Consider $n$ disks $C_{1}; C_{2}; ... ; C_{n}$ in a plane such that for each $1 \leq i < n$, the center of $C_{i}$ is on the circumference of $C_{i+1}$, and the center of $C_{n}$ is on the circumference of $C_{1}$. Define the [i]score[/i] of such an arrangement of $n$ disks to be the number of pairs $(i; j )$ for which $C_{i}$ properly contains $C_{j}$ . Determine the maximum possible score.
1. **Understanding the Problem:**
   We are given \( n \) disks \( C_1, C_2, \ldots, C_n \) in a plane such that for each \( 1 \leq i < n \), the center of \( C_i \) is on the circumference of \( C_{i+1} \), and the center of \( C_n \) is on the circumference of \( C_1 \). We need to determine the maximum possible score, where the score is defined as the number of pairs \((i, j)\) for which \( C_i \) properly contains \( C_j \).

2. **Initial Observations:**
   - If \( C_i \) properly contains \( C_j \), then the radius of \( C_i \) must be greater than the radius of \( C_j \).
   - The condition that the center of \( C_i \) is on the circumference of \( C_{i+1} \) implies a specific geometric relationship between the disks.

3. **Graph Representation:**
   - We can represent the disks and their relationships using a graph \( G \) where each vertex represents a disk and an edge between two vertices indicates that one disk's center is on the circumference of the other.
   - The graph \( G \) will be a cycle since \( C_n \)'s center is on the circumference of \( C_1 \).

4. **Inductive Approach:**
   - We will use induction to determine the maximum score.
   - Base Case: For \( n = 2 \), there is only one pair \((1, 2)\) or \((2, 1)\), and the maximum score is 1.
   - Inductive Step: Assume the maximum score for \( n = k \) disks is \( \binom{k-1}{2} \). We need to show that the maximum score for \( n = k+1 \) disks is \( \binom{k}{2} \).

5. **Constructing the Disks:**
   - Start with a large disk \( C_1 \).
   - Place \( C_2 \) such that its center is on the circumference of \( C_1 \) and its radius is smaller than \( C_1 \).
   - Continue this process, ensuring each subsequent disk \( C_{i+1} \) has its center on the circumference of \( C_i \) and a smaller radius.
   - For the last disk \( C_n \), ensure its center is on the circumference of \( C_{n-1} \) and its radius is smaller than \( C_{n-1} \).

6. **Counting the Pairs:**
   - Each disk \( C_i \) properly contains all subsequent disks \( C_{i+1}, C_{i+2}, \ldots, C_n \).
   - The number of such pairs for \( n \) disks is the sum of the first \( n-1 \) integers, which is \( \binom{n-1}{2} \).

7. **Conclusion:**
   - The maximum possible score for \( n \) disks is \( \binom{n-1}{2} \).

The final answer is \( \boxed{\binom{n-1}{2}} \)