1. On a flat desktop, there are three wooden balls with radii of 1, 2, and 2, each touching the desktop and externally tangent to the other two balls. Additionally, there is a smaller wooden ball with a radius less than 1 between the three balls, touching the desktop and externally tangent to all three balls. What is the radius of this smaller ball? $\qquad$
Two, $1.4-2 \sqrt{3}$.
For two tangent spheres with radii $r_{1}$ and $r_{2}$, the distance between the two points of tangency on the table is $2 \sqrt{r_{1} r_{4}}$.

Let the radius of the smaller sphere be $x$. The problem simplifies to: In $\triangle ABC$, $AB = AC = 2 \sqrt{2}, BC = 4$. $D$ is a point inside the triangle, $AD = 2 \sqrt{x}, BD = CD = 2 \sqrt{2 x}$. Therefore, $AD$ lies on the perpendicular bisector $AE$ of $BC$, as shown in Figure 5.

Thus, $AD = AE - DE = \sqrt{AB^{2} - BE^{2}} - \sqrt{BD^{2} - BE^{2}}$. Substituting the values, we get $2 \sqrt{x} = 2 - 2 \sqrt{2 x - 1}, x < 1$.
Solving this, we get $x = 4 - 2 \sqrt{3}$.