A solid cube of edge 12 cm is cut into 64 identical smaller cubes. How many ${\text{cm}}^2$ of fresh surface does the cutting create?

Since $64=4^3$, the small cubes have edge 12/4 = 3 cm. All of them together show $64\times 6\times 3^2=3456{\text{cm}}^2$, while the original showed $6\times 12^2=864{\text{cm}}^2$. Increase $=3456-864=2592{\text{cm}}^2.(3456{\text{cm}}^2$ is the new total and $864{\text{cm}}^2$ the old one — neither is the increase; 1728 is the volume in ${\text{cm}}^3$, which cutting does not change at all.)