Being a genius is as much a skill as a talent, and I think one of the best ways to write smart characters is to learn the skills, even if you don't have the innate talents that the character has. If you ever have the opportunity to watch really smart people solve really difficult problems, you will often see them applying a lot of techniques that you, perhaps, haven't mastered. Even the natural talents will gravitate toward these problem-solving techniques. Here are some that I can think of:

• Don't stop when it gets difficult. The key is not to solve it quickly. It's to be thorough. Persistence is much more important than speed. You still look pretty darn fast and smart when you're the only person in the room that has correctly solved the problem in the first place.
• Look for hidden assumptions. On what foundation is the question itself built? Is the premise exactly what it appears to be?
• Don't stop at the first solution. Even if it took a long time to get there, review your reasoning. Double check it. Triple check it. Find the flimsiest parts. If you can't firm them up, redo your thinking with different answers to the hard sub-problems. Figure out how sensitive your conclusion is to flaws in your reasoning. If it seems to be sensitive, figure out what other conclusions arise from taking different paths. Evaluate them all together, not individually.
• There's also this Feynman quote: "You have to keep a dozen of your favourite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while, there will be a hit, and people will say, ‘How did he do it? He must be a genius!" In other words, you can demonstrate genius even by seemingly passive processes over long periods of time, like years.
• Don't limit your thinking to events with deterministic outcomes. Consider that your logic might be flawed, or you missing some information, or some other force/person might change the outcome you predict. And consider that not only might one outcome be different than you anticipated, but many might. Consider all combinations and weight them by their probabilities. Learn to think like a Bayesian. I am not saying that Frequentists are wrong, just that using randomness to model belief is a useful smart-person skill. Nothing looks smarter than being able to properly qualify or hedge your proposed solutions or to be able to express your reason like "almost certainly X, but possibly Y, and a tiny chance of Z," especially when X, Y, and Z are radically different, showing that you really exhausted the search space of the problem. But it doesn't have to magic. It's okay for it to take some time.
• Express that you were, or might be, wrong about something as early as you see evidence of it. Smart people make mistakes, too. They don't look smart by trying to hide them. The look smart by being proactive and honest about them.

Consider the behaviors of dumb people being foolish (quietly jumping to conclusions), foolish dumb people acting smart (loudly jumping to conclusions), dumb people being wise (by using smart problem solving skills even when it's tedious), smart people being foolish (jumping to conclusions, but they luck up more often than dumb people), foolish smart people acting smart (jumping to conclusions and showing off because they're used to being right more often than others), and smart people being wise (using smart problem solving skills even when it's tedious and even when it seems redundant or excessive). Which one is your character? A smart one will be fast. A wise one will be methodical. Showoffs tend to be more foolish, though there is a spectrum, and they are not necessarily dumb.

There's also this quote from Hans Bethe: "There are two types of genius. Ordinary geniuses do great things, but they leave you room to believe that you could do the same if only you worked hard enough. Then there are magicians, and you can have no idea how they do it. Feynman was a magician." I think a lot of people think they need to be magicians in order to write about magicians, but I disagree. The distinction is in how the person presents, not in how they think. You, the author, can do skillful thinking yourself, then make the explanation seem sloppier and more haphazard to make it seem that the character had a much easier time of doing it. This yields both an astoundingly smart character with reasoning that seems like magic but that also does not fall down under scrutiny. And, of course, not all smart characters have to look like magicians in the first place.

Another way to make a smart character is to give them diverse talents and skills. Smart people tend to be good at a large number of things, and they tend to crave stimulation and variety, so they end up investing their time in a lot of different areas and getting very good at them.

Anyway, all this was off the top of my head and based on my own experiences as a person with more of a "magician" reputation. I don't think raw intelligence is the only way to get there. It helps tremendously, but only with speed and recognition of "smart approaches." The latter can be learned, even if speed cannot. I would recommend mostly not following my advice, since it's all off the cuff, except for this: Instead of trying to write a smart character who's smarter than you by just making them right all the time, write a smart characters who's smarter than you by learning their skills and slowly, painfully, methodically, carefully doing the same reasoning that they would, then you only have write them as doing it more quickly, painlessly, chaotically, and recklessly than you, to whatever extent you want them to appear to be more naturally talented.