Certainly! Here is a Python function to generate test cases for the "Yamanote-line Game" problem:

```python
import random

def tcgen():
    # N is the number of stations
    N = random.randint(3, 1000)
    
    # d is the fare to ride the Yamanote-line modoki
    d = random.randint(1, 1000)
    
    # p is the list of rewards at each station
    p = [random.randint(1, 1000) for _ in range(N)]
    
    return N, d, p

# Generate a test case and print it in the required format
N, d, p = tcgen()
print(N, d)
print(" ".join(map(str, p)))
```

This function generates random values for the number of stations \(N\), the fare \(d\), and the rewards \(p[i]\) for each station within the provided constraints. It then prints the test case in the required format.

You can use this function to generate various test cases for the "Yamanote-line Game" problem. Here's how it works:

1. `N` is randomly chosen between 3 and 1000.
2. `d` is randomly chosen between 1 and 1000.
3. `p` is a list of \(N\) integers, each randomly chosen between 1 and 1000.

You can run this function multiple times to generate different test cases for testing your solution.