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Veigar is the best snowball AP champion if you ask me. It is all about last killing minions with Baleful Strike:[Q] to get the AP bonus throughout the entire game. Playing veigar means having patience until you can catch champions off guard with massive nukes. If you play him right, he will have more AP than any other by the end of the game. This build deals with veigar's mana problem by alternating between clarity and teleport to keep him in the lane for the longest amount of time.
9 0 21 is alway great for veigar because of his mana needs and cooldowns. getting 21 0 9 isnt really worth the small amount of extra AP.
The best Combo for Veigar is All ability power per lvl, except for magic pen for Marks, supporting the snowball effect, and making him harder as the game grows longer.
I always get Baleful Strike[Q] first to help me last hit minions, and once that has enough dmg to kill them at half life, I focus on getting Dark Matter[W] high, as it is the most valuable ability once it gets up there. I get Event Horrizon[E] early, encase the enemy or my team decides to gank early. Two points to event horrizon is enough stun until late game. Always lvl up Primordial Blast[R] whenever possible, and always aim this at AP heavy champs like annie or ryze.
When playing veigar, I always get clarity and teleport to help with his mana dependence. I switch off using clarity and then teleport whenever mana gets too low, so I can constantly be using [Q] for last hitting minions. (Using teleport first if my health gets low.) I dont believe any others can help him snowball near as well. Flash is an ok optinon. Ignite does work to finish champions off, but it never helps in late game because of the amount of dmg your other abilities do. The more time spent in a lane, the more experience, gold, and AP veigar gets.
No matter which lane I go, I always start with boots of speed and two mana pots. Having the extra mana pots in the beginning will always keep veigar in the lane longer to last hit the most amount of minions. Then I pick between Sorc Boots or Merc Treads depending on if there are stun or slow champions on the other team. Once I get enough gold, I imediately go for Megai's for the stacking effect (unless im doing horrible). After that, I am usually hurting for mana, so I get the Meki Pendant which usually suffices until I get Rod of ages. Late game, I pick up Zhonya's. By this time, these items together with all the AP you get from [Q] is more than you will need. (Should be about 500 AP) If you still find yourself with a lot of gold, I suggest Rylai's Crystal Scepter for the slows and extra health to keep you alive.
Early game: (1-6)
Mid: Focus on auto attacking until you can last hit minions with [Q]. It is very important to start building AP early. Dont attack champions with anything because it usually wastes mana that could be used for AP gain. Also, not attacking gives the enemy a false sense of power and they do not know how much dmg you do. With an aggressive mid enemy champ like kat, focus on just last hitting minions with [Q], but stay way back; remember, veigar is a glass cannon. He is very easy to kill if you are not watching his HP.
I suggest telling your lane partner to not be aggressive; as you veigar is usually not powerful enough to support a gank early game. You need to build AP before doing any dmg to enemy champs. Yes, you can use [E] and [W] to help if you have a great laning partner, but this usually doesnt happen. Besides this, refrain from using [E] or [W] until they are higher lvl because it uses up your precious mana. Veigar is not a great harraser because of his mana needs.
Mid game: (7-13)
Once I hit level 6 or 7 (depending on how im doing), I turn the switch and become a nuking machine. If you are laning, setup a gank in the bushes with your partner. If you are midding, wait till your enemy gets too close to your side and then blast them with a [E][W][R][Q] combo. You can more often then not, land all of these within a couple of seconds if you are quick and get quick kills. (I usually do Q last because of its AP gain if it is a finish, but its also a good idea to lead with [Q] followed by [E] and [W] if you think it can recharge before they get away.) If you are midding, and you have a strong or aggressive enemy, try [E][W] them a couple of times to get their health down, and only use [R] when you are sure you can finish them. Nothing is more dissapointing then using all four attacks and having them continue attacking and possibly killing you. A nuker veigar is also a defenseless veigar after all the moves are on cooldown. If your aggressor in mid is too strong, or you find yourself alone in mid, help your laning partners with a gank every once in a while. Veigar is very strong when he mids, and then attacks the lower levels. When team battles break-out, always be the first to run; you can always come back and do more dmg when the enemy attacks your ally.
Late game: (14-18)
By now, the amount of AP you have aquired has made you infamous. It is nearly impossible to escape your [E][W]combo, and when you nuke: ([E][W][R][Q]) it dominates all champions. It may be a bad idea to solo a lane now, always be traveling with a partner to ensure the least amount of deaths. Continue last hitting minions with [Q] and use [W] on the second group of minion waves when you need to push. During team standstills, I usually wait until someone else initiates, and then I charge in with at least the [E][W][Q] combo; then backing off until my abilities recharge and then [W][Q] again where most of the fighting is to turn the fight. (In the middle of team battles, most enemies don't see the [W] pre-emptive until it is too late.) If they begin to get smart, and kill you first, try hiding in bushes and wait for them to concentrate on your tank until you come in.