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1) unusually smart, I presume by "smart" you are not saying the same thing as "clever", "insightful" or "thoughtful", which IMO leaves an academic understanding of how things work. You show this ...
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#3: Attribution notice added
Source: https://writers.stackexchange.com/a/45128 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
1) unusually smart, I presume by "smart" you are not saying the same thing as "clever", "insightful" or "thoughtful", which IMO leaves an academic understanding of how things work. You show this by finding an opportunity to have the character explain something others (including the reader) may not understand very well. Of course as an author you can "cheat" by doing an hour of research on some question relevant to the storyline, and learning something surprising that your smart character can toss out in a paragraph or so. 2) clever: You show this by the character being inventive on the spot, coming up with a solution to a problem that seems like a good idea. This can be a solution to somebody else's stated problem, or a problem of his own. 3) insightful: Being insightful is usually understanding a problem or situation in such detail that conclusions can be drawn that were not stated or obvious. This is particularly true in relationship issues; an insightful person not only understands another's situation as stated, but understands the _person_ well enough to know what they are not saying, or how they feel about the situation, and uses that. In fiction, they might state that, and elicit agreement from the person they are talking about. 4) thoughtful: This is somebody that thinks ahead to what WILL be needed, or what others will appreciate or need when the time comes. It is related to memory, but also to caring for others. For example: Joan is coming tomorrow, I know Joan, so I will shop today for Darjeeling tea because that is what she will want. Thoughtful involves thinking about others, thinking about the near term future, and taking action in the present to make the near term future better in someway. 5) somewhat reserved: Many highly intelligent people are also insecure and feel a need to show off their intelligence and receive praise or accolades for it. But not **all** of them. Some highly intelligent people know they are intelligent, and know that braggadocio and constantly reminding others they are the smartest person in the room, alienates them. In a way, it is applying their intelligence to understanding the patterns of social interaction. So they ARE somewhat reserved, willing to help but not insisting they know best or should be in charge or that all the stupid people should listen to them. In fact, _some_ smart people, besides knowing they are smart, also know that being a "know it all" is a pejorative; it can be threatening or off-putting to the 99% of people that are not as smart as they are. And knowing that, and being insightful, they will only assert their intelligence when it is really going to matter, when they anticipate significant negative consequences of _failing_ to speak up. In this way, most exceedingly intelligent people portrayed in fiction as socially awkward or lonely are actually be portrayed as _stupid_ or _defective_, for all their brains they cannot figure out how to make friends, woo a romantic partner, or learn to use humor to relieve tension. All of these are learnable skills. 6) mysterious. Highly intelligent people can take actions on opportunities or threats that others do not perceive. Intelligence is about being able to predict the probable future (or for detectives what most likely happened), and the highly intelligent do this better than others. Thus what they are doing can be mysterious; if they don't explain it, we can't figure it out. You show this just by having them do things that seem significant, but are not explained. I know I am not giving you examples; I think the point of this StackExchange is to provide instruction on how to accomplish what you wish in writing.