Problem #PRU-107702

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Features of dissection pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)
 13–16  3.0

Problem

The surface of a \(3\times 3\times 3\) Rubik’s Cube contains \(54\) squares. What is the maximum number of squares we can mark so that no marked squares share at least one vertex?

Make sure you show that both (a) you can achieve this maximum and (b) that you can’t do better than this maximum.