You have an N by N board. Write a function that, given N, returns the number of possible arrangements of the board where N queens can be placed on the board without threatening each other, i.e. no two queens share the same row, column, or diagonal.

This is a popular question that is solved with backtracking. The following is a solution done in Python

def solve_n_queens(n):
    def is_safe(board, row, col):
        # Check if a queen can be placed at the given position without conflicting
        # with any other queens on the board
        for i in range(row):
            if board[i] == col or \
               board[i] - i == col - row or \
               board[i] + i == col + row:
                return False
        return True

    def backtrack(board, row):
        # Base case: all queens have been placed successfully
        if row == n:
            return 1

        count = 0
        for col in range(n):
            if is_safe(board, row, col):
                board[row] = col
                count += backtrack(board, row + 1)
                board[row] = -1

        return count

    # Initialize the board with empty cells
    board = [-1] * n
    return backtrack(board, 0)


n = 4
count = solve_n_queens(n)
print(f"Number of possible arrangements for {n}-queens: {count}")