6 unit radius spheres are arranged so that each one touches exactly four others. What is the volume of the convex body defined by the points of contact?
Let's denote the centers of the spheres by $A, B, C, D, E, F$. We know that two spheres touch each other exactly when the distance between their centers equals the sum of their radii, which in our case is 2 units.

![](https://cdn.mathpix.com/cropped/2024_05_02_8e0d0ad733250ab78f1cg-1.jpg?height=732&width=819&top_left_y=263&top_left_x=642)

We can assume, for example, that the spheres centered at $A$ and $F$ do not touch each other. Then both touch all the spheres centered at $B, C, D, E$. Therefore, the points $B, C, D, E$ are 2 units away from both $A$ and $F$, meaning they lie on the intersection of the 2-unit radius spheres centered at $A$ and $F$, which is a circle. Thus, the quadrilateral $BCDE$ is a cyclic quadrilateral, and since all its sides are 2 units, it must be a square. The points $A$ and $F$ are equidistant from every vertex of the square, so they lie on the line perpendicular to the plane of the square and passing through its center. Since $AB = BC$ and $FB = BC$, the points $A$ and $F$ together with $B, C, D, E$ form the vertices of a regular octahedron. Conversely, it is easy to see that the vertices of a regular octahedron with edge length 2 units, each surrounded by a unit sphere, will each touch exactly four others.

![](https://cdn.mathpix.com/cropped/2024_05_02_8e0d0ad733250ab78f1cg-1.jpg?height=721&width=672&top_left_y=1453&top_left_x=716)

The solid defined by the points of tangency is thus identical to the convex solid defined by the midpoints of the edges of a regular octahedron with side length 2 units. This solid can also be obtained by cutting off a square-based pyramid with all edges 1 unit from each vertex of the octahedron. The desired volume is therefore (using the appropriate volume formulas from the table):

$$
V=\frac{\sqrt{2}}{3} \cdot 2^{8}-\frac{6 \cdot\left(\frac{\sqrt{2}}{3} \cdot 1^{3}\right)}{2}=\frac{5}{3} \sqrt{2} \quad \text { cubic units. }
$$