三个数字的组合总计为1000

Question

I need every combination of three positive integers with the sum of 1000.

This was my attempt but I'm unsure if this is correct since I have no way to validate it.

def getSum():
    l = []
    for x in range(1, 999):
        total = 1000-x
        for y in range(1, 999):
            total = total-y
            if total>0:
                l.append([x, y, total])
    return l

print len(getSum())

I get 28776 different combinations. Is that correct?

Answer 1

Since 1+998+1 and 1+1+998 are not the same thing, there are some incredible amount of combinations:

This line can generate them all:

[(i, 1000-i-k, k) for i in range(1,999) for k in range(1,1000-i)]

Results:

[...
(1, 4, 995),
(1, 3, 996),
(1, 2, 997),
(1, 1, 998),
(2, 997, 1),
(2, 996, 2),
...]

The length of this list is:

498501

Answer 2

No, that number is not correct. The problem with your code is this line:

        total = total-y

Here, you decrease total further and further with each value of y that you try, never resetting it to the value after just subtracting x . To fix it, create a new variable, eg total2 , and use that in the inner loop.

        total2 = total-y

This way, you get 498501 combinations. Also, you can break from the inner loop as soon as total2 < 0 .

---

If you need just the number of combinations: Note that there are N-1 combinations to sum two numbers to N , eg for N==4 : 1+3 , 2+2 , 3+1 (assuming you consider 1+3 and 3+1 different). You can extend this to the case of three numbers as partitioning the number in two parts two times. This way, you only need a single loop. And this can be simplified further to an O(1) formula.

Example, with naive approach using product as reference:

>>> N = 100  # to make reference faster
>>> sum(1 for t in product(range(1, N+1), repeat=3) if sum(t)==N)
4851
>>> sum(N-1-i for i in range(1, N-1))
4851
>>> ((N-2)*(N-1))//2
4851

Of course, also works for N = 1000 (or much, much larger):

>>> N = 1000
>>> sum(N-1-i for i in range(1, N-1))
498501
>>> ((N-2)*(N-1))//2
498501

Answer 3

If you treated [1,1,998] and [1,998,1] the same (no unique integers):

def getSum():
    l = []
    for x in range(1, 999):
        total = 1000-x
        for y in range(1, 999):
            total = total-y
            if total>0:
                z = [x, y, total]
                z.sort()
                if z not in l:
                    l.append(z)
return l

a = getSum()
print(len(a))

If you want 3 unique integers:

def getSum():
    l = []
    for x in range(1, 999):
        total = 1000-x
        for y in range(1, 999):
            total = total-y
            if total>0:
                z = [x, y, total]
                z.sort()
                if (z not in l) and (not((len(set(z)) < len(z)))):
                    l.append(z)
    return l

a = getSum()
print(len(a))

Otherwise your code is (in my sense) ok. I haven't check your answer yet...

EDIT : I have checked it using brutal force. The correct answer is actually 498501 if you treated (1,1,998) and (998,1,1) differently. Currently I don't know why...

Answer 4

Try this:

def getSum():
    l = []
    for x in range(1, 6):
        for y in range(1, 6):
            total = 6-(y+x)
            if total>0:
                s = set([x, y, total])
                if s not in l:
                    l.append(s)
                    print(x, y, total)
    return l

print (len(getSum()))

This is my algorithm, Although there is better ways. In this case I wrote code for number 6 and printed all combinations to show how it works. You can set 1000 or any number instead of 6 in this code(in 3 position) and ignore print() line.