A favorite Knuth story to illustrate: He was re-writing the 8080 code for the processor that ran our new Alphatype CRS Phototypesetter, since the manufacturer's code wasn't suitable. Real-time, 7 levels of simultaneous interrupt stuff, since it controlled two stepper motors that moved the lens around the photographic paper, plus four custom font outline-tracing S100 cards that flashed vertical segments of characters at just the right moment on the CRT that the lens then focused onto the paper. No debugability other than a 4x4 keyboard and a 15(?) character display. I was responsible for the code on the other end of the RS232 port that connected it to the DEC mainframe, sending down character outlines on the fly to the very limited memory on the 8080, along with the properly-sorted character placement info ultimately from TeX.

Anyway, we'd been doing morning, noon, and night debugging sessions together for a number of days. On maybe the 3rd morning, I arrived at his office and was speaking with his secretary of many years, Phyllis Winkler, when Knuth came in. He handed her a stack of maybe a dozen or two pages, saying "Here's a paper I wrote; please type it in." When? When did he write it? It's still a mystery to me.

My brother was something of a prodigy. Whatever it is that people like Carmack have, he had it too (and Carmack was one of his personal heros).

I would say with my brother it wasn't just that he was analytically extremely sharp, it's that he could focus it on things in a way that bordered on unhealthy obcession.

For example during undergrad he took a semester off because he burned himself out taking an insane course load, while also working on research as an undergrad.

Anyhow he spent that semester and the following summer at my parents house, and he spent months in basically the same pattern. Work on his research topic on the family PC until he needs to read a paper or think about something, at which point he'd roll the office chair over to the nearby couch and flop onto that. Every once in a while he'd disappear to his old bedroom to sleep for a few hours or take a shower or such.

And I mean he lived basically half a year like that, just relentlessly drilling into his research topic with every waking moment.

I'll add my stories to the 2 brother comments, in my prepa my best friend was by far the best in class. He did the 2 year program in chemistry in the first 2 months by himself. He arrived once in exams after a full night of re-reading the 3 LOTR books and still nailed the best grade effortlessly. And to his own admission, he was far from the most productive / smarter guys he knew.

I remember listening to him and a teacher, the teacher would give some anecdotes on what he's seen at the ENS. People reciting world chess championship games at night to fall asleep. Reading massive math books (1000+ pages) in an afternoon. Another story I heard in a different context from another teacher, math students were assigned a python project, after an introductory course. They never did any programming before. One student wrote 300 LoC in a single function and showed it to the teacher. It didn't have a main function, so the teacher asked him how he tested it. The student didn't know what he was talking about. They added what's needed to run the function and it worked.