You were failed by people who didn't help you learn intuitions and instead caused you to focus on memorization.

The rules underpinning math are axioms in the end, but they're not completely arbitrary, because if you change them in most cases it just fucks everything up.

The axioms that were chosen were chosen for good reason, and the rules they result in (such as summation and multiplication being commutative so 3x4=4x3 and 3+4=4+3) allow more complex rules to be created.

There's a lot of philosophy of math at the core of all this , but it's not really true that this is all arbitrary.

It's not arbitrary. Really try to think about the problem at hand. The 'why' is quite apparent. Ask yourself why did they go with 99+1+98+2... in the first place? And why is that the same as 101+101...? What was the benefit of simplifying it to that? How did it save the student time?

You can deduce this yourself and literally no memorization is involved to figure this out. No formulas needed either.

Once you have the idea, seeing that it works if often easy. But coming up with ideas like that can be really hard, which is why gauss was the only one in his class who got it. There is no general method, you just have to think about stuff for a while, but you can get better with practice. And it feels really good when you prove something for yourself, even if it's relatively straightforward. You can just try to prove some simple things yourself, if you want, the advanced college courses are just for proving really advanced stuff.

Look up how to solve it by george polya

It's just a matter of breaking the problem down Into an easier problem or set of problems.

All the additions are interchangable, so you could choose to add 1+2+3+4 or 4+1+2+3 and then 4+1=5 and 2+3=5 and then youve got 5+5 which is easy its 10. So you go ok you can do the conversion with 1 and 50 except it's still tough mental math so you say 1 and 100 to get 101 100 times, but that's twice too big so you slap it in half and you get the answer. It's solving a tough problem by splitting it into problems that aren't as tough.

The first step is knowing what tools you have in your belt. The second is knowing how they work in detail. The third part is the inspiration of using them in a way that solves a difficult problem.

I'm not a mathematician, but I've found interesting solutions to problems like this before, and it's fun when you understand your tools and understand the problem and it all comes together to find a solution nobody else would have.