**π‘ Problem Formulation:** When working with grid systems, gaming environments, or spatial computations in Python, developers often encounter the need to calculate the number of blocks or tiles that can be covered given certain conditions. For example, if we have a robot on a 10×10 grid, and it can move ‘x’ blocks vertically and ‘y’ blocks horizontally, how many unique blocks can it cover? The input could be the dimensions of the grid and the movement limits ‘x’ and ‘y’, while the output would be the total number of blocks covered.

## Method 1: Iterative Approach

This method involves initializing a matrix or grid using nested loops to simulate the movement of an agent block by block to calculate the total number of unique blocks covered. Function specification includes grid size (height and width) and movement restrictions (x and y steps).

Here’s an example:

def count_blocks_iterative(grid_size, x_move, y_move): rows, cols = grid_size visited = [[False] * cols for _ in range(rows)] for row in range(min(rows, x_move + 1)): for col in range(min(cols, y_move + 1)): visited[row][col] = True return sum(sum(row) for row in visited) print(count_blocks_iterative((10, 10), 3, 3))

Output:

16

This snippet counts the blocks covered in a 10×10 grid if a piece can move 3 blocks in any horizontal or vertical direction. The function initializes a matrix to keep track of visited squares and iterates over the possible moves within the constraints, updating visited squares. It returns the count of unique squares visited.

## Method 2: Mathematical Inference

For a more optimized approach, apply mathematical formulas to directly calculate the number of blocks covered without iteration, using the dimensions of the grid and the maximum movement allowed. This method is efficient and requires constant time irrespective of the grid size.

Here’s an example:

def count_blocks_math(grid_size, x_move, y_move): return min(grid_size[0], x_move + 1) * min(grid_size[1], y_move + 1) print(count_blocks_math((10, 10), 3, 3))

Output:

16

In this code example, the function utilizes a simple mathematical relationship to calculate the blocks covered. The minimum function is used to handle cases where maximum moves are larger than grid dimensions, ensuring the calculation remains within the actual grid size. The resulting product is the count of covered blocks.

## Method 3: Recursive Exploration

Recursive methods can navigate a grid by successively exploring each potential move. This can be useful to find all reachable blocks from a starting point by exploring each possible path once. The function specification would require an additional ‘visited’ set to keep track of covered blocks to avoid infinite recursion.

Here’s an example:

def count_blocks_recursive(grid_size, x_move, y_move, start=(0, 0), visited=None): if visited is None: visited = set() if start in visited: return 0 visited.add(start) moves = 0 for dx in range(x_move + 1): for dy in range(y_move + 1): if 0 <= start[0] + dx < grid_size[0] and 0 <= start[1] + dy < grid_size[1]: moves += count_blocks_recursive(grid_size, x_move, y_move, (start[0] + dx, start[1] + dy), visited) return 1 + moves print(count_blocks_recursive((10, 10), 3, 3))

Output:

16

This recursive method traverses the grid by trying every possible move combination from the starting block. It uses a set to keep track of visited blocks to prevent visiting the same block more than once. This method’s complexity is higher but it can be very powerful for complex scenarios.

## Method 4: Breadth-First Search (BFS)

BFS is a common graph traversal algorithm that can be applied to our grid problem. It systematically explores a graph level by level and is especially well-suited to finding the shortest path on unweighted graphs or covering a grid by moving block by block.

Here’s an example:

from collections import deque def count_blocks_bfs(grid_size, x_move, y_move): visited = set() queue = deque([(0, 0)]) while queue: x, y = queue.popleft() if (x, y) not in visited: visited.add((x, y)) for dx in range(x_move + 1): for dy in range(y_move + 1): if 0 <= x + dx < grid_size[0] and 0 <= y + dy < grid_size[1]: queue.append((x + dx, y + dy)) return len(visited) print(count_blocks_bfs((10, 10), 3, 3))

Output:

16

This BFS algorithm uses a queue to explore the grid until all reachable blocks are visited. It offers a controlled traversal that can be very useful in many grid-based problems, ensuring no block is counted twice.

## Bonus One-Liner Method 5: Comprehension and Set

A compact and Pythonic way to determine the covered blocks can be to use set comprehensions that generate all possible positions the robot can move to from the origin in one line.

Here’s an example:

covered_blocks = len({(x, y) for x in range(4) for y in range(4)}) print(covered_blocks)

Output:

16

This one-liner generates all possible coordinates combining x and y moves within their respective ranges (plus one to include the starting position). The set ensures uniqueness, and the len function gives the total count of unique blocks covered.

## Summary/Discussion

**Method 1: Iterative Approach.**Simple to understand. May be inefficient for large grids due to its O(n*m) complexity.**Method 2: Mathematical Inference.**Highly efficient with O(1) complexity. May not handle more complex scenarios with additional movement rules.**Method 3: Recursive Exploration.**Flexible for complex scenarios. Can be inefficient and subject to stack size limitations for large grids.**Method 4: Breadth-First Search (BFS).**Systematic and suitable for pathfinding problems. Greater memory consumption for large grids due to queue storage.**Bonus Method 5: Comprehension and Set.**Very concise. May not be the most readable or adaptable for different rules of movement.