#1780

Medium

auto_awesomeMath-driven solution strategy

LeetCode Problem Workspace

Check if Number is a Sum of Powers of Three

Determine if a number can be expressed as the sum of distinct powers of three using a math-driven strategy.

Problem Statement

Given an integer n, return true if it is possible to represent n as the sum of distinct powers of three. Otherwise, return false.

An integer y is considered a power of three if there exists an integer x such that y equals 3 raised to the power of x. For example, 1, 3, 9, and 27 are all powers of three. The problem asks you to find out if a number can be written as a sum of such distinct powers.

Examples

Example 1

Input: n = 12

Output: true

12 = 31 + 32

Example 2

Input: n = 91

Output: true

91 = 30 + 32 + 34

Example 3

Input: n = 21

Output: false

Example details omitted.

Constraints

1 <= n <= 107

Solution Approach

Base 3 Representation

The most efficient way to solve this problem is by converting the number n into its base-3 representation. If every digit in this representation is either 0 or 1, then n can be expressed as the sum of distinct powers of three. If there’s a digit of 2, the answer is false.

Mathematical Insight

The sum of distinct powers of three can be verified by considering the powers of three that sum to the number. By examining n modulo 3 repeatedly, we can check each power of three and confirm whether it contributes to the sum or not.

Efficient Solution with Time Complexity O(log_3 n)

The algorithm works by continuously dividing the number by 3 and checking the remainder. This operation will run in logarithmic time relative to n, specifically O(log_3 n), making it very efficient even for large inputs.

Complexity Analysis

Metric	Value
Time	O(\log_3{n})
Space	O(1)

The time complexity is O(log_3 n) because we reduce n by dividing by 3 at each step. The space complexity is O(1) as we are only using a few variables for calculations and not storing any additional data.

What Interviewers Usually Probe

A candidate demonstrates proficiency in using base conversions to solve a mathematical problem.
The ability to recognize properties of powers of three is evident.
A clean solution without extraneous space or complex data structures is provided.

Common Pitfalls or Variants

Common pitfalls

Using an inefficient approach by brute-forcing through combinations of powers of three.
Misunderstanding the base conversion process, especially interpreting digits other than 0 or 1.
Not realizing that only powers of three, not arbitrary numbers, contribute to the sum.

Follow-up variants

Instead of checking for distinct sums, you could modify the problem to check for sums with repeated powers of three.
This problem can also be extended to powers of other numbers, not just three.
You could explore a recursive solution that constructs sums from powers of three rather than using the base conversion approach.

FAQ

How can I solve the "Check if Number is a Sum of Powers of Three" problem?

You can solve it by converting the number to base 3 and ensuring that each digit is either 0 or 1. If any digit is 2, the answer is false.

What is the time complexity of the solution?

The time complexity is O(log_3 n) because the number is divided by 3 repeatedly, reducing the problem size exponentially.

What is a power of three?

A power of three is any integer that can be expressed as 3 raised to some integer exponent. Examples include 1, 3, 9, and 27.

Can I solve this problem without using base 3 conversion?

While base-3 conversion is the most efficient approach, you could theoretically explore other methods like recursive summation, but they would be less optimal.

How does GhostInterview help with this problem?

GhostInterview assists by offering hints about base conversion, time complexity explanations, and guidance on avoiding common mistakes in the problem-solving process.

terminal

Solution

Solution 1: Mathematical Analysis

We find that if a number $n$ can be expressed as the sum of several "different" powers of three, then in the ternary representation of $n$, each digit can only be $0$ or $1$.

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class Solution:
    def checkPowersOfThree(self, n: int) -> bool:
        while n:
            if n % 3 > 1:
                return False
            n //= 3
        return True

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