Are you up for a challange? Nine years ago, I experimented with some variables and came up with an equation system. I figured out there were solutions to it and worked hard to find generalized ways to solve for the different variables. I only used a note pad, pen and an ordinary "high school" calculator with the most basic stuff on it like logaritms, cosine and so on.
This is the system in question: $$z=x^2-y^2=xy=(x-y)^z $$
Can you solve it without plotting graphs on a calculator (in which case it is not much of a challange)? I don't remember exactly how I solved it, becuase it invloed a lot of testing, intution and homemade numerical methods. If you find solutions it will be interesting to see in what form you present them. Becuase they can be expressed in many different ways, I suspect.
This is way over my head to take on myself, but after we have the solutions with as many decimals as we find suitable the question also arises whether we can prove that the variables are transcendental numbers. (I suppose they are.)
The trick is to solve the relationship for $(x,y)$, then it's easy to compute and check the $z$ from there. You have $$ x^2-y^2=xy\\ y^2+xy-x^2=0 $$ and now the quadratic formula yields $$ y_\pm = \frac{-x \pm \sqrt{x^2+4x^2}}{2} = x \left(\frac{-1 \pm \sqrt{5}}{2}\right). $$