Question 114
Let $C$ be a convex subset of the real space $\mathbb{R}^3$ with positive volume. Let $C_1, C_2, \dots, C_n$ be $n$ sets obtained by translating $C$ (without rotation), such that $C_i \cap C \ne \emptyset$ for every $i = 1, 2, \dots, n$, but for different $i, j$ the sets $C_i$ and $C_j$ intersect at most on their boundaries. Prove that $n \le 27$, and prove that $27$ is the largest possible value of $n$.