I think that the simplest way to get the 16 corners is to start with the two-dimensional square, understand why it has four corners, and go from there. Imagine a square lying on an XY plane. You can then define the four corners as being at the points:
[0,0]
[0,1]
[1,0]
[1,1]
So for every dimension, the coordinate is either 0 or 1. You can convince yourself that this generalizes by working it out for the cube. The number of corners is then 2^n, where n is the number of dimensions. This also has the fun attribute of teaching us that a straight line is the 1-dimensional equivalent of a square.
I don't know what ^ means lol. I won't give up just yet on pursuing this and maths generally, but I'd just have endless questions for you to answer if I really start going towards this initial problem for me to figure out.
As for your educational issues there are no surprises there. I'm sure that in the abstract there is such a thing as greater or lesser mathematical aptitude, but it's not really differentiated by the public school system that much, as the material is not that difficult. What that does differentiate is psychology, confidence, etc. A bad math teacher early on, or a bad experience with math early on, can propagate onwards and snowball into an avalanche of bad experiences. You are not the first, nor the last, person to have been surprised that he wasn't awful at math.
I supplemented my income in graduate school by tutoring undergraduate math. I did have some students who needed moderate amounts of help, and honestly that was often just to help them focus, or to help them understand what the teacher wanted. Often they would end up no longer needing my help.
Hahahaha that first sentence made me think "Oh, well obviously I'm dumb, thanks David". But I understand what you mean from the rest. My maths teacher wasn't exactly a bad teacher but he was just a total nutcase, screaming in frustration that his class (which was the top class in that school) was constantly f*****g about and not listening.
So we were the top class out of 9, you can imagine how he went nuts on the lower end of the spectrum, we'd hear him screaming in particular from our Geography class sometimes, that's when he had an infamous class of nutjobs (most of the males and plenty of the females have done jail time, or even died, stabbed people, etc). They were the 6th on the ranking list, just above the classes that had kids bordering on autism, or total special needs, so this was the cusp of madness and it repeated itself in the several years at that school. It was ALWAYS that class which were the craziest, anything above had some sort of ambition, anything below were too socially inept to really cause problems in such an extreme sense. It's really sad thinking of it, and how that system left a lot of talented people totally fucked from the age of 11 because they were literally put in a caste system.
Anyway it's 14 years since I got my B in maths, and my general calculations on basic math are very quick, so that encourages me to maybe think about trying this again and seeing how I go.
Just a random story that is often brought up from friends- I was talking to my friend and laughing at the back of class, which my teacher picked up on, he had written an algebra equation on the board which we should have been studying the night before to figure out.
Nutjob teacher: "Rudiger! What are you doing?! OK.. tell me the answer to this... how to we get x?!"
Rudiger: "uh.... the question is too vague"
Nutjob teacher: "TOO VAGUE?!"
The explosion from there is legendary, and I got moved to the front of class. People just find it hilarious that it was a specific mathematical problem that can't possibly be vague, and I have no idea why I said that.
But anyway I agree with your summation of public school learning, I understand in the States there are no "levels", all classes are just the same. My high school has apparently done away with the "class system" for several years now but they've simply renamed it differently, so there are still the dumb kids and the smart kids. Keep in mind that this is 5 years of school, starting at 11, and if showing progress in exams at the end of the year then you can actually move up in the class system. I don't think this is a bad thing personally, it sounds barbaric on the face of it, but you have teachers giving lessons to kids on a similar level, rather than a completely mixed bunch.
You may say that maths is too simple in schools (although I'm hearing it's getting considerably more difficult) but for some people it's really not that easy, those of low IQ. Possibly I get an indication from one sentence you wrote that it's more down to dedication and interest than IQ, which I understand as well.
