Problem p03364 - Generation 1

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.
An N x N grid of lowercase English letters.

### Example Input:
```
2
ab
ca
```

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

Test Case Generator

Sure, I can help you craft a test case generator for the given problem statement. The generator will create random inputs that follow the constraints of the problem. Here's the Python code for the test case generator:

```python
import random
import string

def generate_board(N):
    """Generate a random NxN board with lowercase English letters."""
    return [''.join(random.choices(string.ascii_lowercase, k=N)) for _ in range(N)]

def test_case_generator():
    N = random.randint(1, 300)
    board = generate_board(N)
    return N, board

# Example usage:
if __name__ == "__main__":
    N, board = test_case_generator()
    print(N)
    for row in board:
        print(row)
```

### Explanation:
1. **generate_board(N)**: This function generates an NxN board filled with random lowercase English letters.
2. **test_case_generator()**: This function generates the value of \(N\) and the corresponding board.
3. **Example usage**: When executed, this will print a randomly generated test case in the format specified by the problem statement.

### Sample Output:
Here’s what a sample output might look like:

```plaintext
4
abcd
efgh
ijkl
mnop
```

This output represents a 4x4 board with randomly chosen characters. Each execution of the script will produce a different board since it uses random choices.

Extract Arguments

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