Hello, I'm a 14 year old from California who enjoys playing league and making mathematically correct builds. I'm a silver 3 69LP 69 W 69L 69-enthusiast with a Project Vayne skin. See my Qualifications at the end. That's all about me, and let's jump into the build:

Dear reader,
I know there has been a ton of mathematically correct builds recently, but this one stands out from the rest. I’ve actually started theorycrafting this build on May 6th, and wrote my first draft of the build on May 10th, one day before the release of mathematically correct Sett. Since then, I’ve turned this build into a masterpiece: a 45 page report on the build, which proves through numbers and through logic that this build is mathematically superior to all other builds in the meta. This is by far the highest effort build that you've ever seen, and the most advanced mathematics you’ve seen so far (calculus). I converted all my nasty numbers to graphs so you can figure for yourself graphically the superior build. I really hope you see and enjoy the beauty of this build.

My original doc
Original slideshow

Most of Vayne’s power comes from her ability to shred tanks through her w. Combine this with her hyperscaling and attack speed and she can shred anyone from 100 to 0 in seconds. However, being an ADC, she is also extremely squishy. Since she also has relatively low range, she requires extreme mechanics to dish out as much damage as possible while staying alive.

Vayne however is granted safety with her ultimate which gives her invisibility while she q’s as well as reducing it’s cooldown. However, while not invisible, she’s extremely vulnerable to damage, cc, and other nasty things.

Introducing Calculus Optimized Vayne. Ability haste brings her q cooldown down to up to a 0.5 second cooldown with ultimate, allowing her to essentially be permanently invisible. Even without her ultimate, she has extra peel by being able to q every second and up to a 4 second effective cooldown knockback on her E. With this build, she’s basically Shaco, except with a crossbow(the more you think about it it’s more like Kha'zix but Shaco rhymes with crossbow so it sounds better :) ).

Usually buying ability haste items on Vayne would significantly reduce her damage output. However, this build uses a secret weapon: Sheen.

Vayne has a very low cooldown on her Q. With just 40 ability haste, she can essentially use Sheen better than ANY other champion in the game by triggering it every time it’s available. Build this into either an Essence Reaver and Divine Sunderer and your damage is absolutely nuts, as you’ll see later with the graphs. Even just with the base Sheen, one QA deals 3 Auto Attacks worth of damage!

A quite underrated part of Vayne is her dueling. She is known to be a scaling hypercarry, with a weak early game. However, with this build you should have a ton of early presence while also scaling really well.

This build is most effective in solo lanes. With q-max, ability haste, and Hail of Blades, you are able to play extremely aggressively. However in the bot lane, you will have a range disadvantage in almost every game and you won’t have much kill pressure in lane. In the top lane, you will face almost 100% melee champs which allow you to poke, kite, and all in melee champs. Even though Vayne with this build mathematically works best in the top lane, but just don’t. Through very intense calculations, I found that you will lose anywhere between 6969 to 69420 subscribers if you post a video playing Vayne top.

This leaves us with the mid lane. There you’ll find mainly assassins and control mages, who are usually much squishier than top laners, which makes lethality actually somewhat effective. As for matchups, going as an ADC into an assassin on most ADCs usually leaves you 0/10, but that’s often not the case for Vayne. You have poke as good as anyone, and you can get the enemy assassin so low to the point that they don’t have enough hp to go for an all in on you. You also have great self-peel with your e which allows you to knockback whenever he/she tries to jump on you. If the assassin tries to roam, you can even buy an early tiamat to outpush.

The second matchup are mages. Although they are ranged, there is no way they can possibly outtrade you in extended trades and at early levels. Since with ability haste, your q is on a short cooldown so you can use it to dodge any cc and skillshots that come your way. Even if you get hit by cc, you’ll be taking cleanse in order to deactivate the cc. There will be some control mages that you’ll struggle into, but for the most part you outscale them and always win late game.

Hail of Blades gives Vayne the bonus attack speed that Vayne lacks in her core build. Not only that, Vayne’s Q-autos don’t consume HOB stacks! With this, you’ll be able to get much more w procs that you wouldn’t have got with press the attack. HOB gives an incredible amount of trading power early which lets you dominate the lane.

With this build, you’ll be triggering sudden impact every time you q, essentially giving 7 free lethality. At level 1, this is equivalent to dealing 10% additional damage in the mid lane.

Take Eyeball collection since it’s the only viable option.

Ingenious Hunter gives item haste for your spellblade passive. At max hunter stacks, you’ll be able to proc sheen every Q! This rune is pretty much required to maximize sheen proc damage if you have over 40 ability haste. In the Sunderer build, feel free to switch out Ingenious Hunter for Ravenous Hunter.

Secondary options:

Precision gives great options in Triumph/Presence of Mind for Heroism and Legend: Alacrity/Bloodline. Triumph is one of the best if not the best rune in the game, while Presence of Mind gives much needed mana regeneration for the lane. Either are fine depending on the matchup. Alacrity and Bloodline are both very good on Vayne but Bloodline is much better with this build. The extra attack speed from Alacrity shakes up the attack speed of the build with 2-3 items making the rune choice inefficient in most scenarios. Bloodline on the other hand is great as it gives up to 9% lifesteal. In the Divine Sunderer build , you'll have almost 40% lifesteal in addition to the healing from spellblade!

The inspiration tree gives a ton of sustain and fixes a lot of the mana issues. Cookies gives so much early game power and makes laning so much easier. Time Warp Tonic is almost a must when taking Corrupting Potions. However, these two runes don't scale aside from 150 extra mana into the late game. I'd still recommend this build in most scenarios.

The sorcery tree is mainly taken for the Nullifying Orb rune against AP assasins and burst mages. The 40-120 HP shield can save you and sometimes turn fights around in your favor. Transcendence gives Ability Haste and scaling into the late game. Gathering Storm is another option for hyperscaling, and is especially good into farm matchups.

Cleanse is really good on mid-lane Vayne especially into champions that have CC. I go into detail when to take Cleanse in the matchup guide above.

Exhaust is the rune of choice into assassins. Since your very squishy during the laning phase, this item helps a ton with dealing with them.

Although it seems troll at first, ghost is a really fun and very good summoner spell choice. With all the mobility in league right now, ghost allows you to kite and chase enemies. Take this summoner if both exhaust and cleanse aren't viable.

Ignite is only for playing against Sylas and minor threat matchups. That's it.

With how busted Divine Sunderer is, there isn't much reason why Vayne shouldn't build this item. This mythic powerspike gives so much power from the get-go that most ADCs can only dream of. All the stats are great, even the 400 HP which helps Vayne semi-front line as a short ranged in-your-face ADC. The tank-shred on the item pairs really well with Vayne's W and the next item in the build path: Blade of the Ruined King.

BORK synergizes really well with Sunderer as the tank-shred on the two item combination is absolutely nuts. I first theory-crafted this combo in patch 11.10 to not only be good on Vayne but champions such as Kindred and Viego (even Lucian) over a month before the item became meta in 11.13. Not only that, but this two item spike is mathematically efficient as your Q will be exactly up after exactly two AAs without HOB.

The following four items can be bought in essentially any order:

Bloodthirster gives a bunch of AD and a whopping 20% extra lifesteal. As said earlier in the runes section, with Legend:bloodline Vayne gets up to 40% lifesteal! The passive shield gives you even more survivability in teamfights. If you get to this three item spike, you are essentially unkillable. .

Rageblade gives even more on-hit and doubles the total on-hit every 3 auto attacks. You might say rageblade isn't effective without crit, but as long as you already have an average of 40 on-hit per item or higher, the buy is efficient.

Runaan's Hurricane gives a heck ton of attack speed for late game teamfights and essentially TRIPLES on-hit effectiveness when the enemy team is clumped up together.

At this stage, Guardian Angel solves your only weakness, burst. With 40% lifesteal, your healing is unmatched against any sustained damage over time. If assassins try to jump in and one shot you, you'll team CC him/her and you'll quickly heal back to full after you GA pops. As long as your team isn't 10k gold behind, there is essentially no way that you lose the game if you get to this stage.

Essence Reaver is actually huge on Vayne. It gives her a ton of stats: attack damage, crit, and ability haste, mana while also being 104% gold efficient. However, it’s sheen passive as I discussed earlier is what makes it great on Vayne. You might think that the mana isn’t very useful, but when maxing q in a solo lane, it’s actually really important (you run OOM much much quicker than you would expect). Sidenote: Vayne actually has the lowest base mana level 1 in the entire game!

The next item is Duskblade. This item gives a lot of AD and a lot of ability haste. Even though lethality isn’t the greatest on Vayne, it still gives Vayne much more damage to squishies. However, the Nightstalker passive is what makes this item good on Vayne. Nightstalker gives Vayne even more burst, but more importantly Vayne is given invisibility for 1.5 seconds on takedowns. This pairs well with her ultimate, which on takedown extends its duration by 4 seconds. This invisibility also can pair with invis on her q, which can allow her to be fully invisible for up to 2.5 seconds. This can allow her to reposition during a fight and hunt down her next target.

Phantom Dancer gives great sustained damage for teamfights since it gives a ton of attack speed. It also synergizes really well with Hail of Blades, as it’s passive triggers right as HOB runs out. Phantom Dancer’s movement speed also helps you stick on to opponents well, as ult+PD gives the same amount of movement speed as Turbo Chemtank.

Navori Quickblades gives great AD and a ton of Ability haste, as well as crit. With Essence Reaver and Duskblade, you’ll get a total of 100 ability haste, which makes your Q on a 0.5 second cooldown with ultimate. Quickblades' passive massively reduces your E as it only has an effective cooldown of only 4 seconds! (with HOB)

Wit's End is a very great item on Vayne. The 50 MR essentially means that you take 1/3 less magic damage while the 40 AD and 40% attack speed gives so much offensive power. The passive on the other hand, is on a totally different level. It gives up to 80 bonus magic damage on every single auto attack and 50 even if your buying it as your first item. Take this item first into literally every AP mid-laner or comps which have 2-3 sources of AP.

Trinity Force is basically a budget Sunderer with attack speed. The extra attack speed allows you to take defensive boots while remaining mathematically efficient. TLDR build Tri-force and Steelcaps into AD instead of Sunderer and AS-boots.

The stats on this item make it feel very bad to buy so only build this if you really want the QSS. Take this instead of either Bloodthirster or GA as either your 4th or 5th item.

Although this buy seems pretty troll at first, this item is actually quite effective! I saw C9 Perkz build this item on Kalista in C9’s win vs MAD so this item on short-ranged ADCs is actually viable! Vayne has low range, which means that she needs to be up and close in the enemy’s faces to deal damage. This item gives you ability haste, HP for tankiness, and a ton of armor. Assuming not taking any other armor items, the 80 armor reduces physical damage by 30% compared to the 18% on GA. The item’s actives are also quite useful. The slow gives you insane self-peel against assassins and when going in, and the AD reduction essentially gives you an effective 40 armor for 4 seconds. The Rock Solid passive also helps when dealing with the damage from the enemy AD carry(s) as well as AA heavy fighters. You might say that Vayne loses out on damage by going for this item, but think about it this way. Assuming you're taking sustained damage in a teamfight in which you die, by taking 30% less damage you’ll deal 30% more damage! Buy this item when your death recap is 70%+ physical damage and if you feel squishy.

ou should look to buy a grievous wounds item as your last item. In most cases you probably won’t need grievous wounds, so only buy these items if you're into a fed healing carry or into a fed ADC with an enchanter support. As for the items, there are two options: chempunk chainsword and mortal reminder. Chempunk gives more AD, ability haste and HP, while mortal reminder gives little AD but AS, crit and MS. Sepsis is better than Hackshorn on Vayne, so you should look to buy a mortal reminder in most scenarios. If you skipped an ability haste item (if you have less than 100 ability haste), you should buy chempunk chainsword instead.

If you wish to know more about the Calculus behind this build, you can read this slideshow, the full document, and/or check out my spreadsheet. If you made it this far, thanks very much for sticking with me in reading this guide, and I hope you have success trying this build in your games.

Thanks,
Fanchessfan

2018 AP Physics 1 4, 2019 AP Physics 2 5, 2020 AP Physics C 5, 2020 AP Calculus BC 5, 2020 USAPHO qualifier, 2021 MATHCOUNTS Regional #1, 2021 AIME qualifier, 2021 USAPHO Winner, 2021 US Physics Olympiad Team member.

Last Semester I finished Multivariable Calculus and I'm currently taking Linear Algebra and Differential Equations.