Lego Blocks Hackerrank: A Comprehensive Guide to Solving the Brick Tiling Problem

The Lego Blocks hackerrank challenge is a popular problem-solving exercise that requires you to write a program to calculate the number of ways to tile a grid using L-shaped bricks. The problem involves several key concepts, including recursion, memoization, and algorithmic thinking.

Understanding the Problem

In this problem, you are given a grid with N rows and M columns, where each cell can be either blocked (represented by '#') or empty (represented by '.'). Your task is to find the number of ways to tile the grid using L-shaped bricks, which have one side of length three units and another side of length two units. The bricks can be used in any orientation (they can be rotated or flipped).

Brick Tiling Algorithm

To solve this problem, you need to write an algorithm that recursively calculates the number of ways to tile each sub-grid. The basic idea is to divide the grid into smaller sub-grids and then calculate the number of ways to tile each sub-grid.

Here's a high-level outline of the brick tiling algorithm:

1. Initialize a 2D array `dp` to store the number of ways to tile each sub-grid.

2. Set the base cases: if the sub-grid has only one row or column, there is only one way to tile it (i.e., the empty grid).

3. Recursively calculate the number of ways to tile each sub-grid by considering all possible orientations of the L-shaped bricks.

4. Use memoization to store the results of previously calculated sub-grids and avoid redundant calculations.

Pseudocode

Here's a pseudocode implementation of the brick tiling algorithm:

```python

def lego(n, m):

if n < 0 or m < 0:

return 0

elif n == 0 or m == 0:

return 1

else:

dp = ''0 for _ in range(m+1)' for _ in range(n+1)'

for i in range(1, n+1):

for j in range(1, m+1):

if grid'i-1''j-1' == '.':

dp'i''j' = dp'i-1''j-1' + dp'i-1''j' + dp'i''j-1'

else:

dp'i''j' = 0

return dp'n''m'

```

Implementation in Python

Here's a Python implementation of the brick tiling algorithm:

```python

def lego(n, m):

if n < 0 or m < 0:

return 0

elif n == 0 or m == 0:

return 1

else:

dp = ''0 for _ in range(m+1)' for _ in range(n+1)'

for i in range(1, n+1):

for j in range(1, m+1):

if grid'i-1''j-1' == '.':

dp'i''j' = dp'i-1''j-1' + dp'i-1''j' + dp'i''j-1'

else:

dp'i''j' = 0

return dp'n''m'

grid = '''.' for _ in range(80)' for _ in range(n)'

result = lego(n, m)

print(result)

```

The Lego Blocks hackerrank challenge is a challenging problem that requires you to write an algorithm to calculate the number of ways to tile a grid using L-shaped bricks. By understanding the problem and implementing the brick tiling algorithm, you can solve this problem and improve your programming skills.

Remember to always read the problem statement carefully and understand the requirements before attempting to solve it. Good luck!

Lego blocks hackerrank