Example 3 Given 10 points on a circle, color six of them black and the remaining four white. They divide the circumference into arcs that do not contain each other. Rule: arcs with both ends black are labeled with the number 2; arcs with both ends white are labeled with the number $\frac{1}{2}$; arcs with ends of different colors are labeled with the number 1. Multiply all these numbers together, and find their product.
Mark all the black points with $\sqrt{2}$, and all the white points with $\frac{1}{\sqrt{2}}$, then the number marked on each arc is exactly the product of the numbers at its two ends. Therefore, the product of the numbers marked on all these arcs is the square of the product of the numbers marked on all the points, i.e.,
$$
\left[(\sqrt{2})^{6}\left(\frac{1}{\sqrt{2}}\right)^{4}\right]^{2}=4 \text {. }
$$