Sure, I can help you write a test case generator for the given problem "Life Game". Here's the test case generator in Python:

```python
import random

def tcgen():
    # Generate N in the range [1, 6]
    N = random.randint(1, 6)
    
    # Generate M in the range [2, 10^9]
    M = random.randint(2, 10**9)
    
    # Generate L in the range [1, 10^9]
    L = random.randint(1, 10**9)
    
    # Generate the initial state of the grid
    grid = []
    for i in range(2 * N - 1):
        if i < N:
            row_length = N + i
        else:
            row_length = 3 * N - 2 - i
        
        row = [random.randint(0, M - 1) for _ in range(row_length)]
        grid.append(row)
    
    return N, M, L, grid

def format_test_case(N, M, L, grid):
    test_case = f"{N} {M} {L}\n"
    for row in grid:
        test_case += " ".join(map(str, row)) + "\n"
    return test_case

# Generate a single test case
N, M, L, grid = tcgen()
test_case = format_test_case(N, M, L, grid)

print(test_case)
```

This generator will create random test cases according to the constraints provided in the problem statement. The `tcgen` function generates values for `N`, `M`, and `L` within their respective ranges. It then generates the initial state of the hexagonal grid according to the structure specified in the problem statement. The `format_test_case` function formats the generated test case into the required input format.

You can use this generator to create multiple test cases by calling `tcgen` multiple times and formatting each output using `format_test_case`. This will help you test your solution comprehensively against a variety of inputs.