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Problems > High School Math > 2008

Reconstruct the 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 a different set of 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?



3. Is it possible to find a set of 5 numbers in either case above which results in the sums 1-10? 
Find an example or prove it impossible.



4. 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: unknown


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