To begin with, I worked out that as one of the numbers was 0, there had to be a negative number and then the same number but positive to make it. For example, I could've started off with -2 and 2 as that would make 0.
Then I realised that there could only be one negative number as two negative numbers added together makes a negative- there are no negatives in the 10 answers. I couldn't have had -2 and -6 as (-2)+(-6)=(-8). This is the same with all negatives added together.
I also worked out that the series of 5 numbers had to contain both odd an even numbers. To make 15 for example, you always have one odd and one even in an added pair- 3(odd)+12(even)=15
6(even)+9(odd)=15
Once I had established some certain properties that the five numbers would have, it was now simply time for trial and error.
WRONG BUT CLOSE:
-2, 2, 4, 6, 9
That made all the 10 numbers apart from one of the 4s and 9. I had a 10 and a 7.
WRONG BUT CLOSER:
-2, 2, 4, 5, 10
I tried changing the last two digits slightly to see if a difference was made. I then noticed that I seemed to make a few of the answers but I made lots that were one off which led me to the right answer.
CORRECT ANSWER:
-1, 1, 3, 5, 10
Following on from the last one, I tried to change the numbers a little to get 1 less so I changed the -2 & 2 to a -1 & 1 and I also changed the 4 to a 3. It worked and made all 10 answers.
(I attached the word version to. I didn't upload the three pages of trial and error I did or type it into my answer)