Given a convex solid with uniform density, which is less than the density of water, if it's thrown into water, it will float.
How many stable equilibrium points can it settle into?
To think about the question, it helps to think about an energy function:
$E:$(orientation of the solid)$\to$(gravitational energy of the orientation)
Fix some orientation $x$ of the solid, and lower the solid slowly into water. At some height, the displaced water would equal in weight with the solid, and the combined gravitational energy of the solid and the displaced water is $E(x)$.
What is the space of orientations? Guess: it's $SO(3)$. But that's wasteful, since if we rotate the solid around the perpendicular direction, we get a different orientation $x'$, but the same $E(x') = E(x)$. So it should be $S^2$.
So we get a continuous energy function $E: S^2 \to \mathbb R$. The stable equilibrium points are exactly the local minima of $E$.
So we immediately see that the solid should have at least one stable equilibrium and one unstable equilibrium. Further, if the solid has a symmetry group $G$, then the solid has at least $|G|$ many stable equilibria.
The question is, how many? My intuition says that, generically, it is tight: a generic convex uniform solid has exactly $|G|$ stable equilibria.