Corki was super OP back when the only ranged AD carries were Trist, Corki, Ashe, and Sivir. His jump makes him tied with Trist and Cait for situations like fighting a J4/Anivia, but his other damage spells let him do the bursty AD damage you expect from Ezreal and Vayne. He's the champ I play in tournaments when I need a slightly more durable AD, especially if they have a major need for movement mechanics.
Anyway, I know you guys love seeing pretty screecaps of people getting cool scores as Corki. I don't see the merit in doing that whatsoever... anyone can run into a bot game or roll a new account and slap bunch of noobs around. Some people take their ranked stats and stuff too but I also think that's pretty inconsistent, partially because I mess around in plenty of ranked games... also I didn't screencap my Corki stats last season. HOWEVER, the #1 time I do tryhard in games as Corki is in tournaments, so here are a couple of screencaps I could not have faked, that actually say a little bit about my successes.
Pretty simple, take MR blues if they have more than one threatening AP caster; take cooldown blues if not. The ability to regenerate your missiles faster is pretty useful, plus the quicker valkyrie cooldown helps occasionally.
I use ArP reds like every AD player and their dog, but unlike most of them I run AD quints. AD quints and ArP quints are more or less tied in value throughout the possibilities, with ArP being stronger on low armor targets and AD being stronger on tankier targets, so either could be better or worse depending on the phase of game in which you want to excel. However, like MF and Ezreal, Corki has a move that scales to AD yet deals magic damage, therefore gains from AD quints and not at all from ArP quints. For this reason I use AD quints for my general AD pages.
As they were with the previous mastery choices, my masteries are once again unconventional. I run 26/3/1 for my AD carries nowadays and, of course, always adjust as needed based on the enemy team.
I would like to note immediately that the 3 points you see in defense should always be moved appropriately between armor and MR, anywhere from 0-3 against an all-magic team to 3-0 against an all physical team, or usually to 2-1 against a standard team with 1 AP and the rest physical-based.
I like where all of my points are in offense, and I think you will too. Typically when someone goes 21 in AD they omit 1 of 3 things that I have: 5 points for armor pen/attack speed, 5 points for crit chance/damage, or 3 points in havoc, or some mixture thereof, of course. Obviously each of these masteries add damage, but it's to each person's discretion which they prefer. In my case, I refuse to lose any of them, because if I'm going to spend half an hour shooting people with a demented yordle in a small plane, I'm going to do it voraciously.
What I lose from going 26 points into offense is utility/defense choices. Probably the most common of these would be to go 21/0/9 and get buff duration in utility. While that is quite legitimate, I don't feel it's worth 8 points just to get buff duration and 2% movement speed, since the mana is nearly inconsequential. Similarly, getting a few more points in defense can be viable, but it's simply a matter of preference.
In general, I would encourage anyone strongly interested in ADs and/or Corki specifically to experiment with various masteries like I did; 26/3/1, 21/0/9, 21/9/0, 21/8/1, et cetera.
I don't take a manamune on Corki because he doesn't need it. If you know you just can't live with yourself without spamming your spells then go ahead and take one. However, any high ELO player will tell you that spending early game money on mana or mana regen on a champion who doesn't directly benefit from it (like Ryze or Blitz) is a waste of the money you could be spending on lane advantage.
Go straight for high damage and you will kill stuff, just like any other ranged AD. Think of it this way: if you just farm as well as the enemy AD and get the same items as them, then you will do more straight DPS than them, because of your stupidly strong passive. Most standard ADs in lane only have moves that put them on par with Corki's spell damage, which is automatically going for you. Add to that your passive and all you need to be doing is thinking of yourself as just another ranged AD.
With the exception TriForce. Corki does stupidly strong with TriForce if you can learn to use it well. Get the sheen part first almost always because that is the very reason you're getting TriForce: the sheen proc. Corki does tons of damage mid-game if you staggar sheen procs and his missles.
After TriForce you can just go standard AD stuff and kill stuff. I usually go a more AD-centered build because of his three abilities that scale with AD. Beyond that, I typically get a veil mid-game because of the general obvious survivability.
Oh yea, and I didn't list lvl-2 boots in my build. Take either Merc boots, Ionian boots, or berserker's greaves. It completely depends on your opponents so I don't want to confuse anyone by suggesting you should always get one or the other.
Update 11/10/2011: I'd like to remind everyone that on about 75% of champions, you should not take a skill at spawn, and Corki is among them. Taking your skill at spawn or before you actually use it is a terrible practice that gets people killed or makes them miss opportunities all the time, and it's a noob mistake. Do not take your skill on Corki until you intend to use it; this means don't skill phospho bomb until you are in lane about to shoot it, or doing a well-positioned team invade. On the flip side, if **** gets nasty in the first couple minutes, you can save your flash by skilling valyrie and jumping a wall.
Not too complicated. Phospho bomb may not scale to AD but it is the move you use almost entirely for damage early- to mid-game. After that's stacked the game is late enough for valkyrie to have absolutely no value as a damaging move, so stack the gun that scales to AD second.
Flash is the best summoner spell, so you should almost always take it. Some people will tell you that you shouldn't take flash on Corki because he already has a similar move, so you can fill that summoner spot with a different move, like ignite. That's like saying "If you're going skydiving, one parachute is good enough; turn the other one into a stylish hat." They're losing sight of how ludicrously good flash can be, in favor of one extra gadget. Flash is almost always worth taking.
As for your second summoner, heal is the latest rage because people finally noticed that the AD tends to take more aggro than the support, so the heal is better on them. If your support has heal, you have options between ignite, exhaust, cleanse, surge (if you're a badass), and a second heal. Otherwise, you probably want heal.
Corki is exceptional for farming with autoattacks because of his passive. Also, unlike Tristana and a couple others, he has an instant damage move for picking up cs you might otherwise miss with just autoattacks. Therefore, you DO NOT have to spam moves to cs at all. One phospho bomb every couple waves to grab an extra one you'd miss, otherwise keep yourself nice and high on mana, hence the not needing a manamune. Later on, you can use combinations of missles and phospho bombs on the fly to tear waves of creeps apart and push your lanes as needed. With blue buff you can go nuts, but that's just for fun.
Most AD's has a very decisive shift in their playstyle from early- to mid- to late-game. Corki's is almost identical to Tristana's: Early-game he can occasionally use valkyie offensively, even for damage, and in general should only use mana for poking other champs in lane and grabbing otherwise missed CS. Mid-game he can participate in the fray of fights to deal all of his damage and contribute to soaking the non-AoE damage of the enemy team, and in general he is good on mobility mid-game and can run around clearing waves and jungle packs. Late-game he stays wayyyyy the hell in the back and does not get himself spanked by the enemy APs. Stay on the safe end of teamfights, throwing in your spells and attacks on the nearest logical targets until most cooldowns are down and he can safely participate more.
So one of my decent tournament games was recorded a while back, it occurs to me if anyone wants to see video this is the best example I can show since it does what I couldn't in a custom demonstration versus bot or whatever.
The missile kill on Chauster a minute in was spectacular timing, gotta say.
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