Sure, let's create a test case generator for the given "Flick Input" problem. 

Here's the generator in Python:

```python
import random

def generate_flick_input():
    """
    Generates a random test case for the 'Flick Input' problem.
    """
    # Character sets
    digits = '0123456789'
    flicks = 'TLURD'
    
    # Generate the length of input string (between 2 and 1000)
    length = random.randint(2, 1000)
    
    # Generate the flick input string
    flick_input = []
    for _ in range(length // 2):
        digit = random.choice(digits)
        flick = random.choice(flicks)
        flick_input.append(digit)
        flick_input.append(flick)
    
    # Ensure correct length if it's odd
    if length % 2 == 1:
        digit = random.choice(digits)
        flick = random.choice(flicks)
        flick_input.append(digit + flick)
    
    return ''.join(flick_input)

# Example usage
for _ in range(5):
    print(generate_flick_input())
```

This code will generate valid test cases for the "Flick Input" problem by creating a string with the required pattern of alternating digits and flick directions. The length of this string is ensured to be between 2 and 1000 characters.