Mathematical Brute-Force
We normally get the impression that mathematics is used to reduce the amount of brute-force we need in some other area, say, computer science or physics. It tries to reduce human effort in the long run.
But, of course, there’s no free lunch. Brute-force approaches are just as common in mathematics as in any other area, and, if not done properly, can result in more unnecessary effort than seemingly “brute-force” approaches.
Many people doing math hate brute force; that also includes a large number of professional mathematicians. After all, it’s thought of as physical, not mental labor. It defies the elegance and beauty of mathematical knowledge.
However, as we will see, brute-force can be highly technical and almost innovative. It depends on the level of brute-force; brute-force can be brute-forced, but that’s something for later.
Now, I’ll show you my silly thought process at an interesting (and suprisingly simple) problem.
The Problem:
The Solution:
A surprisingly silly-looking problem. Breaking things down so that they look more comfortable,
And now it’s decomposed into manageable bites. I like structure, so it naturally rearranges to
I made an accidental mistake here and thought to use AM-GM… but that would likely prove a lower bound, rather than finding a desired upper bound. As a lazy person, I proceed to do casework.
Case 1: x ≥ 0, and claim is true
Case 2: x ≤ -1, and claim is true
Case 3: -1 < x < 0, and, it seems to be true. Now, if we tried bounding each individual term from here, it’s almost impossible.
Perhaps we can move -1 and get only 0 on one side. That way, we can find negative squares and prove the inequality. Or, at least we can decide if this approach works instead of staring at a blank piece of paper.
We begin factorization, hopefully into negative squares.
Almost, but not quite. We set our hopes back on Case 3:
And… I’ll just give up the lazy approach. Using these tiny tricks isn’t going to work; we need to view the problem from a higher level, focusing on entire expressions rather than small details.
We’ll probably need things like
A nice negative square. We may need negative exponents to deal with the pesky term
Or, maybe let’s try more “normal” stuff first.
Seems to work just fine. If we just add the two expressions together at this point,
That’s a big mess. Simplifying, we find we’re missing a power of 6! But, if we observe carefully, the source of error is not from the pesky term; it’s from
because the square gave rise to unnecessary terms that cancelled out the power of 6. If we separate the powers of 2 and 3 into two squares, it will work out much better, because no terms multiply to give rise to powers of 6. Let’s do that right now:
An easy fix, which works! This expression is always nonpositive, as desired.
TL;DR
If we just look at the last line, it seems like a random brute-force factorization attempt. But, it really isn’t!
People who are experienced in the field of algebra may see this factorization immediately, so no effort is needed.
And, for inexperienced people like me, we can use mathematics to obtain this factorization in manageable steps, albeit with some small errors and highly misleading motivations.
