Some Facts and Conjectures about Mandelbrot Polynomials

We show here some facts about the Mandelbrot iterates $z_k(c)$ where $z_0(c)=0$ and $z_{n+1}(c) = z_n^2(c) + c$, which are polynomials in $c$. Some of the facts have proofs, and some other ``"``facts" only have experimental evidence but no proof. We invite you to try your hand at filling in the gaps.