Flights are arranged between 13 countries. For $ k\ge 2$, the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$, from $ A_{2}$ to $ A_{3}$, $ \ldots$, from $ A_{k \minus{} 1}$ to $ A_{k}$, and from $ A_{k}$ to $ A_{1}$. What is the smallest possible number of flights such that how the flights are arranged, there exist a cycle?

$\textbf{(A)}\ 14  \qquad\textbf{(B)}\ 53  \qquad\textbf{(C)}\ 66  \qquad\textbf{(D)}\ 79  \qquad\textbf{(E)}\ 156$
1. **Label the countries**: Let the countries be labeled as \( C_1, C_2, \ldots, C_{13} \).

2. **Understanding cycles**: A cycle is a sequence \( A_1, A_2, \ldots, A_k \) such that there is a flight from \( A_1 \) to \( A_2 \), from \( A_2 \) to \( A_3 \), ..., from \( A_{k-1} \) to \( A_k \), and from \( A_k \) to \( A_1 \).

3. **Counting possible flights**: The total number of possible flights between any two countries is given by the combination formula \( \binom{13}{2} \), which counts the number of ways to choose 2 countries out of 13:
   \[
   \binom{13}{2} = \frac{13 \times 12}{2} = 78
   \]
   This means there are 78 pairs of countries, and each pair can have at most 2 flights (one in each direction).

4. **Applying the pigeonhole principle**: If we have 79 flights, by the pigeonhole principle, at least one pair of countries must have both flights (one in each direction), forming a 2-country loop. Therefore, with 79 flights, a cycle must exist.

5. **Constructing a flight arrangement without cycles**: To show that 78 flights can be arranged without forming a cycle, consider the following construction:
   - Arrange flights such that if \( i > j \), then there is a flight from \( C_i \) to \( C_j \).
   - This means each country \( C_i \) (for \( i > 1 \)) has flights to all countries \( C_j \) where \( j < i \).

6. **Verifying no cycles in the construction**: In this arrangement, any sequence \( A_k \) will have strictly decreasing indices, and thus it is impossible to return to the starting country, ensuring no cycles are formed.

7. **Conclusion**: Since 78 flights can be arranged without forming a cycle, but 79 flights must form a cycle, the smallest number of flights required to ensure a cycle exists is 79.

The final answer is \(\boxed{79}\).