I know I'm probably going to get a lot of downvotes for this, but PLEASE LISTEN

AP Lee Sin is actually VIABLE. Here's how it works:

Your abilities have 2 uses, so you can constantly proc Lich Bane.

Flurry scales well off attack speed, and slightly off CDR since you can spam Tempest more, which is an auto-attack reset.

And finally, Resonating Strike gives you free 8% Missing Health damage so you don't even need AD to do damage! Awesome!

Since Lee Sin doesn't do magic damage until he gets items, don't run any magic penetration runes.

Instead, run flat AD so you can be stronger early game and bully around your enemy laner.

Flat Armor is for in case you're against an AD assassin mid, but otherwise it's still good for mitigating auto-attack harass. Lee Sin is prone to this because of his short range.

You can switch out the Quints for Move Speed quints if you think you can abuse your mobility to bully certain lane opponents, like Syndra or Nidalee.

Run 21/0/9.

Take hybrid penetration in the offensive tree. Taking Spellsword is a MUST.

Anything else is up to you, really.

Start with:

Boots of Speed and 4 Health Potions. These let you bully around those noobs who dare fight you in mid lane because you have the mobility advantage.

Doran's Blade is good too, but you're AP Lee Sin and you don't need AD.

Doran's Ring is stupid because Lee Sin has no problems and 1 AP Scaling.

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Next, you want to rush a Sheen. It synergizes very well off Broken Wings, and it's good for harass since it basically doubles your AD for one attack.

Your next item is Berserker's Greaves, because it gives Attack Speed and Lee Sin doesn't benefit off Spell Penetration. With increased movespeed and attack speed, you're able to bully the other laner with Flurry more easily.

The first big item is Nashor's Tooth. Not only does it give attack speed so you can pull off more Flurry procs, BUT the passive that converts 15% of your AP into bonus on-hit damage is OP. I mean, if you do the math, 2000 AP gives you 300 damage! Who needs AD when you can go AP?

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Next items: Lich Bane, Zhonya's Hourglass, and Rabadon's Deathcap.

Lich Bane first, because you can constantly proc it with Sonic Wave for a lot of damage. Also, with the S3 change, it has a base value, so you don't need to stack 1000000 AP before it starts doing damage.

Zhonya's Hourglass next, because you can Ult > Sonic Wave > Resonating Strike > Hourglass > Safeguard > Flash for a super trolly duration of untargetability and mobility against people. Talk about unkillable!

Rabadon's Deathcap is last, so your Lich Bane and Nashor's Tooth procs will start to hurt a lot!

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Situational:

Liandry's Torment - If the other team is stupid and they keep letting you stun their tanks, then buy this so while they're stunned you can shred their health really really fast :D

Twin Shadows - Are people still getting away from you? Then uninstall you ****ing idiot.

Tear of the Goddess/ Archangel's Staff/ Seraph's Embrace - If there's a lot of burst on the other team, consider buying an early Tear and charging it so that you have an available shield to protect you from burst. Lee Sin also charges it quickly because of the short cooldown of Everything! Lee Sin is also constantly oom so he needs ways to replenish his mana quickly, and Tear does that really well!

Iceborn Gauntlet - I don't like this item, yeah the stats are nice but IMO it's better on tanks and Lee Sin has enough chasing power already. I guess this can replace Lich Bane if the other team is 5 AD.

Guinsoo's Rageblade/ Hextech Gunblade - These aren't good because they're hybrid items, but buy Guinsoos if you want more AS and Gunblade for more chasing power + damage + lifesteal/survivability.

Deathfire Grasp - You don't really do magic damage but the stats are nice. 120 AP is a sizable amount, and CDR to be an annoying stealthing ***** is nice too.

In mid, make sure you're against a champion who

1) Can't gapclose very well,
2) Has a lot of skillshots, or
3) Is controlled by an unskilled summoner (preferably Bronze or unranked)

Lee Sin's main strength is his mobility, so if you're against a Nidalee you can just Safeguard away from her Spears then counterattack.

If you're against someone with all targetted abilities like Ryze, you're ****ed you might as well bend over and ask your jungler to camp.

Which he won't do, because you're AP Lee Sin.

You want to play passively because you aren't that strong early on, but if you see them commit to CS, Safeguard towards them, resetting your AA timer, and you're able to get a free hit on them.

At level 6/7, your power spikes tremendously because you have obtained your ultimate. If you see them out of position, you want to Sonic Wave > Dragon's Rage > Resonating Strike, then keep auto-attacking them while trying to avoid their skillshots, and hopefully you force them out of lane via death or them being low on health.

Late game, remember that you're AP Lee Sin and your only real strength is being annoying with shield + mobility, slight damage, and carry shredding Flurry. Use Dragon's Rage to peel for your ADC, or you can use it to pin down priority targets to walls and hopefully you can finish them off with your team.

AP Lee Sin is quite an underrated pick with a Huge Burst, good sustained damage, and an AMAZING Low CD Shield. I mean, why pick Shen when you can go AP Lee Sin?

I went 5/7/3 as AP Lee Sin, but I lost because of noob teams! go figure!