Short Problem Definition:

Find the number of ways that a given integer, X , can be expressed as the sum of the Nth powers of unique, natural numbers.

For example, if X = 13 and N = 2, we have to find all combinations of unique squares adding up to 13. The only solution is 2^2 + 3^2.

Link

The Power Sum

Complexity:

time complexity is O(N!)

space complexity is O(1)

Execution:

This solution does not use DP or memoisation, but it does not time out. We know that it needs to explore all P! possible combinations where P is floor(sqrt(X, N)). Since maximum X is 1000, this should still compute in real time.

Keep in mind that all integers in the solution have to be unique. So for X = 3, the solution 1^2 + 1^2 + 1^2 is not acceptable.

Did I mention that I dislike both NP-Hard problems and recursion? No? So, now you know.

Solution:

def powerSum(X, N, current = 1):
    pw = pow(current, N)
    if pw > X:
        return 0
    elif pw == X:
        return 1
    else:
        return powerSum(X, N, current+1) + powerSum(X-pw, N, current+1)