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I saw many Poppy builds here, but only few of them had AP with focus on survivability. Here is my AP/Tank one. In this build, Poppy isnt BEST nuker or BEST tank, but clever combination of them (I dont think she is both best nuker and tank xDDD), which make her very effective early-mid-late game ;))

FIRST BUILD IS FOR DEFENSIVE START, SECOND BUILD IS FOR AGGRESIVE START, depends on your playstyle and your enemies/lanemate

Feel free to say anything good or bad to comments..

Pros:

+ She can deal heavy amounts of damage (if you consider fact, that she is AP/Tank Hybrid)
+ Tough
+ Fast as hell
+ Had high chance to win 1v1 fight
+ probably the BEST towerdiver at game (if have ulti)
+ Awesome teamfight initiator, when ultimate is ready
Cons:

- Really horrible farmer
- Isnt good pusher
- Low mana pool
- Havent any Ranged ability

Runes with focus on:
Magic penetration : (you are AP in this build), you need this to make your Impacts of your shiny Hammer even more Devastating...
Armor: because there wasnt space for +Armor item, but dont worry, just press W and keep your stacks alive for more Armor...
Mana regeneration: Poppy is Mana-starving almost whole game, I know, that isnt enough, but with Morello E.T. you will be at least more durable, becasue going to base after each Nuking isnt much good for you...

classic 9/21/0 ... what to say? i focused on Defense, which is needed if you want absorbe some damage, becasue
MORE SURVIVABILITY = MORE TIME = MORE DAMAGE TO DEAL
and also some points on AP/CDR/M.Pen, which is standard for every offensive caster...

Boots of Swiftness : poppy needs to be fast, all other boots are waste, all these bonus from them i have at Masteries/Summ.Spells/Runes/Other items, main is movement (you can buy Ninja Tabi, if you fight against DPS-team)

Warmog's Armor : what to say? maybe only "MUAHAHAHAHAHAH" (If you rather use Frozen mallet, you can exchange Warmog for it, Frozen Mallet is also good thing for Poppy)

Force of Nature : nice magic resist (Making you UNABLE to be slain by Nukers with combo with your PASSIVE!), also HP regen and great movement speed bonus, (If your team has no Mages, then buy Randuin Omen)

Lich Bane : Core item for my AP Poppy, (Outcome: E(+stun), lich bonus, Q, lich bonus, Q, li.. wot?.. you are already dead?)

Morello's Evil tome : nice thing for AP, mana regen and also CDR, for Q-spamming

Mejai' Soulstealer : good source of AP + if stacked, gives more CDR (if you arent sure, you will handle all 20 stacks and want to be less squishy against DPS-ers, buy Thornmail..)


PS: at start, buy golden heart, will help you with fact that Poppy is terrible farmer, and also makes her some survivability early-game

Your Q: Bread'n'Butter ability, makes you very dangerous.., also have very low Cooldown in my build, so feel free to spam with..
Your W: DEMAAACIAAAA ability, this is very useful thing, you will save much money with that (+35armor/Dmg on rank5), and +MS, this is something like your THIRD summoner spell, im calling it "Nano-Ghost"
Your E: Tricky ability, learn how to use it to make adittional dmg and STUN!, great thing to initiate your hammer storm
Your R: This is most awesome skill ever, which makes Poppy Poppy.. tons of uses (anti-gank, initiate a TF, towerdive, pushing, 1v1...)

For skill sequence look up
PS: rank up E as last, because using on rank 5 it will punish your mana early game, and main effect is STUN, ans stun will be not affected by rank (and some DMG+...., your Q is main source of DMG..)

Flash - is great for landing your E to follow him make it stun, escaping, getting closer to your target... so much uses (better than ghost because you have W for ghosting
Cleanse - great for Poppy, without that thing, your havent chance against heavy-CC enemies and almost all champions have some CC, or Frozen Mallet at least, so it will never be wasting..

(if you want Exhaust instead of Cleanse: Modification Ghost/Flash)

If your have any questions on my build, plz write it as comment, I would like or either Good or Bad suggestion

Also excuse my gramatics, im not native speaker :))

Have fun with Poppy ;)

...COMMENT/RATE PLEASE...

Updates:
2011-04-14 : Created basic idea
2011-04-15 : Added much more informations and some cosmetic adjustments
2011-04-16 : Added picture of Chinese-Noxus-Poppy as intro and added tree for Exhaust/Flash
2011-04-17 : Added defensive/agressive build for different playstyles and few tips to Items