A solid cube has side length $4$ inches. A $2$-inch by $2$-inch square hole, centered on each face with edges parallel to the cube's edges, is cut all the way through the cube. What is the volume, in cubic inches, of the remaining solid?

1. For a cube of side $N$ with square tunnels of width $w$ through the center of each face, each tunnel has volume $w^2N$, each pairwise (and the triple) overlap is the central cube of volume $w^3$.
2. By inclusion-exclusion the union has volume $3w^2N-2w^3$, so the remaining solid has volume $N^3-3w^2N+2w^3$.
3. With $N=4,w=2$: remaining volume $=64-3(4)(4)+2(8)=64-48+16=32$.