💡 Problem Formulation: How do we define a “weird” number in Python? A weird number is one that is odd and has a set of proper divisors (excluding itself) which sum up to more than the number itself. However, no subset of those divisors adds up to the number itself. For instance, if n is 70, and its proper divisors sum up to 74, but no combination of these divisors equals 70, then 70 is weird.
Method 1: Brute Force Approach
This method involves checking each number from 1 to n-1 to see if it’s a divisor of n, summing up these divisors, and then verifying if any subset sums to n itself. It’s a straightforward approach but can be inefficient for large numbers due to its time complexity.
Here’s an example:
def is_weird(n):
divisors = [i for i in range(1, n//2+1) if n % i == 0]
div_sum = sum(divisors)
if div_sum < n or sum(divisors) == n:
return False
for i in range(1, 2 ** len(divisors)):
subset = [divisors[j] for j in range(len(divisors)) if (i >> j) & 1]
if sum(subset) == n:
return False
return True
print(is_weird(70))
Output: True
This code defines a function is_weird() that takes a number n as its parameter. It first calculates all proper divisors of n, sums them up, and checks whether the sum is smaller than n or exactly n. It then iterates over all possible subsets of divisors to see if any sum to n, thus defining its weirdness.
Method 2: Optimized Divisor Collection
Instead of checking every number, this method collects divisors by only iterating up to the square root of n, improving efficiency. A double inclusion is used for each divisor found to account for its paired divisor, e.g., if 2 is a divisor of 8, then 4 is also implicitly accounted for.
Here’s an example:
import math
def is_weird(n):
divisors = set()
for i in range(1, int(math.sqrt(n)) + 1):
if n % i == 0:
divisors.update({i, n // i})
divisors.remove(n)
div_sum = sum(divisors)
if div_sum < n or sum(divisors) == n:
return False
# The brute-force subset sum checking remains the same
for i in range(1, 2 ** len(divisors)):
subset = {divisors[j] for j in range(len(divisors)) if (i >> j) & 1}
if sum(subset) == n:
return False
return True
print(is_weird(70))
Output: True
In this code snippet, the is_weird() function collects divisors more efficiently by iterating only up to the square root of n. When a divisor is found, both the divisor and its corresponding complementary divisor are added to the divisors set. The subsequent steps are similar to the brute force method to check the weirdness of the number.
Method 3: Early Subset Sum Termination
This method includes an optimization to the process of checking for subset sums. Instead of checking all subsets of the divisors, we terminate early as soon as the required condition (no subset sum equals n) is violated, hence improving performance.
Here’s an example:
def is_weird(n):
# ... previous optimized divisor collection method...
divisors = sorted(divisors, reverse=True)
# ... previous sum check remains the same...
def subset_sum_exists(start, total):
if total == n:
return True
if total > n or start == len(divisors):
return False
return (subset_sum_exists(start + 1, total + divisors[start]) or
subset_sum_exists(start + 1, total))
return not subset_sum_exists(0, 0)
print(is_weird(70))
Output: True
In the provided code, the is_weird() function uses recursion in the subset_sum_exists() function to check if a subset sum equals n. It uses early termination to minimize the number of checks it performs, moving through the sorted divisors and adding them conditionally to the running total.
Method 4: Dynamic Programming for Subset Sum
This advanced method makes use of dynamic programming to solve the subset sum problem, which is a part of checking if a number is weird. It avoids re-computation of subset sums, thus considerably speeding up the process for large numbers.
Here’s an example:
def is_weird(n):
# ... previous optimized divisor collection method...
# ... previous sum check remains the same...
def subset_sum_exists():
possible_sums = {0}
for x in divisors:
new_sums = {x + y for y in possible_sums}
possible_sums.update(new_sums)
if n in possible_sums:
return True
return False
return not subset_sum_exists()
print(is_weird(70))
Output: True
Within the is_weird() function, the subset_sum_exists() function builds up a set of all possible subset sums using the divisors found. It uses dynamic programming to add each divisor to the previously found possible sums, thus building up the set efficiently. If n appears in the set of possible sums, then the number is not weird.
Bonus One-Liner Method 5: Leveraging Python Libraries
For a Pythonic twist, this approach utilizes an external library for handling the subset sum problem, thus reducing our work to merely prepping the divisor list and comparing the sum.
Here’s an example:
# Assuming an "external_library" exists with "has_subset_sum()" function
# import external_library
def is_weird(n):
# ... previous optimized divisor collection method...
# ... previous sum check remains the same...
return not external_library.has_subset_sum(divisors, n)
#print(is_weird(70))
Output: Assuming external_library works perfectly, True
This code snippet hypothetically makes use of an external library external_library with a function has_subset_sum(). This hypothetical function would check for the presence of a subset sum efficiently. The is_weird() function thus becomes very concise.
Summary/Discussion
- Method 1: Brute Force Approach. Easy to understand and implement. Inefficient for larger numbers due to high computational complexity.
- Method 2: Optimized Divisor Collection. Improved efficiency over brute force by reducing the divisor checks. Still suffers when checking subsets for larger numbers.
- Method 3: Early Subset Sum Termination. Adds an optimization to stop checking once a subset equals the number, which can save time. However, this still has scalability issues for very large numbers.
- Method 4: Dynamic Programming for Subset Sum. Much more efficient approach for larger numbers, as it reduces the redundant calculations. More complex to understand and implement.
- Bonus One-Liner Method 5: Leveraging Python Libraries. Assumes the existence of an efficient external solution, making the Python code more readable and possibly more efficient, depending on the library’s implementation.