Based on various books and movies and TV shows, I have identified a few ways to introduce a character and make it immediately apparent that s/he is a genius (even if s/he is not).
#1 Trust me, I know what I'm talking about.
It’s a sunny day in the park. Kids are playing soccer, young couples are walking their dog, some old ladies sit at a bench and gossip. Suddenly, a man collapses, clutching his head and screaming. People turn and stare, someone shouts “Call 911!” From the gathering crowd, our protagonist emerges, leans down over the thrashing victim. A moment later, the protagonist stands and declares that, “There’s a foreign body lodged in his vitreous body.”
This guy clearly knows what he’s talking about. I mean, would you be able to determine that just by looking? Do you even know what a vitreous body is? This guy must be a brilliant doctor! But let’s try it again, except with layman’s terms:
“Suddenly, a man collapses, clutching his head and screaming. People turn and stare, someone shouts “Call 911!” From the gathering crowd, our protagonist emerges, leans down over the thrashing victim. A moment later, the protagonist stands and declares that, “There’s something stuck in this guy’s eye.”
Both versions of that statement say pretty much the same thing. In fact, the latter would probably be more useful simply because the people the protagonist is addressing aren’t medical professionals.
This is a good way to display professionalism or an area of expertise, but unless it is backed up by actual evidence of genius (not just competence in a specific field), it’s not enough.
#2 The Smartest Guy in the Room
To best demonstrate why this is wrong, I’ll use a brief anecdote about how I accidentally convinced an entire class that I was some kind of mathematical genius. In my Writing about Literature class, we read a play called Proof by David Auburn. In the first scene, as a proof of the protagonist’s genius, she immediately recognizes the number 1,729 as being the smallest number “expressible as the sum of two cubes two different ways.” I pointed out in class that this was The Hardy-Ramanujan number and explained the story behind it. After that, I was asked for my opinion on the veracity of every mathematical assertion in the play. I’m not a mathematical genius. I just happened to watch an episode of Q. I. that discussed that number a couple nights prior to the class. The reason it seemed to be an indicator of genius is that I was amongst English Majors, just like the young upstart FBI agent is among investigators.
You wouldn’t expect a FBI investigator to know the prevalent themes in T. S. Eliot poems any more than you would expect an English Major to know about the Hardy-Ramanujan number. The protagonist here isn’t necessarily smarter than the other people in the room, rather, the protagonist knows things that are generally trivial at best for the group being addressed.
#3 The Smartest Person in the Room (Version B)
Same situation as above. The young upstart FBI agent comes in and tells the other agents what they missed. After hours of racking their brains for clues, the genius protagonist figures out what they all missed within seconds.
Even taking into account what was discussed in the previous section, the agents in this example would probably have at least skimmed the books, or found the Cliff Notes on them, or called an English professor, or something. All too often, the smartest man in the room ends up being the least incompetent man in the room. Just look at any crime procedural where the protagonist has some special ability (e.g. Psych, Monk, The Mentalist, The Dead Zone, Sherlock, etc.). Ninety-nine percent of the time, they only solve the crimes because they either don't operate by police procedure (breaking and entering, warrant-less searches, etc) or because the detectives missed something that should have been obvious to them, not because of any particular genius on behalf of the hero.
#4 The Smartest Man in the Room (Version C)
The protagonist is face to face with the serial killer in an empty warehouse. A fistfight, a struggle over a gun, a gunshot, and the serial killer crumples to the ground. A moment later, the FBI agents rush in. They had received a note from the killer earlier in the day that hinted towards a carrot farm as his safe house. The protagonist tells the FBI agents that carrots do not actually improve eyesight. In World War Two, the British fighter pilots had found a way to put radar devices in their planes, and were therefore able to shoot down the German planes at night. Not wanting to give up their secret, they claimed that eating carrots improved their night vision. The protagonist knew that someone so obsessed with eyes would know this, and that the carrot farm must have been a distraction.
This has a lot of similarity to the first “Smartest Man in the Room” entry, but is so prevalent I think it deserves its own spot. Busting an urban legend or breaking out a neat piece of history is an incredibly common way to show a character’s intelligence by demonstrating that they don’t think like everybody else. Putting aside the reasons in the earlier “Smartest Man in the Room” entries (which are perfectly valid), you should avoid this one because when people do this they are, very frequently, wrong. The two big ones that I see a lot are the “truth” behind Kennedy’s “Ich bin ein Berliner” speech and the NASA multimillion dollar pen vs. Russian pencil stories. The former story says that when John F. Kennedy said “Ich bin ein Berliner,” he was really saying “I am a jelly donut.” While there was a jelly donut called a Berliner, this would be like hearing someone say “I am a New Yorker” and assuming that they’re calling themselves a magazine. This is repeated not only in countless TV shows, but even in Pulitzer Prize nominated books like Kingsolver’s The Poisonwood Bible. The (condensed) NASA pen story goes as such: NASA spent millions developing a pen that would write in zero gravity. The Russians just used a pencil. NASA didn’t actually spend money on developing the pen. Additionally, pencils are terrible in a zero gravity situation, because the tips break off and you get graphite and wood dust whenever you write with them. I guess my point is, this particular method of displaying genius is usually done so poorly that it has the opposite effect of what’s intended.