Solving a system of non-linear equations (not differential equations)

In my free time, I fooled around with the "crossing ladders" puzzle:

http://www.softlab.ntua.gr/~ttsiod/ladders.html

I did figure out a solution, but in the process, I got interested in
ways to figure it out in a completely automated manner. Is there a
generic way to solve a system of non-linear equations (not
differential equations) using numerical methods? As you can see in my
article above, I did try some things on my own, but didn't succeed.

Thanks for any help,

Thanassis.
.

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