Mathematical Olympiad Rioplatense

Imagine the eight corner parts yellow and the rest of the cube white. The two horizontal cuts produce three parallelepipeds. The middle one is white and has the same vertical dimension as the central piece. Assume the latter dimension to be $1$ and remove the middle part. A $10\times 10\times 9$ parallelepiped is obtained with $4$ yellow parts instead of $8$. This is because the initial $8$ yellow parts come into $4$ pairs with equal horizontal dimensions (the two of them) in every pair. So after removing the middle white part every pair becomes a single yellow piece. Repeat the same with the two cuts in direction left-right. They also produce three parts; the middle one is white. Its width equals the front-back dimension of the central piece which we assume to be $2$. So removing the middle part yields a $10\times 8\times 9$ parallelepiped in which the yellow portion consists of two parallelepipeds separated by a white parallelepiped $3\times 8\times 9$. Remove this white piece; the entire yellow part remains in the shape of a parallelepiped $7\times 8\times 9$, hence its volume is $7\cdot 8\cdot 9=504$.