15. On the number line, points with coordinates
$1,2, \cdots, 2006$ are called marked points. A frog starts from point 1, and after 2006 jumps, it visits all the marked points and returns to the starting point. What is the maximum length of the total path the frog has jumped? Explain your reasoning.
15. Let the points the frog reaches in sequence be $X_{1}, X_{2}, \cdots, X_{2006}$, with $X_{1}=1$. The total length of the path jumped over is
$$
S=\left|X_{1}-X_{2}\right|+\left|X_{2}-X_{3}\right|+\cdots+\left|X_{2006}-X_{1}\right| \text {. }
$$

For each point $X_{i}$, there is one entry and one exit, so $X_{i}$ appears 2 times in the above expression.

On the other hand, each term $\left|X_{i}-X_{i+1}\right|$ in the expression, when expanded, takes a “+” sign for the larger value and a “-” sign for the smaller value. Therefore, in the entire expression, the points that take a “+” sign and those that take a “-” sign each total 2006.

Assume that $\{1,2, \cdots, 1003\}$ is the upper half, and $\{1004,1005, \cdots, 2006\}$ is the lower half. If the points in the upper half are made to take a “-” sign twice during the calculation, and the points in the lower half are made to take a “+” sign twice, then we have
$$
\begin{aligned}
S & \leqslant 2(1004+1000+\cdots+2006)-2(1+2+\cdots+1003) \\
& =2 \times 1003^{2} .
\end{aligned}
$$

In fact, as long as each jump is from the upper half to the lower half, or from the lower half to the upper half, the points in the upper half will take a “-” sign, and the points in the lower half will take a “+” sign (clearly, there are many such jumping schemes), and the length of the path jumped over in this process is exactly $2 \times 1003^{2}$. Therefore, the maximum length of the path the frog can jump over is $2 \times 1003^{2}$.