Comment by ktallett
6 days ago
I sense we may just have a different experience related to colleagues skill sets as I can think of 5 people I could send some questions too and I know they would do them just fine. Infact we often have done similar problems on a free afternoon and I often do similar on flights as a way to pass the time and improve my focus (my issue isn't my talent/understanding at maths, it's my ability to concentrate). I don't disagree that some level of training is needed but these questions aren't unique, nor impossible, especially as said training does exist and LLM's can access said examples. LLM's also have brute force which is a significant help with these type of issues. One particular point is that Math of all the STEM topics to try and focus on probably is the best documented alongside CS.
I mean these problems you can get better with practice. But if you haven't solved many before and can do them after an afternoon of thought I would be very impressed. Not that I don't believe you, it's just in my experience people like this are very rare. (Also I assume they have to have some degree of familarity of some common tricks otherwise they would have to derive basic number theory from scratch etc and that seems a bit much for me to believe)
I think honestly it's probably different experiences and skillsets. I find these sort of things doable bar dumb mistakes by myself, yet there will be other things I'll get stressed and not be able to do for ages (some lab skills no matter the number of times I do them and some physical equation derivations that I regularly muck up). I maybe sometimes assume that what comes easy for me, comes easy for all, and what I struggle with, everyone struggles with and that's probably not always the case. Likewise I did similar tasks as a teen in school and assume that is possibly the case for many academically bright so to speak but perhaps isn't so that probably helped me learn some tricks that I may not have otherwise. But as you say I do feel that you can learn the tricks and learn how to do them, even in older age (academically speaking) if you have the time and the patience and the right guide.
Here you go — you did this type of problems as a kid/teenager. 1) you likely have a talent for it 2) you have some training.
I did participate in math/informatics olympiads as a teenager and even taught it a little and from my experience, some type of people just _like_ that sort of problems naturally, they tickle their minds, and given time this people would develop to insane levels at it.
'Normal people', in my experience, even in math departments, don't like that type of problems, and would not fare well with them.