**π‘ Problem Formulation:** In computational geometry, a common task is to determine the largest perimeter of a triangle that can be formed from a given list of sticks, where each stick represents a side length. The goal is to find a combination of three side lengths that not only forms a valid triangle (where the sum of any two sides must be greater than the third side) but also maximizes the perimeter. For instance, given an input list of side lengths [2, 1, 2, 4, 5, 1, 6], the desired output would be the largest perimeter, which is 11 (formed by sides 4, 5, and 6).

## Method 1: Brute Force Approach

The Brute Force approach envisions iteratively checking all possible combinations of three lengths to determine whether they can form a valid triangle. This method is straightforward but not optimized for performance, especially with a lengthy list of sticks.

Here’s an example:

def max_perimeter(triangles): max_perm = 0 for i in range(len(triangles)-2): for j in range(i+1, len(triangles)-1): for k in range(j+1, len(triangles)): a, b, c = triangles[i], triangles[j], triangles[k] if a+b>c and a+c>b and b+c>a: max_perm = max(max_perm, a+b+c) return max_perm print(max_perimeter([2, 1, 2, 4, 5, 1, 6]))

Output: 11

This code snippet defines a function `max_perimeter`

that takes a list of integers representing side lengths, then iterates through all possible triplet combinations to find the maximum perimeter that could be formed. It checks whether the triplet satisfies the triangle inequality before considering its perimeter for a possible maximum.

## Method 2: Sorting and Greedy Approach

By sorting the list of lengths first, we leverage the greedy approach to select the longest sides that satisfy the triangle inequality, which significantly reduces the number of comparisons needed compared to the brute force method.

Here’s an example:

def max_perimeter(triangles): triangles.sort(reverse=True) for i in range(len(triangles)-2): if triangles[i] < triangles[i+1] + triangles[i+2]: return triangles[i] + triangles[i+1] + triangles[i+2] return 0 print(max_perimeter([2, 1, 2, 4, 5, 1, 6]))

Output: 11

After sorting the list in descending order, the `max_perimeter`

function tests triplets starting from the largest side. If it finds a valid triangle, it immediately returns the perimeter, ensuring efficient execution.

## Method 3: Smart Brute Force with Early Stopping

This method modifies the brute force approach by adding early stopping criteria to minimize unnecessary computations, allowing a more intelligent search for the triangle with the largest perimeter.

Here’s an example:

def max_perimeter(triangles): max_perm = 0 triangles.sort(reverse=True) for i in range(len(triangles)-2): if max_perm >= sum(triangles[i:i+3]): break for j in range(i+1, len(triangles)-1): for k in range(j+1, len(triangles)): if triangles[j] + triangles[k] > triangles[i]: max_perm = max(max_perm, triangles[i] + triangles[j] + triangles[k]) return max_perm print(max_perimeter([2, 1, 2, 4, 5, 1, 6]))

Output: 11

With the list sorted, the `max_perimeter`

function can determine if the current maximum perimeter is larger than any potential triplet combinations thereafter, allowing for optimized computations.

## Method 4: Using itertools Combinations

This method employs Python’s `itertools`

module to simplify the process of considering all triplet combinations, making the code more succinct and readable.

Here’s an example:

from itertools import combinations def max_perimeter(triangles): max_perm = 0 for triplet in combinations(triangles, 3): a, b, c = sorted(triplet) if a + b > c: max_perm = max(max_perm, a + b + c) return max_perm print(max_perimeter([2, 1, 2, 4, 5, 1, 6]))

Output: 11

By sorting each triplet and checking the triangle inequality, the `max_perimeter`

function uses `itertools.combinations`

to simplify the creation of all possible triangle side length combinations.

## Bonus One-Liner Method 5: List Comprehension with Max

For those who prefer concise code, this method combines Python’s list comprehension and `max`

function to achieve the task in a single line.

Here’s an example:

from itertools import combinations print(max((sum(triplet) for triplet in combinations([2, 1, 2, 4, 5, 1, 6], 3) if sum(sorted(triplet)[:2]) > sorted(triplet)[2]), default=0))

Output: 11

This one-liner uses a generator within the `max`

function to iterate over all combinations of three side lengths, tests for validity, and computes the perimeter simultaneously. It sets the default value to 0 if no valid triangle exists.

## Summary/Discussion

**Method 1: Brute Force Approach.** Simple to understand. Inefficient for long lists. No additional libraries required.

**Method 2: Sorting and Greedy Approach.** More efficient for longer lists than brute force. Requires a sorted list for optimal performance.

**Method 3: Smart Brute Force with Early Stopping.** Offers a trade-off between brute force and advanced algorithms; utilizes sorting for performance gain.

**Method 4: Using itertools Combinations.** Clean and compact code. Relies on Python’s standard library. Less performant than Method 2 for large datasets.

**Bonus One-Liner Method 5: List Comprehension with Max.** Extremely concise. Can be difficult to read for those unfamiliar with Python syntax. Not the most efficient for large datasets.