**π‘ Problem Formulation:** We are faced with a scenario where we must determine the legality of a king’s move on a chessboard that has been altered by the presence of ‘n’ knights. With Python, we wish to assess if there exists at least one valid move for the king. An example input would be the coordinates of the king, the coordinates of each knight, and the size of the board. The desired output is a boolean value indicating the possibility of a valid move.

## Method 1: Brute Force Checking

This method involves checking all possible moves for the king and determining if any are valid given the positions of knights. We create a function that iterates through each potential move, checks for boundary conditions, and then checks for collisions with knights on the board. If no collisions are detected and the move is within bounds, it is valid.

Here’s an example:

def can_king_move(king_pos, knights, board_size): moves = [(0, 1), (1, 0), (0, -1), (-1, 0), (1, 1), (1, -1), (-1, 1), (-1, -1)] for move in moves: new_pos = (king_pos[0] + move[0], king_pos[1] + move[1]) if (0 <= new_pos[0] < board_size) and (0 <= new_pos[1] < board_size) and new_pos not in knights: return True return False print(can_king_move((4, 4), [(5, 5), (6, 4)], 8))

The output:

`True`

The above function defines the possible moves a king can make, iterates through them, checks if the resulting position would be within bounds of the board and not occupied by a knight. The function returns True upon finding a valid move, proving that the king can indeed move.

## Method 2: Using Set Operations

Set operations can streamline the process of checking for valid moves by treating the potential king moves and knight positions as sets. We calculate the set of all possible moves for the king and then subtract the set of positions occupied by the knights. Any remaining moves are valid if they also fall within the board’s boundaries.

Here’s an example:

def can_king_move(king_pos, knights, board_size): moves = {(king_pos[0] + dx, king_pos[1] + dy) for dx in [-1, 0, 1] for dy in [-1, 0, 1] if (dx, dy) != (0, 0)} knights_set = set(knights) valid_moves = moves - knights_set valid_moves = {pos for pos in valid_moves if 0 <= pos[0] < board_size and 0 <= pos[1] < board_size} return len(valid_moves) > 0 print(can_king_move((4, 4), [(5, 5), (6, 4)], 8))

The output:

`True`

In this snippet, we create sets for the king’s moves and the knights’ positions and perform a set difference operation to find unoccupied moves. Then, we filter out moves that step off the board, returning True if there are any moves left, thus being more efficient than individually checking each move.

## Method 3: Vectorized Move Checking

By using a more mathematical approach with vectors and matrices, we can visualize the chessboard as an array and the possible moves as vectors. This method is especially efficient if working with libraries like NumPy, which can handle such operations natively at high speed.

Here’s an example:

import numpy as np def can_king_move(king_pos, knights, board_size): board = np.zeros((board_size, board_size), dtype=bool) for knight in knights: board[knight] = True moves = [(x, y) for x in range(-1, 2) for y in range(-1, 2) if not (x == y == 0)] for move in moves: new_pos = np.add(king_pos, move) if all(0 <= np.array(new_pos) < board_size) and not board[tuple(new_pos)]: return True return False print(can_king_move((4, 4), [(5, 5), (6, 4)], 8))

The output:

`True`

Here, we generate a NumPy array to represent the board and mark the knights’ positions. We create a list of move vectors for the king and check if any vector, when added to the king’s position, results in a move inside the board and not to a square with a knight. If such a move exists, the function returns True.

## Method 4: Object-Oriented Approach

An object-oriented approach allows us to encapsulate the behavior of the chess pieces and the board into classes, resulting in code that is more modular and easier to manage, especially for more complex game logic.

Here’s an example:

class ChessBoard: def __init__(self, board_size): self.board_size = board_size self.knights = [] def add_knight(self, position): self.knights.append(position) def can_move(self, king_pos): moves = [(0, 1), (1, 0), (0, -1), (-1, 0), (1, 1), (1, -1), (-1, 1), (-1, -1)] for m in moves: new_pos = tuple(np.add(king_pos, m)) if all(0 <= np.array(new_pos) < self.board_size) and new_pos not in self.knights: return True return False board = ChessBoard(8) board.add_knight((5, 5)) board.add_knight((6, 4)) print(board.can_move((4, 4)))

The output:

`True`

This code defines a `ChessBoard` class to manage the board and the knights. The `can_move` method checks possible king moves in the same manner as previous methods but provides better abstraction. It iterates through the potential moves, adding them to the king’s position and validating the end positions.

## Bonus One-Liner Method 5: Functional Approach

Embracing Python’s functional programming features, we can construct an elegant one-liner to solve the problem using filter, map, and any functions combined with lambda expressions. This method provides a concise solution but may sacrifice some readability for brevity.

Here’s an example:

can_king_move = lambda kp, kn, bs: any( all(0 <= np.array((kp[0] + dx, kp[1] + dy)) < bs) and (kp[0]+dx, kp[1]+dy) not in kn for dx, dy in [(i, j) for i in [-1, 0, 1] for j in [-1, 0, 1] if i or j] ) print(can_king_move((4, 4), [(5, 5), (6, 4)], 8))

The output:

`True`

The lambda function `can_king_move` creates a generator of all possible king moves, filters out moves off the board or into knights, and checks if any valid moves exist using `any`. Despite its compactness, this approach maintains the same logical checks as the other methods but in one line.

## Summary/Discussion

**Method 1:**Brute Force Checking. This method is straightforward and easy to implement but may not be the most efficient for large boards or a high number of knights.**Method 2:**Using Set Operations. More efficient than brute force as it utilizes Python’s set theory capabilities to quickly find potential valid moves.**Method 3:**Vectorized Move Checking. Highly efficient with libraries like NumPy, making it suitable for computationally demanding scenarios or large-scale simulations.**Method 4:**Object-Oriented Approach. Offers a modular code structure that is easy to manage and extend, ideal for integrating into larger projects or games.**Method 5:**Functional Approach. Provides a condensed and elegant solution, but may come at the cost of readability, especially for those not familiar with functional programming.