Question

I’m trying to come up with an algorithm that will print out all possible ways to sum N integers so that they total a given value.

Example. Print all ways to sum 4 integers so that they sum up to be 5.

Result should be something like:

5 0 0 0
4 1 0 0
3 2 0 0
3 1 1 0
2 3 0 0
2 2 1 0
2 1 2 0
2 1 1 1
1 4 0 0
1 3 1 0 
1 2 2 0
1 2 1 1
1 1 3 0
1 1 2 1
1 1 1 2

Answer 1

This is based off Alinium’s code.
I modified it so it prints out all the possible combinations, since his already does all the permutations.
Also, I don’t think you need the for loop when n=1, because in that case, only one number should cause the sum to equal value.
Various other modifications to get boundary cases to work.

def sum(n, value):
    arr = [0]*n  # create an array of size n, filled with zeroes
    sumRecursive(n, value, 0, n, arr);

def sumRecursive(n, value, sumSoFar, topLevel, arr):
    if n == 1:
        if sumSoFar <= value:
            #Make sure it's in ascending order (or only level)
            if topLevel == 1 or (value - sumSoFar >= arr[-2]):
                arr[(-1)] = value - sumSoFar #put it in the n_th last index of arr
                print arr
    elif n > 0:
        #Make sure it's in ascending order
        start = 0
        if (n != topLevel):
            start = arr[(-1*n)-1]   #the value before this element

        for i in range(start, value+1): # i = start...value
            arr[(-1*n)] = i  # put i in the n_th last index of arr
            sumRecursive(n-1, value, sumSoFar + i, topLevel, arr)

Runing sums(4, 5) returns:
[0, 0, 0, 5]
[0, 0, 1, 4]
[0, 0, 2, 3]
[0, 1, 1, 3]
[1, 1, 1, 2]