Problem p03364 - Generation 3

Orig Description

Score : 500 points
Problem StatementSnuke has two boards, each divided into a grid with N rows and N columns.
For both of these boards, the square at the i-th row from the top and the j-th column from the left is called Square (i,j).
There is a lowercase English letter written in each square on the first board. The letter written in Square (i,j) is S_{i,j}. On the second board, nothing is written yet.
Snuke will write letters on the second board, as follows:
First, choose two integers A and B ( 0 \leq A, B < N ).
Write one letter in each square on the second board.
Specifically, write the letter written in Square ( i+A, j+B ) on the first board into Square (i,j) on the second board.
Here, the k-th row is also represented as the (N+k)-th row, and the k-th column is also represented as the (N+k)-th column.
After this operation, the second board is called a good board when, for every i and j ( 1 \leq i, j \leq N ), the letter in Square (i,j) and the letter in Square (j,i) are equal.
Find the number of the ways to choose integers A and B ( 0 \leq A, B < N ) such that the second board is a good board.
Constraints
1 \leq N \leq 300
S_{i,j} ( 1 \leq i, j \leq N ) is a lowercase English letter.
InputInput is given from Standard Input in the following format:
N
S_{1,1}S_{1,2}..S_{1,N}
S_{2,1}S_{2,2}..S_{2,N}
:
S_{N,1}S_{N,2}..S_{N,N}
OutputPrint the number of the ways to choose integers A and B ( 0 \leq A, B < N ) such that the second board is a good board.
Sample Input 12
ab
ca
Sample Output 12
For each pair of A and B, the second board will look as shown below:
The second board is a good board when (A,B) = (0,1) or (A,B) = (1,0), thus the answer is 2.
Sample Input 24
aaaa
aaaa
aaaa
aaaa
Sample Output 216
Every possible choice of A and B makes the second board good.
Sample Input 35
abcde
fghij
klmno
pqrst
uvwxy
Sample Output 30
No possible choice of A and B makes the second board good.

Extracted Specification

An integer N (1 ≤ N ≤ 300), representing some quantity or size.
A list of N strings, each of length N, consisting of lowercase English letters.

### Example Input:

```
2
ab
ca
```

### Function Signature:
Write a function f(N, S) that takes in the input.
def f(N: int, S: List[str]):
    ''' 
    N: an integer 
    S: a list of strings
    '''

Test Case Generator

Sure! Let's create a test case generator for the given problem statement. The goal is to generate random test cases that fit the problem constraints.

Here's the test case generator:

```python
import random
import string

def generate_random_letter():
    return random.choice(string.ascii_lowercase)

def tcgen():
    # Generate a random N within the constraint
    N = random.randint(1, 300)
    
    # Generate the first board S with random lowercase letters
    S = []
    for _ in range(N):
        row = ''.join(generate_random_letter() for _ in range(N))
        S.append(row)
    
    # Return the generated test case
    return N, S

def format_test_case(N, S):
    # Format the test case for output
    testcase = f"{N}\n"
    for row in S:
        testcase += row + "\n"
    return testcase.strip()

def main():
    # Generate a test case
    N, S = tcgen()
    
    # Format the test case for easy display
    formatted_test_case = format_test_case(N, S)
    
    # Print the formatted test case
    print(formatted_test_case)

if __name__ == "__main__":
    main()
```

### Explanation

1. **Generate Random `N`**:
    - `N` is randomly chosen between 1 and 300.

2. **Generate the Board `S`**:
    - For each row in the board, a string of `N` random lowercase letters is generated.

3. **Format the Test Case**:
    - The test case is formatted as required by the input format described in the problem statement.

4. **Output**:
    - The generated test case is printed in the required format.

You can run this script to generate random test cases for the problem statement. This will ensure that you have a variety of test cases to work with when solving the problem or testing your solution.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    S = [fh.readline().strip() for _ in range(N)]
    return N, S