B4 In a cube with edge 6, there is a sphere with diameter 6. The sphere thus touches the six faces of the cube internally. We imagine the cube divided into 216 unit cubes of $1 \times 1 \times 1$. How many of these unit cubes lie entirely within the sphere? A unit cube with a vertex on the sphere's surface and otherwise inside the sphere is counted as lying entirely within the sphere.

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B4. Place the center of the sphere at the origin and consider one-eighth of the cube with one-eighth of the sphere inside it. It is clear that the cubes on the outside do not lie entirely within the sphere. Do all the remaining cubes $(2 \times 2 \times 2)$ lie entirely within the sphere? The vertex farthest from the origin lies at twice the body diagonal length of the origin, which is $2 \times \sqrt{3}$ and that is more than 3. The remaining 7 unit cubes lie within the sphere because the three vertices farthest from the origin lie at a distance $\sqrt{2^{2}+2^{2}+1^{2}}=3$ from the origin and thus exactly on the surface of the sphere. In total, there are therefore $8 \times 7=56$ unit cubes within the sphere.

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