**π‘ Problem Formulation:** We want to program a solution in Python to calculate the percentage of possible moves a knight can make on an 8×8 chessboard without leaving the board. Starting from any given position, the desired output is the percentage (0 to 100%) of valid moves out of the possible eight a knight can make in a single turn.

## Method 1: Brute Force Approach

This method involves checking all possible moves a knight can make from its current position and counting the ones that remain within the bounds of the board. By calculating the ratio of valid moves to possible moves, we obtain the required percentage. It’s straightforward and easy to implement.

Here’s an example:

def knight_moves_percentage(x, y): moves = [(2, 1), (2, -1), (-2, 1), (-2, -1), (1, 2), (1, -2), (-1, 2), (-1, -2)] valid_moves = 0 for move in moves: if 0 <= x + move[0] < 8 and 0 <= y + move[1] < 8: valid_moves += 1 return (valid_moves / 8) * 100 # Example usage: percentage = knight_moves_percentage(3, 3) print(f"Valid move percentage from position (3, 3): {percentage}%")

Output:

Valid move percentage from position (3, 3): 100.0%

In this code snippet, we define the function `knight_moves_percentage(x, y)`

that takes the knight’s position as parameters and returns the percentage of legal moves. It calculates this by iterating over each potential move, checking if the new position is within bounds, and counting the valid moves.

## Method 2: Using a Chessboard Matrix

This approach defines the chessboard as a matrix and assigns boolean values to each cell indicating whether the cell is within the board limits. This method provides a visual representation of the board and could be easily expanded for more complex board-related problems.

Here’s an example:

def knight_moves_percentage_matrix(x, y): board = [[True for _ in range(8)] for _ in range(8)] moves = [(2, 1), (2, -1), (-2, 1), (-2, -1), (1, 2), (1, -2), (-1, 2), (-1, -2)] valid_moves = sum(1 for move in moves if board[x + move[0]][y + move[1]] == True if 0 <= x + move[0] < 8 and 0 <= y + move[1] < 8) return (valid_moves / 8) * 100 # Example usage: percentage = knight_moves_percentage_matrix(3, 3) print(f"Valid move percentage from position (3, 3): {percentage}%")

Output:

Valid move percentage from position (3, 3): 100.0%

The `knight_moves_percentage_matrix(x, y)`

function also returns the valid move percentage like the brute force method but using a chessboard matrix to check for validity of moves. The matrix is not necessary for this problem but benefits visualization and adaptability in strategy.

## Method 3: Optimized Move Check

Instead of defining all moves, this method checks only the possible moves based on the knight’s position relative to the board’s boundary. It leads to slightly faster computations, especially useful for large-scale applications or simulations.

Here’s an example:

def knight_moves_percentage_optimized(x, y): valid_moves = 0 for dx in (-2, -1, 1, 2): for dy in (-2, -1, 1, 2): if abs(dx) != abs(dy) and (0 <= x + dx < 8) and (0 <= y + dy < 8): valid_moves += 1 return (valid_moves / 8) * 100 # Example usage: percentage = knight_moves_percentage_optimized(3, 3) print(f"Valid move percentage from position (3, 3): {percentage}%")

Output:

Valid move percentage from position (3, 3): 100.0%

The function `knight_moves_percentage_optimized(x, y)`

loops through the potential changes in the x and y coordinates, checks if the resulting position is within the board, and only considers the moves that result in a valid position, thus optimizing the process.

## Method 4: Functional Programming Approach

By using functional programming, this method simplifies code and makes it more concise by applying filters and maps to generate a list of the valid moves within the board boundaries, then computing the percentage based on the length of this list.

Here’s an example:

def knight_moves_percentage_functional(x, y): is_valid_move = lambda dx, dy: 0 <= x + dx < 8 and 0 <= y + dy < 8 moves = filter(lambda move: is_valid_move(*move), [(2, 1), (2, -1), (-2, 1), (-2, -1), (1, 2), (1, -2), (-1, 2), (-1, -2)]) valid_moves = len(list(moves)) return (valid_moves / 8) * 100 # Example usage: percentage = knight_moves_percentage_functional(3, 3) print(f"Valid move percentage from position (3, 3): {percentage}%")

Output:

Valid move percentage from position (3, 3): 100.0%

The `knight_moves_percentage_functional(x, y)`

employs a lambda function to encapsulate the validity logic and uses a filter to produce only valid moves. Then, we calculate the length of the resulting list of moves and derive the percentage.

## Bonus One-Liner Method 5: List Comprehension

A one-liner approach using list comprehension can dramatically reduce the amount of code needed while maintaining readability. This method combines the move generation and validation steps into a single, succinct expression.

Here’s an example:

def knight_moves_percentage_oneliner(x, y): return sum(1 for dx in (-2, -1, 1, 2) for dy in (-2, -1, 1, 2) if abs(dx) != abs(dy) and (0 <= x + dx < 8) and (0 <= y + dy < 8)) / 8 * 100 # Example usage: percentage = knight_moves_percentage_oneliner(3, 3) print(f"Valid move percentage from position (3, 3): {percentage}%")

Output:

Valid move percentage from position (3, 3): 100.0%

In the one-liner function `knight_moves_percentage_oneliner(x, y)`

, list comprehension with a condition generates the count of valid moves directly. Itβs the epitome of Python’s philosophy of simplicity and elegance in code writing.

## Summary/Discussion

**Method 1: Brute Force Approach.**Simple and straightforward. Suitable for small-scale problems. Can be slow for very large datasets.**Method 2: Chessboard Matrix.**Visual representation of the chessboard. Easy to understand. Not as efficient as other methods since the entire board is represented.**Method 3: Optimized Move Check.**Faster computations by reducing the number of checks. Less intuitive than the brute force method. Best used where performance is critical.**Method 4: Functional Programming.**Concise and readable. Leverages Python’s functional capabilities for elegant code. May be less familiar to those new to this programming paradigm.**Bonus Method 5: List Comprehension One-Liner.**Extremely concise and pythonic. Best for situations where code brevity is valued. Might be slightly harder to understand for beginners.