## Finding Five Numbers

1. 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?
2. 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:

1. Is it possible to find a set of 5 numbers in either case above which results in the sums 1-10?
2. 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.
Solutions were received from David Bachtel, Joseph DeVincentis, Dan Chirica, Denis Borris, Kirk Bresniker, Jeremy Galvagni, Claudio Baiocchi, Mark Moyer, Kitsuki Ikeda, Dan Dima, Philippe Fondanaiche, Jaume Simon Gispert, Graeme McRae, and Santo.
1. -3,3,4,5,7
2. -3, -1, 4, 7, 8
3. There is no way to make 10 in a row.  Joseph DeVincentis' summary:
The sum of 1-10 is 55, so (using pairs) the sum of the five numbers would have to be 13.75. The largest and smallest pairs add to 1 and 10 so the middle number is 2.75. Then from the sums 2 and 9 we know the second smallest and second largest are -0.75 and 6.25. This makes the smallest and largest numbers 1.75 and 3.75, a contradiction since our "second smallest" and "second largest" are more extreme. And likewise our sums do not add up.
Using triples, the sum of the five numbers is 55/6, so subtract each of the ten sums from this number to get -5/6, 1/6, 7/6, ... as the sums of the pairs. Now the same logic as above applies; shifting the set of sums by a fixed amount does not change the result; it only changes the numbers you calculate along the way to the contradiction by half that fixed amount.
4. For 9 in a row, there are many answers.  Jeremy Galvagni provided these.  The five numbers: {-1, 0, 1, 3, 6} can sum by twos to give -1, 0, 1, 2, 3, 4, 5, 6, 7, 9.  This is a sequence of 9 in a row (-1 through 7).  The same five sum by threes to 0, 2, 3, 4, 5, 6, 7, 8, 9, 10.  This is a sequence of 9 in a row (2 through 10).

Mail to Ken