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CORKI AP FLEX
An AP Corki can be a fantastic support player. Hang out behind a tank or solid DPS character and bombard enemies with magic damage Missiles and Phosphorus Bombs. This build provides around 500 AP at level 18, once all bonuses and runes have maxed out. This means that each missile deals 475 magic damage, and his Phosphorus Bomb deals 530! And with the cooldown provided by Frozen Heart, Corki can fire one missile every .9 seconds and a Phosphorus Bomb every 6. That means that in six seconds, Corki could deal at least 3910 damage [/i]to multiple enemies[/i]. Pair him with a good crowd control character (Anivia, for example), and you'll be raking in the kills. But the real beauty of this build is the item sequence flexibility. But first of all, I'll explain the logic behind the "default" build:
We start off building towards the Archangel's Staff. It has everything we're looking for: AP, Mana, and Mana Regen, PLUS the all-important passive which turns 3% of our max mana into AP. Rod of Ages is next -- great AP, health, and mana, which now gets turned into AP. Plus, this is presumably still rather early in the game, so you'll have plenty of time to get the most out of its passive. Once we have Frozen Heart, we're ready to be a bit more aggressive. We have Armor, HP, and good AP, plus our cooldowns are lowered considerably. Last is Zhonya's Ring, which provides a HUGE AP boost at this point.
But here's where the flexibility comes in. Blasting Wand is an ingredient in all three AP items in the build, which means any of them could be built first. And the Sapphire Crystal is used in Rod of Ages, Archangel's Rod, and Frozen Heart. You could also use your Blasting Wand to build towards a Void Staff if the enemy is building up Magic Resist, or Abyssal Scepter YOU need magic resist. You could even build a Guinsoo's Rageblade if you wanted a bit more balance and some synergy with Corki's Big Bombs.
ALSO, you'll note I didn't include boots in my build. I STILL RECOMMEND THEM! I merely think that which boot you want depends entirely on the game. Magic penetration is redundant if the enemy has terrible Magic Resist, so in that scenario I would go with Berserker's Greaves over Sorcerer's Shoes. Boots of Swiftness can be extremely useful with Corki, as he can create some distance and really make use of the range of his missiles.
Other possible items not mentioned: Manamune, Soulstealer, Nashor's Tooth, Rylai's Crystal Scepter.
I'm a guest right now but I may be making an account and copying this build later under my LoL character name, A Blank Stare. :)