Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.0 AIME Find the answer South Africa

Anelize put $120$ identical cubes together to form a rectangular prism. She then painted all $6$ faces of the prism. Once the paint had dried she disassembled the cubes and found that $24$ of the cubes had not been painted on any face. The surface area of the prism she formed was

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Solution

Let $l$ , $b$ , $h$ denote the numbers of cubes in the length, breadth and height respectively. Then $lbh = 120$ . To find the unpainted cubes we must remove a layer at both ends of the length, breadth and height, giving $(l-2)(b-2)(h-2) = 24$ . By trying various factorizations of $120$ , we soon find that $l$ , $b$ , $h$ are equal to $4$ , $5$ , $6$ in some order. The surface area is $2(lb + lh + bh) = 2(20 + 30 + 24) = 148$ . (Trying to solve the equations algebraically is harder than using trial and error.)

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