Rumble is (luckily) a champ that is hardly played, although he got a lot of potentail and power, especially in early- and midgame. A well played rumble on a dual lane is mainly a guaranteed lane domination, because Rumble is a heavy spammer that can deal high amounts of damage, especially while chasing enemies down.
Rumble is also able to do good initiates with his ultimate ability The Equalizer, that has an awesome long range. I place him as an Offtank because your damage will not suffer significantly if you buy the "tanky" AP-items.
Although Rumble has no hard CC, his slows are truly good enough to dominate a lane early on and his tankiness in earlygame makes him a really hard enemy, although he got nerfed a bit. And last but not least, not everyone knows Rumbles abilities, because he is rarely played. Especially below Level 30 you might be able to run a game easily.

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Marks: Magic damage is Rumble's main damage source, so I prefer Greater Mark of Magic Penetration

Seals: Laning top, most enemies will deal physical damage. Additional armor by [[Greater Seal of Resilience will increase your survivability notably

Glyphs: For some early damage I like Greater Glyph of Ability Power, but if you want to scale into lategame, take Greater Glyph of Scaling Ability Power

Quints: In my opinion there are two viable choices: Greater Qunitessence of Fortitude for notable early durability or Greater Qunitessence of Potency for even more damage.

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I prefer to start with Doran´s Shield. It gives you an allround bonus with 120 HP, 12 Armor and 8 HPregen/5s. If you want to build more tanky you can also take a Regrowth Pendant and build it into Force of Nature but thats not the common way. But if you experience to meet a heavy harasser dealing non-physical damage, go for the pendant in any case (e.g. Galio), though you might sell it later.
And if you feel that you will dominate the lane wihtout much effort ( Nasus etc.), take Boots and 3 Health Potion.

If you get back get the Mercury's Treads because this are the best boots in the game with the 35 Tenacity bonus and get a Hextech Revolver if you think that you will stay in the lane for a longer time. Also, try to afford a Giant's Belt making you almost unkillable in combination with the revolver.

Then start to build your midgame core; after about 35 minutes it hsould look somewhat like this (excuse me if the time scale is not that good):


This gives you about 1000 health, 45 armor, 25 MR, 2 MS, 120 AP and 12% Spellvamp. You will be really tanky and deal awesome amounts of damage because the basedmg of your skills is set really high.

Now you have the full amount of health bonus, the much needed slow on that Flamespitter (that is really beasty!) and a basic amount of AP.
What you buy next depends on your team and you enemies. If you do not have another guy in your team who buys a Will of the Ancients, you will definetly be the one, since you already have a Hextech Revolver.
In almost every game, I go on with Abyssal Mask since it gives a great amount of MR and 70 Ap for a price that is quite acceptable (2650 i think). Besides, it has an aura cutting down enemies' MR which gives you the much needed Magic Penetration. Your APCarry will love you for that.
For a RAPID boost of your own damage I suggest you get a Rabadon's Deathcap preventing you to fall off in lategame. This gives no durability at all, but damage like nothing else.
If you did not already build Will of the Ancients i recommend to build a Hextech Gunblade now, since its stats bonusses (is that right?) are great and the active is beyond useful. You can really set anything into slow motion here.

Some remaining alternative items are a Void Staff if you already have a tanky team and need to deal more damage, a Force of Nature if you have to be the tanky one and another one that is really good is a Guardian Angel but in most cases I find the items I listed above more useful.

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I recommend to start with one point in Electro-Harpoon since it is the best tool for early ganks and can be used to lasthit on range that you will need from time to time. Besides it deals more damage than a level 1 Flamespitter.

At level 2 i mostly spend a point in Scrap Shield, depending on the ability of my enemy to harass ( Parrrley, Cull the Meek, Siphon of Destruction) and the little but notable movement bonus is also not bad.

Then i take 3 points in Flamespitter in a row, since it is your main damage source and max it first.

At level 6 i skill (of course) my Ultimate, The Equalizer; in lane i mostly use it to finish off enemies that are low anyway. I set it in their Movement direction and follow with am activated Flamespitter and try to get some shots with the Electro-Harpoon on them before they get off the AoE of The Equalizer. It is also a great ability to initiate a gank if your opponent doesnt have flash.

Well then i go on maxing Flamespitter followed either by Scrap Shield if you are harassed a lot or by Electro-Harpoon if you burn your enemy down. But in most cases i get these two skills up balancedly.

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My personal favorites are Ghost and Teleport.

I recommend Ghost because your main job in the entire game is chasing. I dont think there is a better chaser than Rumble since you have an ability to boost your own MS and you have a lot of slows... basically you slow with everything you do since you have Rylai's.

And I recommend Teleport, which is kinda Must-have for laning top. You have no spells to heal yourself early on as most opponents will and you are very item dependant, so going back for shopping is no problem anymore. Besides, you can set up some great Tp ganks ;)

Since the nerf for Flash i don't like it anymore and prefer Ghost instead, well, at least on most champs and on offtanks as good as always.

Ignite is also okay to secure early kills and MAYBE Exhaust but i really do not recommend it (another slow.. WHY?!) . And forget about the rest.

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You take solotop. Nothing else. Better: You only pick Rumble if you have ensured to go solotop. In rare cases you can go mid to counter Mordekaiser or even better Akali. But because of the bigger distance between the tower and your need to chase, laning top is better than mid.

You must harass your enemy out of lane. Rumble has no comparable lategame potential like Nasus or Irelia or even Sion (AD) and is mainly a counterpick against champions who need a long farm in order to be good.

Try to dominate the lane early on and harass strongly with Electro-Harpoon and Flamespitter. Once your enemy is low health, finish him off with the combo using The Equalizer I described above.

If you see any kind of incoming damage, activate the Scrap Shield quickly.

For a quick harass, hit once or twice with the Electro-Harpoon, chase with the Flamespitter and turn away with Scrap Shield.

And always watch your heat! not being able to use your ultimate because you overheated is a pity.

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In teamfights, always try to damage as many enemies as you can with Flamespitter and The Equalizer and chase those who flee with Electro-Harpoon.

You have an OK initiate shining by its range but there are recently better ones; if you can enter the fight second because you are disabled by any kind if snares and movement control; you have to come up close to deal damage.

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Well I just reworked the build all over and nothing stayed same, because it was simply outdated. I didnt even play the way i described on my own anymore because it was simply not that good. I hope you like it as it is now! :D