There are $n$ points, $A_1$, $A_2$, $\cdots$, $A_n$ on a line segment, $\overline{A_0A_{n+1}}$. The point $A_0$ is black, $A_{n+1}$ is white, and the rest points are colored randomly either black or white. Prove: among these $n+1$ line segments $A_kA_{k+1}$, where $k=0, 1, \cdots, n$, the number of those with different colored ending points is odd.
Mark a white point as 1 and a black point as -1. For each line segment, we can then multiply the values of its two ending points. If the result is 1, then the ending points have different colors; otherwise it will have the same colored ending points. If we multiply all these $(n+1)$ results, the aggregated result will be -1. This is because all the internal points, $A_1$, $A_2$, $\cdots$, $A_n$ will be used exactly twice, therefore regardless of their values, the result will be 1. But $A_0$ and $A_{n+1}$ are used exactly once each. Their result is $-1$. The aggregated result of $-1$ implies there are odd number of line segments whose ending points are different colors.