Those are non-linear equations.

These are all polynomial equations, so we can likely solve them by computing a Grobner basis over the set and using the Elimination Theorem.

I don't know what any of those are but I know how to write 15 nested for loops and let it run for a few minutes. Hacking code seems fine if that's what someone knows...

That's the failure of maths education. Plenty of names for formal things and ideas, yet mostly inaccessible to the lay person.

So I know 6 programming languages, maybe 600 libraries, devops, dozens of tools.

So I could go back to math class and learn some more formal proofs. What part of my programming knowledge am I going to drop to learn that? Will it actually help my career?

I have found little to no use for advanced mathematics in 10 years of programming. I am not saying it had no use, but I have not needed it.

Most academic types majorly overestimate the usefulness of math in programming. I can code circled around people who rock out at math.

I don't mean to demean your programming experience or toolkit decisions in anyway. Egg on my face--it appears that's how it was received! Let me try again.

> I have found little to no use for advanced mathematics in 10 years of programming.

I reckon you use or stumble upon the algorithms often, perhaps all the time. Be it with Boolean logic, or network design, or optimal stopping, or fuzzy matching, or statistics in logging, or what have you. These principles show up often. My point wasn't you can't or shouldn't be able to code circles around people who rock out at math, but that maths education often fails to emphasize the logical methods of figuring things out from first principles over the already solved proofs, methods, or equations.

> Most academic types majorly overestimate the usefulness of math in programming.

To some degree, I agree here. If there are well developed libraries and mature paradigms needed (RDBMS, web dev, etc.) then there is little need for mathematical forte. Nothing wrong with that!

Math often provides a much easier and elegant solution to things:

http://www.evanmiller.org/mathematical-hacker.html

There are only so many hours in the day to learn things.

OK now you've got a variety over Q, good luck finding the integral points in the ___domain that you care about.

According to OP these are "12 equations in 18 variables, where the variables were restricted to integers in the range [1, 26]". Can you solve them using the elimination theorem?