A solid cube of edge 15 cm is cut into 27 identical smaller cubes. By how many square cm does the total surface area increase?

The original cube has surface $6\times 15^2=1350$ sq cm. Twenty-seven equal cubes means three along each edge, so each small cube has edge 5 cm and surface $6\times 25=150$ sq cm, giving $27\times 150=4050$ sq cm in all. The increase = 4050 - 1350 = 2700 sq cm. The trap 4050 is the new total quoted where an increase was asked; 1350 is the original; 2025 comes from using the small cubes' VOLUME $27\times 125=3375$ in place of their surface area. Cutting never changes volume but always adds surface, because fresh faces are exposed.