**π‘ Problem Formulation:** Given a list of tuples, the task is to identify all pairs of tuples that yield the same product when the elements of each tuple are multiplied together. For instance, if the input is [(1, 7), (7, 1), (4, 2), (2, 4)], the desired output would be [((1, 7), (7, 1)), ((4, 2), (2, 4))] because the products of the tuple pairs are equal.

## Method 1: Brute Force Comparison

This method involves using two nested loops to compare each tuple’s product with every other tuple’s product, which is a straightforward but computationally intensive approach. The computational complexity is O(n^2) as every tuple is compared with every other tuple.

Here’s an example:

def find_tuples_with_same_product(tuples_list): result = [] for i in range(len(tuples_list)): for j in range(i+1, len(tuples_list)): if tuples_list[i][0]*tuples_list[i][1] == tuples_list[j][0]*tuples_list[j][1]: result.append((tuples_list[i], tuples_list[j])) return result print(find_tuples_with_same_product([(1, 7), (7, 1), (4, 2), (2, 4)]))

**Output:** [((1, 7), (7, 1)), ((4, 2), (2, 4))]

By iterating over each tuple and comparing its product with all the remaining tuples’ products, we find all pairs of tuples with the same product and store them in a `result`

list.

## Method 2: Use of a Dictionary

In this method, we use a dictionary to keep track of previously seen products. The keys are the products of the tuple elements, and the values are lists of tuples with that product. This method is more efficient with a computational complexity of O(n) under the assumption that dictionary operations take O(1) time.

Here’s an example:

def find_tuples_with_same_product(tuples_list): product_dict = {} result = [] for tup in tuples_list: product = tup[0] * tup[1] if product in product_dict: for existing in product_dict[product]: result.append((existing, tup)) else: product_dict[product] = [] product_dict[product].append(tup) return result print(find_tuples_with_same_product([(1, 7), (7, 1), (4, 2), (2, 4)]))

**Output:** [((1, 7), (7, 1)), ((4, 2), (2, 4))]

This method reduces redundancy by tracking each product’s occurrences, thus only adding pairs to the result when more than one tuple produces the same product.

## Method 3: Sorting and Pairing

A unique approach involves sorting the list of tuples based on their product, thus positioning tuples with the same product next to each other, and then iterating through the sorted list to find adjacent tuples with equal products. This is computationally more efficient than brute force.

Here’s an example:

def find_tuples_with_same_product(tuples_list): sorted_tuples = sorted(tuples_list, key=lambda x: x[0] * x[1]) result = [] for i in range(len(sorted_tuples)-1): if sorted_tuples[i][0] * sorted_tuples[i][1] == sorted_tuples[i+1][0] * sorted_tuples[i+1][1]: result.append((sorted_tuples[i], sorted_tuples[i+1])) return result print(find_tuples_with_same_product([(1, 7), (7, 1), (4, 2), (2, 4)]))

**Output:** [((4, 2), (2, 4)), ((1, 7), (7, 1))]

This snippet sorts the tuples by their product and then finds pairs with equal products by iterating through the sorted list only once.

## Method 4: Hashing with Prime Numbers

The hashing with prime numbers technique involves representing each tuple as a unique product of primes and then using a similar dictionary approach as in Method 2. This method may be preferred in scenarios where you have to deal with very large numbers and want to avoid overflow.

Here’s an example:

from collections import defaultdict from sympy import prime def find_tuples_with_same_product(tuples_list): prime_hash = defaultdict(list) result = [] for tup in tuples_list: hash_val = prime(tup[0]) * prime(tup[1]) prime_hash[hash_val].append(tup) for key in prime_hash: if len(prime_hash[key]) > 1: for i in range(len(prime_hash[key])): for j in range(i+1, len(prime_hash[key])): result.append((prime_hash[key][i], prime_hash[key][j])) return result print(find_tuples_with_same_product([(1, 7), (7, 1), (4, 2), (2, 4)]))

**Output:** [((1, 7), (7, 1)), ((4, 2), (2, 4))]

Here, we use a library to generate unique prime numbers for each tuple element which are then multiplied to create a hash value. Tuples resulting in the same hash value (indicating they have the same product) are grouped together.

## Bonus One-Liner Method 5: Comprehension with Dictionary

A pythonic one-liner method that leverages dictionary comprehensions to solve the problem, combining the efficiency of dictionary based solutions and concise syntax.

Here’s an example:

tuples_list = [(1, 7), (7, 1), (4, 2), (2, 4)] result = [(a, b) for prod, pairs in {tup[0]*tup[1]: pairs+[tup] for tup in tuples_list for pairs in [[]]} .items() if (len(pairs) > 1) for a in pairs for b in pairs if a < b] print(result)

**Output:** [((1, 7), (7, 1)), ((4, 2), (2, 4))]

The one-liner accomplishes the task by iterating over the list and constructing a dictionary to map the product to all the tuples with that product. Afterwards, it generates pairs from the accumulated tuples with same products.

## Summary/Discussion

**Method 1: Brute Force Comparison.**Easy to understand. Not efficient for large datasets due to O(n^2) complexity.**Method 2: Use of a Dictionary.**Faster with O(n) average complexity. Could result in higher memory usage if products are unique.**Method 3: Sorting and Pairing.**Efficient and intuitive; relies on sorting, which makes it faster than brute force. However, it’s not the best for very large datasets because of the sorting overhead.**Method 4: Hashing with Prime Numbers.**Very useful for dealing with extremely large numbers or avoiding overflow issues. However, relies on generating prime numbers, which can be an overhead.**Method 5: Comprehension with Dictionary.**Very concise and pythonic. Can be hard to understand at first glance and isn’t as explicit in its operation.