Donatello starts with a marble cube. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length. If the side length of the original cube is $6$, then find the volume of the resulting polyhedron.
When Donatello slices off the corners, the vertices of the resulting polyhedron are the midpoints of the original cube's edges. Donatello makes six of these slices, so our polyhedron has $\frac{2}{3}$ of the volume of the original cube. The resulting polyhedron thus has volume $\frac{2}{3} \cdot 6^3=\boxed{288}$.
