About the summoners , i take Clarity , with my build you wont have a big mana pool at any given time , and this will help alot and also at early game you will go all out farming alot easyer i never ran out of mana even while using Q whenever is off cd and Clarity myself as soon as its needed so i get it on cd and by the time i go oom again my clarity cd is about to finish .

Just want to say here that i take W on lvl 4 ( i see many guides who ignore it ) simply because Dark Matter takes 1 to 1 ap dmg ratio so even on lvl 1 when u have 400 ap at about 22-24 mins your meteor will do 500ish hit which is still alot and you have alot of points to take onto Q and as we know Q is spammable and W is not . and yet it will do the trick

I get doran's ring first and with its bonus mp5 and Veigar's passive i just focus on last hits and not on harassing at all before lvl 7 . You can still get a hit on him whenever possible but focus on last hits . by the time you go back the first time (if you don't get yourself killed of course) you will directly get sorcerer shoes for sure. Then just rush out the Blasting wand and Needlessly large rod , but instead of rushing those 2 into a rabadon (normally you'll be at that point on 20th min mark -ish in a regular game) You get the Giant's belt because:
first - you will have so low hp at that point its not even funny , and you are an easy target to gank.
second - with the 85ap - lvl 18 runes you will gain 5 ap/lvl + your farm gives you enough ap to be well above the enemy and not requiring on the rabadon that fast .
After getting the giant's belt go for that precious rabadon and you will go up to ~380-420 ap that's around 23-24 min mark in a regular game . Of course those numbers depend on how many Q last hits you managed to take.
From there on continue building as the guide says and you will grow so fast that you'll have 600 before 30min mark . 1k before 40 mkn mark , and about the time you get 1k ap you should have the lichbane and then , those physical attacks after a cast will be as strong as another meteor.

Many people say get a deathfire grasp , yes DG's active with veigar's enormose ap is just insane but its not required to oneshot the guy you want to and i find lichbane much more useful (take them both you say, well thats dumb cuz then you lose on one of the really important items like voidstaff and rylaj) and lets see now

Lichbane does 100% of your ap as magic dmg + 100% of your normal attack as physical dmg on your next normal attack and the lichbane has a 2sec CD . DG has 60 sec cd

Lichbane with 1000ap will do 1000 magic + 120ish physical dmg every 2 sec (with only masteries cdr the Q will be 2.6 cd on lvl 6 that means every 2.6 you get lichbane buff)
Deathfire Grasp with 1000ap will do 30% of their cur hp + 35 percent from 1000ap (100 per 3.5 percent)
Thats current hp that means if we encounter a dude with 2200 hp and about 45 percent MR you will do like half his hp then you throw Q and R and the target is dead , and your on 1 min cd
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If we encounter the same guy with lichbane on we will do Q R Physical attack and again the target is dead but in just 2.6 sec we will have another 1200 hit ready and the chances DG to do more then 1200 are next to 0 (we are talking without minding the mr of your target and mr pen of veigar)

Bottom line is the target will die regardless of lichbane or Deathfire Grasp possesion but with Lichbane you will have antoher bonus spell every 2 seconds and imagine how many Lichbane procs there are in 1 min and you'll see and besides DG is mainly to burn a tanks hp or tanky dps hp ( tanky dps wont have 180+ resistances so he will take dmg good enough with lichbane proc + Q R to die ) and a tank will resist so much that youl just tickle him with you full burst regardless of DG or Lichbane use , and your not ment to kill tanks so useless !

When you fill the 6 slots sell the dorans and get an defensive item depending on the situation . Thornamil , Frozen Heart , Banshees veil etc . whatever suits the situation .

Have fun and like this if you find it useful , its my first guide and i\m not a native english speaker so don't mind me if i have some typos and whatnot , i hope this helps . see ya ingame