Check if Number is a Sum of Powers of Three

Check if Number is a Sum of Powers of Three

What is this problem about?

The "Check if Number is a Sum of Powers of Three" coding problem asks you to determine if a given integer $n$ can be represented as the sum of distinct powers of three. For example, if $n = 12$, you can write it as $3^1 + 3^2$, which are distinct powers. However, if $n = 21$, you might try $3^2 + 3^2 + 3^1$, but since the powers must be distinct (meaning each power can be used at most once), you must decide if such a combination exists.

Why is this asked in interviews?

Companies like Meta, Google, and Amazon ask this question to evaluate a candidate's grasp of number systems and mathematical intuition. It is essentially a test of whether you can think outside the base-10 decimal system. Understanding how numbers are represented in different bases (like binary, ternary, or hexadecimal) is a fundamental skill for low-level programming and algorithm optimization.

Algorithmic pattern used

This problem follows a Math and Number Base interview pattern. Specifically, it relates to the ternary (base-3) representation of a number. In base-3, any number is represented using the digits 0, 1, and 2. If a number can be formed by the sum of distinct powers of three, its ternary representation must only contain the digits 0 or 1. If a '2' appears in the base-3 representation, it means that a specific power of three would need to be used twice, which violates the "distinct" rule.

Example explanation

Suppose we want to check if $n = 91$ can be a sum of distinct powers of three.

• We repeatedly divide by 3 and check the remainder:
• $91 \div 3 = 30$, remainder 1. (OK)
• $30 \div 3 = 10$, remainder 0. (OK)
• $10 \div 3 = 3$, remainder 1. (OK)
• $3 \div 3 = 1$, remainder 0. (OK)
• $1 \div 3 = 0$, remainder 1. (OK) Since all remainders were either 0 or 1, the ternary representation is $10101_3$. This translates to $3^4 + 3^2 + 3^0 = 81 + 9 + 1 = 91$. Thus, the answer is true.

Common mistakes candidates make

A common error is overcomplicating the solution with backtracking or dynamic programming to find the exact powers. While those methods might work for small numbers, they are inefficient. Another mistake is failing to realize that "distinct" simply means we cannot have a '2' in base-3. Candidates also sometimes forget that $3^0 = 1$ is a valid power of three.

Interview preparation tip

When you see a problem involving "sums of distinct powers of X," immediately think about base-X representation. Practice converting numbers between bases manually, as it helps build the mental model needed to solve these types of mathematical logic puzzles quickly.