Insufferable Genius/Playing With

Basic Trope: A character who, although extremely arrogant, is always right as well.

• Straight: Carnegie, The Smart Guy, is arrogant to everyone he meets and insults their intelligence at regular intervals. However, he's so smart that in times of crisis, they have no choice but to turn to him.
• Exaggerated: Carnegie is completely incapable of saying anything nice at all, and literally has a superpower allowing him to anticipate everything.
• Justified: Carnegie knows he's a genius, and he knows that his team has no choice but to listen to him.
  • Carnegie is insufferable so people won't pester him with so many questions.
• Inverted: Carnegie is humble, but so stupid he can barely see the forest for the trees.
  • Alternatively: Carnegie is brilliant, but has crippling self-esteem problems stemming from the constant mockery of his peers. In times of crisis, however, he is forced to offer solutions even though he's afraid to even speak.
  • Jerk Jock
• Subverted: Carnegie decides to become a nicer person...
  • Alternatively: Carnegie is only condescending to those he doesn't like. He's got a lot more patience for his friends.
• Double Subverted: ...but gets over it.
  • ... However, he's really insulting them on a higher level then they can understand, laughing behind their backs.
• Parodied: When anyone comes to Carnegie with a problem, he insults them for several hours. When they finally get fed up and walk away, he says "Wait! Don't go! Here's the solution!"
• Deconstructed: Everyone that works with him hates Carnegie - he may be smart, but he's too much of an asshole to deal with.
  • Carnigie's arrogance comes from a crippling need to assure himself he's really that smart. On the occasions he does get something wrong, he collapses into a Heroic BSOD.
• Reconstructed: Everyone that works with him admits that Carnegie may be an asshole, but he's still one of the smartest guys around.
• Zig Zagged: Carnegie's niceness varies from day to day.
• Averted: Intelligence is not a determinant of kindness.
• Enforced: "Let's have The Smart Guy be a real asshole to everyone! That's how it works in real life, right?"
• Lampshaded: "Ah, yes, your problem. Rest assured I've known all about it, long before you could even conceptualize it. By the time you started considering the possibility of a solution, I had already thought of several viable ones."
• Invoked: Carnegie has been an asshole all his life. He decided to start rigorously studying so that people would have to put up with him.
  • Alternatively: Carnegie, who has always been gifted with intelligence, is deliberately rude to people so that other people will leave him alone.
• Exploited: Ben, needing a partner for a science project, immediatly asks the resident asshole.
• Defied: Carnegie is told by those around him that they won't hesitate to kick him out if he doesn't shape up and start acting nicer.
• Discussed: "Dammit, I wish there was someone else who could fix our problems. I hate going to that jackass for help."
• Conversed: "Wow, that guy sure is a jackass. Why can't smart people be nice?"

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I foresaw that you philistines would come to me with questions about examples of this trope. Fortunately, I have catalogued several at Insufferable Genius so that you could understand them.