Why quarters of potatoes fall the way they do? For the purpose of this question, potatoes are spheres, their quarters are obtained by cutting each sphere in two equal parts, and then cutting each of the obtained parts in two equal parts such that the cut goes through the centre of the circle produced by the previous cut.
My understanding of what should happen is based on the observation that the surface area of the "round" part of the potato will be $\frac{4\pi R^2}{4}=\frac{\pi R^2}{2}+\frac{\pi R^2}{2}$ equal to the sum of areas of two semi-circles being its other sides.  In practice, however, it's either I'm extremely unlucky, or there's a problem with this line of thought, because potato quarters seem to land almost exclusively on the "round" part. I'm afraid this has to do with the centre of mass, or maybe some other physical properties, but maybe I'm missing something obvious that doesn't require physics to explain this?
 A: The following is for a "soft" potato falling onto "soft" ground: We may assume that the potato is rotated in a random fashion around its centre of gravity (COG) and after first contact merely tilts into the closest stable position. With this model, the probabilities of the three surface parts are not proportional to their surface shares. Instead, we should project the surface from the centre of gravity onto a sphere around that and measure surfaces there. I didn't do the calculations, but this certainly produces a very different result.
For more general shapes we would additionally have to consider which faces are stable after all and add instable faces to a basin of attraction for other faces.
The model above is still not perfect for shapes that are not convex (or at least star-shaped from the centre of gravity?): If there were a very long, thin, practically weightless protruding stick, it might severely influence the probabilities because it would be disproportionally likely to be the end that made "first contact".
Even the convex potato shape may have this problem if the rotation movement is not small compared to the vertical movement upon contact.
Another problem in modelling this is that the potato and/or floor may be "hard" and bouncing may occur. With a sufficient amount of bounciness, we should consider Boltzmann's law, according to which the probability of a state should be proportional to not only its "a-priori-probability" (the COG-projected surface area as above) but also to a factor that depends exponentially on the energy (here the potential energy due to different heights of the COG) of the state. (The temperature also enters here; ask a physicist how to interprete that in this context)
