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Not Updated For Current Season
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Veigar was the first character I became good with, so I would like to share everything I have learned about him with you.
Early game is all about farming ap. This is done with your Q and is fairly easy. Getting last hits will raise your ap pretty significantly and get you plenty of gold. For this to work best you are going to want to go mid. Going mid makes it easier to get last hits and also results in you leveling faster. This is good for later when you gank since they will be lower levels and you will crush them. Also when I go mid I tend to find myself up against fairly squishy people, so killing them is pretty easy.
Try to lane until you ding to 6. This gives you your ultimate which gives you some killing power. Go back to base and buy. You probably won't have enough for the Deathfire yet, but you should at least be within 1k of getting it.
I will call this part of the game the mid game because I change my tactics. Now when you are in mid you will want to start harassing the guy (but still last hit creeps to keep building your AP). This is another reason I like middle. With it is just being 1v1 they can't hide behind people, and they feel it is their responsibility to stay in the lane. They don't want to be the first to lose a tower, so harassing them down to half life is pretty easy. I use my Q as harassment and I occasionally stun them. Clarity makes this possible. Without it you'd be dry of mana super fast.
If you get them to below half health (preferably near 1/4 health) its time to go for the kill. By the time you have harassed them down you should probably have dinged to 7 or 8. Hopefully you will have your dark matter, but it isn't really a necessity, especially if you have them down to quarter health. Just toss out a stun, preferably so they run into it. You have to stun them to use Dark Matter! Cast dark matter just beyond the ring so they will run into it as soon as they become unstunned. Immediately after you cast your Dark Matter hit them with Baleful Strike and your Ulti. They should be dead. If not they should be running back to base with nearly no health left, which can be frustrating, but at least you have the lane to yourself.
After killing them you should have enough money to get your deathfire, but I try and lane until I have enough to get my boots too.
I rush to get the deathfire because it essentially allows you to skip the harassment part of mid game. Since it takes out a third of their health you only need to hit them once or twice before you can go in for the kill. If you are lucky enough to be going against an AP build like Ryze, Annie, or Heimer you do not need to harass. You can hit them in one sequence.
The new sequence, which you will want to become extremely proficient at is as follows:
Cast your stun so they run into it
Cast dark matter so they run into it when they become unstunned
Deathfire them (which should be your 1)
Or the key sequence of:
Five key strokes and they are dead.
Go back and get a book. Start building up your AP with it and you will quickly become unstoppable.
I like to gank their spell casters, because you can slay them with ease and they are the ones usually causing the most trouble in lanes. So you kill them and then they can push the lane no problem.
Late game I still play very similar to that of mid game, except you will be dealing more with team fights. Your stun is your best spell to help others. You can help everyone escape by planting it in just the right spot and you can catch a fleeing enemy who would otherwise have gotten away. Your team will be most grateful to have you now. Keep in mind though people will be going after you now. If you have been killing them mid game they hate you and since you are Veigar they know you are squishy. So try and stay behind your team. If you have not been slaying them yet then they will probably ignore you. This is also good since you will start slaying them late game instead haha.
The other items
Deathfire, Boots and the book are your core items. Everything else is optional. The 2 others I have listed up there (Hextech Revolver and Void Staff) I like because they offer magic penetration and spell vamp. You should be dealing at least 1.5 if not 2 k damage when you use your sequence and that is about 10% of your health you can recover from the spell vamp. Magic penetration I feel is pretty self explanatory. This leaves you with one item slot so get what ever you want. I would normally getting something that gives more cool down. This allows you to constantly nuke the people.
All I would say is don't get the Crystal Scepter as your last item. A: you have a stun and B: you should be killing people you cast on. Slowing is pointless.
If there was any confusing on the spells here is a little explanation.
Clarity: Clarity is a must. This lets you regen your Mana which is a must for harassing etc and it lets you skip mana regen items and go straight for AP.
Teleport: Teleport is nice because Veigar is so squishy. If you are taking some serious Flak you can just teleport back to base, heal and buy and be back in battle in about 10 seconds. Also he is one of the slowest characters so you don't want to be running to a lane.
I wouldn't say super advanced tactics, but some little tricks and tips that can be a big help.
When playing against super aggressive people they will want to go for you. You can use this to your advantage. Hang out by the tower and try stunning them as soon as they get within the towers range (if they have no creeps in the towers range) or as soon as they hit you in the towers range. This could result in an easy kill or at least a one for one if you are low on health.
With clarity use it often! I can't tell you how many time I would start my sequence and not get the final blow in because I was short 20 mana for my ulti.
Sometimes with teleport it easier to gank people if you teleport to a tower or a minion group behind your allies. This way the enemy doesn't see you teleport in and you can go through the jungle and get behind them in no time.
If you know the enemy is baroning you can go and try for a baron kill steal. This can be done by positioning yourself above the Baron's area and using Dark Matter. Hopefully you can get the last hit. I've only done it once but it was super funny when it happened. Always worth a shot.
Have fun with this build. It is super fun and nuking is very rewarding! Thanks for reading, and this is my first build here so sorry if the formatting looks terrible.