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Finding Five Numbers

- When I sum five numbers in every possible pair combination, I get the
values: 0,1,2,4,7,8,9,10,11,12. What are the original 5 numbers?
- When I sum five numbers in every possible group of 3, I get the values:
0,3,4,8,9,10,11,12,14,19. What are the original 5 numbers?

Extensions:

- Is it possible to find a set of 5 numbers in either case above which
results in the sums 1-10?
- If the above problem is not possible, what is the longest series of
sequential sums you can find? For example, problems 1 and 2 have six
and five sequential sums, (7-12) and (8-12) respectively.

Source: Alan O'Donnell, though I've seen other sources.

Solution

Mail to Ken