A solid cube of edge 12 cm is cut into 8 identical smaller cubes. What is the ratio of the total surface area of all 8 small cubes to the surface area of the original cube?

Eight identical cubes cut from a 12 cm cube each have edge 6 cm. The original surface is $6\times 12^2=864$ sq cm; the eight small cubes give $8\times 6\times 6^2=8\times 216=1728$ sq cm. The ratio is 1728 : 864 = 2 : 1. The trap 1 : 1 assumes surface area is conserved the way VOLUME is, but cutting exposes new faces; 8 : 1 multiplies by the number of pieces, and 4 : 1 squares the linear factor 2.