💡 Problem Formulation: The task is to determine whether a given integer can be expressed as the sum of unique powers of three. For instance, inputting the number 91 should yield a positive result since 91 = 3^0 + 3^2 + 3^4.
Method 1: Iterative Approach
This method involves iterating over the powers of three and subtracting them from the number to see if it reaches zero. It uses a while loop to check the condition and subtracts the highest power of three possible at each iteration.
Here’s an example:
def is_sum_of_powers_of_three(num):
power = 1
while power 0:
power //= 3
num -= power if num >= power else 0
return num == 0
print(is_sum_of_powers_of_three(91))Output: True
This code checks if a number can be expressed as a sum of powers of three by working backwards from the largest power of three that is less than or equal to the number. By continuously subtracting the highest available power, we check if the number can be reduced to zero.
Method 2: Recursive Approach
The recursive method breaks the problem down into smaller subproblems, reducing the number by the highest power of three at each step and recursively calling the function on the new value.
Here’s an example:
def check_sum_of_powers_of_three(num):
if num < 1:
return num == 0
power = 1
while power * 3 <= num:
power *= 3
return check_sum_of_powers_of_three(num - power)
print(check_sum_of_powers_of_three(91))Output: True
This snippet uses a recursive function to continuously subtract the highest power of three possible from the number. It demonstrates the divide-and-conquer technique often used in recursive algorithms.
Method 3: Base 3 Conversion
This method leverages the base 3 representation of the number to determine if it can be expressed as a sum of powers of three. A valid number, in this case, should not have digits in base 3 other than 0 and 1.
Here’s an example:
def is_valid_base_three(num):
base3 = ''
while num > 0:
base3 = str(num % 3) + base3
num //= 3
return all(d in '01' for d in base3)
print(is_valid_base_three(91))Output: True
By converting the number into its base 3 representation and then checking that all digits are either 0 or 1, this code ensures the number is a sum of unique powers of three.
Method 4: Bitwise Operations
This method uses bitwise operations to test if the number can be a sum of powers of three. Similar to base conversions, if a number is a sum of powers of three, its binary (base 2) representation will not have consecutive ‘1’s.
Here’s an example:
def is_sum_of_powers_of_three_binary(num):
# Find next power of 3
next_power = 1
while next_power <= num:
if next_power & num:
num -= next_power
next_power *= 3
return num == 0
print(is_sum_of_powers_of_three_binary(91))Output: True
Through bitwise ‘&’ (and) operation, this function efficiently checks whether the bits corresponding to the powers of three can account for the number.
Bonus One-Liner Method 5: Set Representation
For a more concise solution, this one-liner uses the properties of sets to determine if the number is a sum of powers of three. It leverages the fact that sums of unique powers of three will not duplicate base 3 digits.
Here’s an example:
is_sum_of_powers = lambda num: len(set(int(digit) for digit in base3(num))) == 2 print(is_sum_of_powers(91))
Output: True
This snippet converts the number to base 3 and then creates a set from its digits. If the set’s length is 2 (including ‘1’ and ‘0’), the number is a sum of unique powers of three.
Summary/Discussion
- Method 1: Iterative Approach. Straightforward and easy to understand. However, iterative subtraction can be inefficient for large numbers.
- Method 2: Recursive Approach. Offers a clear logical progression. But, it may hit recursion limits for very large inputs and isn’t as efficient as non-recursive methods.
- Method 3: Base 3 Conversion. Base conversion is a clever way to solve the problem. It’s more robust but can be slower due to string operations involved.
- Method 4: Bitwise Operations. Very efficient for computation. It might be hard to understand without a clear grasp of bitwise operations and base conversion logic.
- Bonus Method 5: Set Representation. Extremely compact and elegant. However, it might be seen as ‘too clever’ and lack readability for those unfamiliar with Python’s functional programming capabilities.