Isolated cubicles

What this 3D connectivity puzzle tests

This is a medium-difficulty spatial reasoning problem that asks you to think about the structure of a lattice cube and the minimal graph connectivity required to link interior nodes to a boundary. It rewards systematic thinking about layers, distances, and redundancy.

To approach problems like this, candidates typically model the cube as a graph where unit cubes are vertices and walls are edges. The key insight is recognizing which interior cubes are "farthest" from the surface and working backward to determine how many walls must be removed to ensure every cube has a path to at least one surface cube. The challenge lies in avoiding unnecessary removals while guaranteeing coverage.

• Graph connectivity and minimum spanning structures
• Manhattan distance and layer-based analysis
• Optimality arguments (proving you cannot do better)