2004 AIME II Problems/Problem 3

Problem

A solid rectangular block is formed by gluing together congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of

Solution

231 cubes cannot be visible, so at least one extra layer of cubes must be on top of these. The prime factorization of 231 is , and that is the only combination of lengths of sides we have for the smaller block (without the extra layer). The extra layer makes the entire block . Multiplying gives us the answer, .