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Geometry Difficulty 3.0 AMC 10/12 Prove it Canada

A cube has edges of length $n$ , where $n$ is an integer. Three faces, meeting at a corner, are painted red. The cube is then cut into $n^3$ smaller cubes of unit length. If exactly 125 of these cubes have no faces painted red, determine the value of $n$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.