Sure! To create a test case generator for this problem, we need to generate three integers \( A, B, \) and \( C \) each between 1 and 9. Here is a Python function that accomplishes this:

```python
import random

def tcgen():
    # Generate three integers between 1 and 9 inclusive
    A = random.randint(1, 9)
    B = random.randint(1, 9)
    C = random.randint(1, 9)
    
    return A, B, C

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

This function will create different test cases every time it is called. Each test case contains three integers between 1 and 9 inclusive, suitable for the problem constraints.

Here is a breakdown of the process:

1. **Generate three random integers**: Each integer \( A, B, \) and \( C \) is generated within the range of 1 to 9.
2. **Return the generated integers**: The function `tcgen()` returns these integers as a tuple.

You can use the `print(tcgen())` to see the generated test cases. Each call to `tcgen()` will produce a new set of integers which you can then use as inputs to test your problem solution.