Changes

I've issued some changes on the items. I replaced Atma's Impaler by Frozen Mallet and moved Trinity Force up a bit. Also, I replaced Youmuu's Ghostblade for Last Whisper, because I think Last Whisper makes you deal way more damage. The only reason for you to get Youmuu's Ghostblade anyway is the CDr.

What makes this build innovative?

I have been looking around on mobafire, and I have not seen a Gangplank build with Sheen in it. Why is that? Sheen is the best item for gangplank. It is the key to playing him. Why, you ask?
Because it makes your Parrrley so much better! It doubles your AD when you use Parrrley, so it gives you much more damage.

Level 3 with this build AD(54*3*3) = 63; bonus damage = 55. Damage: 55+63*2 = 181. No crit chance. Level 9 with this build AD(54*9*3) = 81; bonus damage = 120. Damage: 120+81*2 = 282. Crit chance (30%): 282*1.30 = 366.6.

Level 3 with most other builds AD(54*3*3) = 63; bonus damage = 55. Damage: 55+63 = 118. Crit chance (20%): 118*1.2 = 141.6. Level 9 with most other builds AD(54*9*3) = 81; bonus damage = 120. Damage: 120+81 = 201. Crit chance (42%): 201*1.42 = 285.42.

Difference at level 3 181-141.6 = 39.4 more damage. 39.4/141.6 *100 = 27.8% more damage. Difference at level 9 366.6-285.42 = 81.18 more damage. 81.18/285.42 *100 = 28.4% more damage.

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Pros Lots of damage Excellent staying power Killer, even in early game Ultimate ulti range Good heal and cleanse

Cons Needs a lot of money Squishy when focused Doesn't have a good CC The ulti is very random

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Runes

9x Greater Mark of DesolationGreater Mark of Desolation: as proven and explained in Rune tips - what secondary and primary means and more, armor penetration is better than critical chance. In any case, except if you buy Last Whisper, which we don't, because we already have the armor pen. (lolwut..? :P)
9x Greater Seal of Evasion: I think this is the best seal in this case. We don't need HP/lvl and we don't need M/5/lvl, because Sheen gives us enough mana.
9x Greater Glyph of Focus: We have an ulti with a really long cooldown, so this CDr is very nice. Also, I don't think that MR or MR/lvl are superior to CDr in our case.
3x Greater Quintessence of DesolationGreater Quintessence of Desolation: same as for the marks.

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Items:

1. Start: Sapphire Crystal ( Health Potion = optional)
2. 860g: Sheen (important!)
3. 850/1200: boots ( Ninja Tabi > Mercury's Treads)
4. 750/1630g: Avarice Blade (+ Cloak of Agility)
5. 880g: Cloak of Agility if you dont have it yet
6. Then complete your Infinity Edge
   
7. Phantom Dancer; start with Cloak of Agility or Zeal

At this point you have either won the game, or are really overfed. Buy something to boost your damage output even more, like Bloodthirster or Sword of the Occult.

Sheen - this item is really good on Gangplank throughout the game. It gives us 100% crit chance with Parrrley or after the use of another ability like Raise Morale or Remove Scurvy. Although it does not give a real crit with Parrrley, as the bonus damage is not doubled, it is only 1260g, which is very early in the game, and gives us mana to harass a lot in the laning phase. Also, it's passive stacks with normal crits, which makes us critting double!

Ninja Tabi - You can take Mercurys Threads against heavy CC teams, but they'll have to be really heavy. I think Ninja Tabi is best, as Mercurys Threads are not used that much, because you can just clear all CCs immediately with Remove Scurvy. Also, Ninja Tabi is much less expensive than Mercurys Threads. Lastly, I don't take Berserkers Greaves, as we do most of our damage with Parrrley early in the game, and I think you need more survivability in fights. Boots of Swiftness are also possible if you really want more speed.

I think the rest of the items are obvious. You want them for crit and damage.
We don't take Lich Bane of course, because we don't need AP; it only makes the healing done by Remove Scurvy a bit better.

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Exhaust A great spell. You do not have a CC ability, so let's pick a CC spell.

Ghost Another great spell. Use this to run to fleeing enemies, or away from a gank.

NO REVIVE You will probably hear this a lot. I will say it again. This summoner spell is bad. It is just really bad. This spell is why you die, not why you do not spend time dead. Nope. You do not need it. You will not take it. Sheen gives us enough mana. If you still have mana problems, you either spam your abilities too much, or you should just recall. You can also replace Greater Seal of Evasion with Greater Seal of Scaling Mana Regeneration.

Lolwut..? I am pretty much insulting you now by thinking you would even consider this when playing Gangplank. I still have to say it, though. You already have a heal, dummy. Again? Ok, this is the last time I will insult you. You have a Cleanse under W, just use it properly.

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How to play

Laning phase: You can try to get first blood with your laning partner. This will give you 400g (or 200g) very early in the game, and you'll almost be ready to buy Sheen, which is our most important item. You can deal at least 200 damage: 70 with ignite, 80 with Parrrley and about two normal attacks which will deal 2*60 and then your poisondamage. If you're up against a Veigar, you can easily kill him.
If you don't go for first blood, you can still go in the brush often, and shoot the enemy with Parrrley, and get a kill after some harassing. Remember: with the 200 extra mana the Sapphire Crystal gives you, you can harass a LOT.
Don't forget to last-hit minions with Parrrley. It gives you quite a bit of extra gold, and boosts your development.
If you are dominating your lane, play aggresively, but if you do not, hug your turret like it's a huge lasershooting teddy bear. You know, the one you sleep with.

Mid/late game: Try to kill the lizard when you have the time. That buff makes you look like you're some kind of pirate... Wait, what?

Teamfights: Use Raise Morale to make your team do more damage. Also time your Remove Scurvy correctly. For instance: when in the rim of Veigar's cage, or when you get Exhausted.


Ultimate

• When in a teamfight, use your ulti in a way that all enemies (might) get hit, and try to aim it a bit to the direction you think the enemies will go. If they are losing, aim your ulti a bit more to the direction of their escape route.
  
• You can also use your ulti to finish off an escaping enemy. Do this by aiming the rim of your ulti at the enemy, with the other end in front of him. That way, they'll have to run through the entire thing.

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Conclusion

Thanks a lot for reading this far. I think this guide is the best guide to Gangplank, because it may be the only one that uses Sheen. And as I calculated, Sheen makes your Parrrley a whopping 28% more effective! I think this guide will make you an OP Gangplank. Plank on, everyone.

Changes

I switched atmas impaler for Frozen Mallet. I have read the comments and I have decided that Frozen Mallet provides better benefits. You can also buy a Bloodthirster instead of Frozen Mallet.

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Credits:
Rune tips - what secondary and primary means and more is a very helpful guide. You should read it. Also, Searz has helped me out a lot. Thanks for that. :)