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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.

 

 Dec 26, 2023
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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 $\frac{2}{3} \cdot 6^3=\boxed{288}$.

 Dec 26, 2023

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