South African Mathematics Olympiad

Question

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 (A) 148 (B) 150 (C) 152 (D) 154 (E) 156

Accepted Answer

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.)