Example 3.3.4 There are 2005 same-sized circular paper pieces placed on a table without overlapping. How many different colors are needed at least to ensure that each piece can be colored with one color, so that any two touching circular paper pieces are colored differently?
Solution: Let the number of colors that can ensure the requirement be $k$, then $k>3$.

That is, three colors cannot guarantee the requirement. For example, in the figure, $A, B, E$ can only be colored with 1 and 3, and $A$ is one of them, while $B, E$ are the other. At this point, $C, D$ can only be colored with 2, which is a contradiction.
Next, we prove that $n$ circular pieces of paper can be colored with four colors.
For $n=4$, it is obvious;
If $n$ can be colored $\Rightarrow n+1$ can be colored.
Consider the convex hull of the centers of $n+1$ circles, let $A$ be one of the vertices of this convex hull, then circle $A$ can touch at most three other circles. Removing circle $A$, the $n$ circles can be colored with four colors, and circle $A$ can touch at most three circles, so it can always be colored with the fourth color that is not used by the three circles, i.e., $n+1$ circular pieces of paper can be colored with four colors.