Question 6
A cube of $(2N)^3$ unit cubes is pierced by axis-parallel needles, each passing through exactly 2 N unit cubes. Every unit cube is pierced by at least one needle. A set of needles is regular if no two pierce the same unit cube. (a) Prove there is a regular set of 2 N² needles using at most two directions. (b) Determine the largest guaranteed size of a regular set. The official original says “at most two directions”; the book’s phrasing is less precise.