Ways to Express an Integer as Sum of Powers

Ways to Express an Integer as Sum of Powers

What is this problem about?

This problem asks for the number of ways a given integer n can be expressed as a sum of unique positive integers raised to the power of k. For example, if n=10 and k=2, we look for unique numbers whose squares sum up to 10 (e.g., 1^2 + 3^2). Since the order doesn't matter and we must use unique integers, this is a combinatorial problem that requires finding all possible subsets that satisfy the condition.

Why is this asked in interviews?

Big tech companies like Microsoft and Meta use this problem to evaluate a candidate's proficiency in Dynamic Programming (DP). It is a variation of the classic "Subset Sum" or "Partition Problem." It tests your ability to define subproblems, identify the base cases, and optimize the solution using memoization or a tabular approach to avoid redundant calculations. It also tests how you handle large outputs using modulo operations.

Algorithmic pattern used

The problem is solved using Dynamic Programming, specifically the 0/1 Knapsack variation. For each candidate integer i, you have two choices: either include i^k in the sum or exclude it. By iterating through all possible integers whose k-th power is less than or equal to n, you build up the total number of ways to reach the target value.

Example explanation

Target n = 5, Power k = 2. Candidate squares are 1^2 = 1 and 2^2 = 4 (3^2 = 9, which is too big).

• To make 5:
  • Can we make 5 using only 1? No.
  • Can we make 5 using 1 and 4? Yes (1+4=5).
• There is only 1 way. If n = 10 and k = 2:
• Candidates: 1, 4, 9.
• Ways: {1, 9} only. Total: 1 way.

Common mistakes candidates make

• Reusing integers: Using the same integer multiple times (e.g., 1^2 + 1^2 + ...) when the problem requires unique integers. This turns it into a "Coin Change" problem rather than a "Subset Sum" problem.
• Incorrect range: Not correctly identifying the upper bound for the base integers (it should stop when i^k > n).
• Missing Modulo: Forgetting to apply the modulo constraint frequently, leading to integer overflows in many programming languages.

Interview preparation tip

Practice the Knapsack pattern frequently. When you see a problem asking for "the number of ways" to reach a sum using a set of items, it's almost always a DP problem. Master the 1D-array optimization for Knapsack to save space and impress your interviewer.