We are given a circle with 21 points marked around its circumference, labeled 0 through 20, and we are tasked with marking $n$ of these points with an asterisk such that no two pairs of marked points are the same distance apart. We need to determine the maximum value of $n$.

Understanding the Problem:

The points on the circle are numbered cyclically from 0 to 20, so the distances between points are taken modulo 21. When we select two points, say $p$ and $q$, the distance between them can be expressed as:

$d(p, q) = \min(|p - q|, 21 - |p - q|)$

This distance is always between 0 and 10, because any larger distance can be wrapped around the circle to yield a smaller distance.

The requirement is that for any two pairs of marked points, the distance between the points in each pair must be distinct. In other words, if you mark two points at positions $p_1$ and $p_2$, and another two points at positions $p_3$ and $p_4$, we must have:

$d(p_1, p_2) \neq d(p_3, p_4)$

Steps to Find the Maximum Value of $n$:

1. List Possible Distances: The possible distances between any two points are the integers from 1 to 10 (because a distance of 11 or more can be reduced by subtracting from 21). Therefore, there are exactly 10 distinct possible distances.

2. Maximize the Number of Marked Points: Since there are 10 distinct distances, we can mark up to 10 points such that the distance between any two marked points is distinct. If we attempt to mark more than 10 points, we would have to repeat one of the distances, violating the condition that no two pairs of points have the same distance.

3. Choose the Points: To ensure that all distances between pairs of points are distinct, we must place the marked points in such a way that the distances between them cover all 10 possible values (1 through 10) without repetition. A possible strategy is to mark the points at positions that are spaced apart by distinct distances, ensuring that each distance is used exactly once.

4. Conclusion: The maximum number of points that can be marked, such that no two pairs of marked points are the same distance apart, is 10. This is because there are 10 distinct possible distances, and marking more than 10 points would force some distances to repeat.

Thus, the maximum value of $n$ is $\boxed{10}$.

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Would you like more details on how to select the specific points or how to prove this is the best arrangement? Let me know if you have further questions!

Here are some related questions to explore:

1. How would the solution change if we had more or fewer points on the circle?
2. What strategies can be used to arrange the points optimally?
3. Can we generalize this problem for a circle with any number of points?
4. What other combinatorial problems are related to selecting points with distinct distances?
5. How does the modulo operation impact the distances in cyclic problems like this one?

Tip: When dealing with cyclic combinatorics, always consider the modulo structure and how distances wrap around, as this can simplify the analysis significantly!