## Sunday, December 28, 2008

### A math problem

Okay, so here's an example of where dropping out of math classes after Differential Equations is coming back to bite me a bit. I'm working on a kind of puzzle, mostly for fun, but possibly to incorporate into a future software product.

At its most basic, the solution process for the puzzle turns out to be solving a system of simultaneous equations. Which is something I learned how to do way back in High School, and for linear equations, I can even find off-the-shelf algorithms and libraries for doing so.

The catch, of course, is that these aren't linear equations. They use modular arithmetic, which is something I understand at a basic level, like anybody who programs for a living probably does, but I don't know where to even start breaking this down to solve a non-trivial version, and Google isn't helping me.

5x + 7y + 4 = 0
4x + 3y + 1 = 0

Use whatever method you like, and you get:
x = 0.3846
y = -0.846

Piece of cake. Now, what if the equation looks like this?
5x + 7y + 4 = 0 (modulo 16)
4x + 3y + 1 = 0 (modulo 16)

If we want to find a few integer solutions for x and y, how do we find them? I could write a program to just guess every integer between 1 and 1,000,000 for each of the coefficients, and that'd find me a solution, but it doesn't scale well if I have a large number of variables. In the example equations given, there are rather a lot of solutions ([9,9],[9,25],[25,25]...), but I suspect that some other (carefully chosen?) sets of coefficients would have a much smaller set of solutions. Actually, that's kind of the point of the whole exercise.

Anyone out there got some hints for me?
Googling "simultaneous modular equations" got me:
http://en.wikibooks.org/wiki/Discrete_mathematics/Modular_arithmetic#Simultaneous_equations
and