Unit cubes touched by paint on a 4-cube

Question

A $4\times4\times4$ cube made of $64$ unit cubes has its outside painted. How many of the small cubes have at least one painted face?

Short answer

56. Only the inner $2\times2\times2$ core is unpainted, so $64-8=56$ cubes carry paint.

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q8.3, p.276; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics

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