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The Story of Veigar:
To most, yordles do not conjure images to be feared. The easygoing half-pint race, though fierce, is often regarded with some degree of joviality. Every now and again, however, a yordle turns so bad that, even at its small stature, it strikes terror into the hearts of others. Veigar is such a twisted yordle - a master of the magical black arts and a corruptor of cosmic energy, making him one of the most feared sorcerers on Valoran. Veigar was once one of those unusual yordles who chose to trade with the outside world. Imprisoned in the city-state of Noxus for years, Veigar slowly went mad in isolation from his people. He eventually escaped, but instead of returning to Bandle City, he sought tutelage from dark wizards across the land. Now he seeks to end all conflict on Valoran, by bringing all of the city-states to their knees, regardless of affiliation.
Trivia:
Veigar, Riot's QA guy, is where this name comes from. He is "quite tall".
Veigar's original and White Mage skins closely resemble the Black Mage and White Mage from the Final Fantasy series. He also possess a strong similarity to the popular mage Vivi of Final Fantasy IX.
His quote "I will swallow your soul!" and "Even now, your loved ones suffer!" are very similar to quotes from The Evil Dead franchise.
The Greybeard skin is a direct reference to Gandalf the Grey from the Lord of the Rings trilogy written by J. R. R. Tolkien.
All of his skills, except Baleful Strike, refer to space. Event Horizon being the edge of a black hole, the point at which light cannot escape. You get sucked in, and you cannot escape. Dark Matter being the undetectable matter in space. Primordial Burst being the big bang. Equilibrium being the balance between everything in the universe. Baleful Strike meaning menacing, wicked, or evil strike.
Veigar and Blitzcrank share the same walking quote "The magic it calls to me."

All Potency, for the specific reason of the stats the Potency's give.

9/0/21
Masteries corresponding to Veigar's AP.

AP based items. If you can get the full build, you will not be able to be killed. Your ult will possibly be a 1-Hit K.O., or something very close to that.

Skills in order of needed.
The stun only needs 1, because you only need it for small amounts at a time.

They help Veigar do the things that are necessary.

I know its not TOO, good, but the build is, so just pay attention to that, kay?