6. A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that each face of the cube forms the base of a pyramid. The volumes of five of these pyramids are 200, 500,1000,1100 and 1400 . What is the volume of the sixth pyramid?
SolUTION
600

Each of the cube's faces forms the base of one of the six pyramids. Any two pyramids formed from opposite faces will have a combined volume of $\frac{1}{3} \times$ area of cube face $\times$ distance between opposite faces $=\frac{1}{3} \times s^{3}$, where $s$ is the cube's side-length.
Pairing the known pyramid volumes we see that $200+1400=500+1100=1600$. The missing pyramid therefore has volume $1600-1000=600$.