1. The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?
Solution: $6 \sqrt{3}$

If the side length of the cube is $s$ inches, then the condition implies $s^{3}+3 \cdot 12 s=2 \cdot 6 s^{2}$, or $s\left(s^{2}-12 s+36\right)=s(s-6)^{2}=0$. Therefore $s=6$, and the long diagonal has length $s \sqrt{3}=6 \sqrt{3}$.