Calculus by Michael Spivak is really, really good if you want a thorough understanding of the concepts. The problems are all useful and challenging, and you learn a lot more than calculus in the process.
However, I've heard it's quite hard compared to other calculus books, and you won't like it...
With absolute values, only the magnitude of the change matters. It doesn't matter if the change is positive or negative. So H must change more than TS.
I calculated the first 8 and put them in to OEIS, and got: oeis.org/A072752.
What you're after is not the gaps, but the difference, so it's one more than the terms in the sequence I linked to.
I'm not sure about an efficient algorithm, my jumbled together program could only do 8 before taking...
So it seems to be (area of circle) * (circumference) * (1/2). It's less a question of why it doesn't work, than why your brother thought it would work. Maybe if you posted the derivation, we could point out the problem with it.
99 = 3 * 33, so 1 and 11 aren't the only values x or y can take.
There's 1 solution to your problem.
Don't be surprised that you're getting "ad hoc" methods, you've thrown together a few random arbitrary conditions, especially the "twin number" bit.
I think that's equivalent to \sum_{p=2}^{n} \frac{\left \lfloor n/p \right \rfloor}{p} , where the square brackets represent the floor function, and p runs through the primes less than or equal to n.
I don't know if that helps at all, and no doubt it can be simplified more so.
Now I hate this kind of discussion, so I won't really get involved, but mathematics does not attempt to prove anything "in terms of the physical world", it has no concern for silly things like atoms and quantum systems.