We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$. Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?

[img]https://i.imgur.com/AHqHHP6.png[/img]
1. **Define Variables and Given Information:**
   - Let \( a \) be the length of the cube's edge.
   - The height of the cone is \( h = 6 \).
   - The radius of the base of the cone is \( r = \sqrt{2} \).
   - The half-angle of the cone, \( \theta \), can be found using the tangent function:
     \[
     \tan \theta = \frac{r}{h} = \frac{\sqrt{2}}{6}
     \]

2. **Vertical Cross Section Analysis:**
   - Consider the vertical cross section of the cone that passes through the apex and a diagonal of the top face of the cube.
   - The length of the diagonal of the top face of the cube is \( a\sqrt{2} \).
   - The height from the apex of the cone to the top face of the cube is \( x \).

3. **Relate the Height and Edge Length:**
   - The height of the cone minus the height from the apex to the top face of the cube gives the height of the cube:
     \[
     6 - x = a
     \]
   - Therefore, we can express \( x \) in terms of \( a \):
     \[
     x = 6 - a
     \]

4. **Use Trigonometry to Relate \( x \) and \( a \):**
   - The tangent of the half-angle \( \theta \) can also be expressed using the triangle formed by the height \( x \) and half the diagonal of the top face of the cube:
     \[
     \tan \theta = \frac{\frac{a\sqrt{2}}{2}}{x}
     \]
   - Substitute the known value of \( \tan \theta \):
     \[
     \frac{\sqrt{2}}{6} = \frac{\frac{a\sqrt{2}}{2}}{x}
     \]

5. **Solve for \( x \):**
   - Simplify the equation:
     \[
     \frac{\sqrt{2}}{6} = \frac{a\sqrt{2}}{2x}
     \]
   - Multiply both sides by \( 2x \):
     \[
     2x \cdot \frac{\sqrt{2}}{6} = a\sqrt{2}
     \]
   - Simplify further:
     \[
     \frac{2x\sqrt{2}}{6} = a\sqrt{2}
     \]
     \[
     \frac{x\sqrt{2}}{3} = a\sqrt{2}
     \]
   - Divide both sides by \( \sqrt{2} \):
     \[
     \frac{x}{3} = a
     \]
   - Therefore:
     \[
     x = 3a
     \]

6. **Substitute \( x \) Back into the Height Equation:**
   - Recall \( x = 6 - a \):
     \[
     3a = 6 - a
     \]
   - Solve for \( a \):
     \[
     3a + a = 6
     \]
     \[
     4a = 6
     \]
     \[
     a = \frac{6}{4} = \frac{3}{2}
     \]

The final answer is \( \boxed{\frac{3}{2}} \).