If someone has a very, very high IQ...

>is that person instantly able to understand and do every math problem perfectly without needing to study or practice
Yes, if you are Von Neumann or Ramanujan.

>Actual geniuses are getting their PhD dissertations without even showing up to lectures or even being enrolled in a class.
Wittgenstein did that.

underrated

Middle Left or Bottom left work given the following
Boxes 4 5 6 are functions of 1 2 3.
Calling grey 0 and blue 1, you can rule out all solutions but center left which must be the answer.

Listing solutions as first row 1, 2, second row, 3, 4, etc., let's look at each one.

Our function we wish to understand is f(0, 0, 0). Assume an 'inverse' of an element is part of the binary set, i.e. (0,0,0) has an inverse of (1,1,1), (0,0,1) has inverse (1,0,0), etc.


#1: (0,0,1) According to the given, f(1,1,0) = (0,0,1), so this rules out #1.

#2: (1,0,1), according to given, f(0,1,1) = (0,1,0) so assuming the inverse relationship holds, f(1,0,0) = (0,1,0) so this rules out #2. Additionally, no given contradicts our inverse rule.

#3: there is no given set which results in (0, 1, 1) or (1,0,0) so we cannot rule this answer out.

#4: (1,1,0), given says f(1,1,0) = (0,0,1) so by inverse f (0,0,1) = (1,1,0) so #4 is ruled out.

#5: exact same arguement for #3

#6: given f (0,1,1) = (0,1,0). Rules out #6

Therefore, based on these assumptions, i.e. f(a,b,c) is one-to-one and the set it acts upon has an inverse, #3 or #5 work.
Of course you could define n-arbitrary rules to get any answer which is why this is a dumb question.

Sorry I meant to say middle left or bottom left in second line.

Yes, and the inverse is true, too.

You could "actually understand how and why the symbols at play relate to each other" and still suck at solving integrals, be slow, forget some tricks, etc.

I've only met one person in my life who can read textbooks on math/physics cover to cover without practicing and understand/apply the principles indefinitely thereafter. He told me he hit the score ceiling on a professional test (160s or something) and got 175 on an unprofessional Hoeflin test, so there you go.