Answer
$ 6\ cm$ or $ 12\ cm$
Work Step by Step
Step 1. Let $x$ be the length of the edge of the cube. The dimensions after the cuts will be $x+6$ by $x+12$ by $x-4$
Step 2. The new volume is twice the old, we have $(x+6)(x+12)(x-4)=2x^3$ or $2x^3-(x+6)(x+12)(x-4)=0$
Step 3. We can solve the equation graphically as shown in the figure to get the positive solutions $x=6\ cm$ or $x=12\ cm$
