Cube Damage Counting

This brain teaser describes a three-dimensional block built from unit cubes, then asks how many of those units end up as the "damaged" outer layer that falls away. The puzzle is framed visually, encouraging you to picture peeling off the surface of a solid object and to connect that picture to a clean counting argument. It sits in the family of discrete-geometry riddles where a large structured object is built from identical small pieces, and a simple modification is applied uniformly to its boundary.

To reason it out, you need to recognize the remaining shape as a smaller, similar cube and express both the original and remaining structures in terms of their side lengths. The solution relies on understanding how volume scales with side length in three dimensions and translating that into counts of unit elements. Interviewers use this type of question to see whether you naturally reduce a wordy geometric story to powers and differences, and whether you can generalize the pattern beyond the specific numbers mentioned.

What it tests

In problems involving removing the outer layer from a higher-dimensional object built from unit elements (like cubes or squares), the key is to recognize that the remaining structure is a smaller, similar object whose side length is reduced by twice the thickness of the removed layer (once for each side). The count of removed elements is the difference between the total number of unit elements in the original and the remaining structure. This pattern holds because each face loses a layer, and in $n$ dimensions, the count of unit elements is the side length raised to the $n$th power. The subtraction captures only the elements exposed on the surface, as the interior remains untouched.