25-1260) * 5% = 56.9125
Total %MS at 4 stacks = (400+56.9125) * 135% * 12% + 8% +5% + 35% = 122.019825%
Total Movement Speed + Cap = (430 * (100 + 32 + 122.019825) %) * 0.5 + 230 = 725.3386588

Summary of the speed calculation:

Before calculating effects from runes, the best item build for Lillia to reach maximum Movement Speed is Everfrost, Boots of Swiftness, Mejai’s Soulstealer (25 stacks), Seraph’s Embrace, Rabadon’s Deathcap, Dead Man’s Plate (full Shipwreck stack) with 743.111174 before counting runes.

If you don’t want to snowball, you can replace Mejai’s Soulstealer with Lich Bane to reach 713.60122 movement speed.

If you want to give memes to your main build, you can build Knight’s Vow to replace Everfrost so that you can reach reach 756.1077735 movement speed or Dead Man’s Plate to reach 753.879902 towards low health allies. If you replace Mejai’s Soulstealer with Knight’s Vow, you can reach 725.3386588 Movement Speed instead.

Hope this mathematical build is useful for you all. If you have any questions or advices (example: the support version of this build) feel free to tell me, I'll help what I can.
Prance around across the battlefield with incredible speed with Running from Math Classes Lillia! Enjoy!