Certainly! Here is the test case generator for the given problem statement:

```python
import random

def tcgen():
    # Generate a random number of flags within the given constraints
    N = random.randint(2, 10**4)
    
    # Generate N pairs of coordinates (x_i, y_i) within the given constraints
    coordinates = [(random.randint(1, 10**9), random.randint(1, 10**9)) for _ in range(N)]
    
    return N, coordinates

# Example usage
if __name__ == "__main__":
    N, coordinates = tcgen()
    print(N)
    for x, y in coordinates:
        print(x, y)
```

**Explanation:**

1. **Number of Flags:** The number of flags \( N \) is generated randomly between 2 and \( 10^4 \) (inclusive) to meet the problem's constraints.
2. **Coordinates:** For each flag, two possible coordinates \( x_i \) and \( y_i \) are generated randomly between 1 and \( 10^9 \) (inclusive).

This script generates a test case, prints the number of flags, and then prints each pair of possible coordinates. You can run this script to generate a new test case each time.