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 dimension.
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.
```python
def f(N: int, S: List[str]):
'''
N: an integer
S: a list of strings
'''
```
Test Case Generator
Sure! Here's a Python function that generates test cases for the given problem statement:
```python
import random
import string
def generate_board(N):
return [''.join(random.choice(string.ascii_lowercase) for _ in range(N)) for _ in range(N)]
def tcgen():
N = random.randint(1, 300)
board = generate_board(N)
return N, board
# Example of generating a test case
N, board = tcgen()
print(N)
for row in board:
print(row)
```
This function generates a random test case for the problem. The `generate_board` function creates an N x N board filled with random lowercase English letters. The `tcgen` function generates a random value for N within the given constraints (1 ≤ N ≤ 300) and a corresponding board.
The example at the bottom demonstrates how to use the `tcgen` function to generate and print a test case.