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It's certainly true that for a some mathematical problems memorizing the steps to the solution is often the most practical way forward (especially if you're planning on passing an exam), but there is definitely more to it. Most of the work proving things is actually exactly what you say - going down a different route. That's why mathematicians have careers, they spend days/months/years going down different routes and taking different appraoches to solve something. Naturally, they don't explain all the ways they didn't solve the problem when they finally do solve a problem. You don't need to know the answer to get to the solution. This is just like saying "The problem with a maze is you need to know the route through it before you can get through the maze"- no, what you do is you search for the route through the maze and once you've finished searching then you know the answer and suddenly you can walk through the maze repeatedly very easily.

And similarly, whilst one proof might have limited practical applications, it's very difficult to know which proofs do have practical applications and it's almost impossible to know which results might later yeild further proofs that do have practical applications.

None of this is really a matter of intelligence, it's a matter of putting in work (although obviously there is intelligence involved to find particularly neat ways of appraoching problems).

And why do mathematicians learn proofs? Often, to see the approaches that you can use to solve other problems.

> Naturally, they don't explain all the ways they didn't solve the problem when they finally do solve a problem.

I do think something related to this is less obvious than you're suggesting. A proof generally presents a train of thought gets you from point A to point B. It's almost never a representation of the train of thought that the author actually followed to get from point A to point B. This is obvious to people with experience in mathematics, but it's much less obvious before you have that experience. Even if you know it's true in principle, appreciating it viscerally and adapting the way you interact with mathematics to account for it is basically a lifelong task.

A big component of good mathematical communication is explaining what motivates a solution to a problem (or even what motivates the problems in the first place). Sure, you don't have to do that every time, but it's fair criticize a broad lack of this. A lot of mathematical communication is... not good in this sense. Notably, the problem/solution in the OP isn't really motivated by the author. A skilled reader will think about it themselves, but they need to learn that skill somewhere.

More or less on topic, here's an example where an experienced mathematician tries to reproduce one of Sylow's theorems without looking it up, and writes down a more-honest account of the thought process involved: https://gowers.wordpress.com/2011/12/10/group-actions-iv-int...

I like the 3blue1brown youtube series for explaining mathematical problem solving, like this example explaining a difficult problem from the Putnam:

https://www.youtube.com/watch?v=OkmNXy7er84

"And those of you who follow the channel know that rather than just jumping straight to the solution, which in this case will be surprisingly short, when possible I prefer to take the time to walk through how you might stumble upon the solution yourself. That is, make the video more about the problem-solving process than the particular problem used to exemplify it."

I bailed out of math around calculus, and these videos helped me at least appreciate what I was missing.

I think a bit of disillusionment with mathematics is healthy, but it's misguided to call this kind of thing "a fundamental flaw in mathematics".

For one thing, in my experience, working on real life problems involves a whole lot more "memorizing problems" than working on mathematics.

And thank goodness for that! When I drive over a bridge or install an app, I really don't want to hear that whoever made it has an aversion to memorizing problems and solutions. Solving a real world problem is usually boring. Making a real life thing well is usually about correctly putting together many pieces that some other people have put together, in a way that's roughly similar to how somebody else has already put a lot of those pieces together before. Most of the work is well-trodden and uncreative.

I think part of the reason mathematics feels like it's about "memorizing problems" is that getting something deeper than that out of it is a habit/skill. A very important aspect of the vague notion of "mathematical maturity" is a habit of looking at a solution to a problem with the mindset of "how could I have come up with this?". That is, unpacking a problem and solution into some deeper understanding or way of thinking that led to it. As opposed to filing the problem and solution away "as is" into some toolbox to be referenced later. A lot of "gotcha!" solutions to mathematical problems are unsatisfying precisely for this reason.

In a lot of cases once you've read a problem and read a solution to that problem, and understood the solution, you've done about 10% of the work. The remaining 90% is this difficult work of unpacking and repacking lessons from that solution into something that actually deepens your understanding of what the problem is about. Sometimes reading the solution is actually doing negative work.

In other words, if you look at this problem and look at this solution and think "neat, but I don't get anything out of it beyond 'neat'", that's normal and fine, and probably correct. But that doesn't mean there isn't anything in it beyond 'neat', and a big part of learning mathematics well is to dig deeper than that even when the problem and solution don't force you to. Whether that digging is worth it is up to you. (It's often kind of a crapshoot in terms of payoff.)

But that's a more complicated situation than "this shows a fundamental flaw in mathematics".

I'm curious, what draws you to thinking about this problem so much lately?

Teachers/Professors always pretend that you can find it, eg. you get it as a homework. And it works that way, yes, kind of - as an excercise and for "relatively simple" challenges. But for tough problems it's a "memorization of the solution challenge" and not one where you lose too much time trying to figure them out. This particular one makes it pretty obvious as well: Even understanding the solution is a challenge.

Maybe you are a student, so let me give you some advice. The working out of the problems is not a meaningless task - understanding the problem solving methods and the underlying structure of the mathematics is important. You are right, engineers in the real world are not expected to do the calculus, they have charts and graphs and rules of thumb, many shortcuts to the "right answer". But the reason we train engineers the math is so they don't make a big mistake and kill somebody. Being able to work the shortcuts alone is not good enough, it's scary, because it means you don't really know what you're doing.