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Ten to five (Posted on 2018-07-30) Difficulty: 3 of 5
You are given a set of 10 numbers that represent all possible pairwise sums of certain five numbers.

Devise a strategy to find the original five numbers.

Provide a numerical example.

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

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Solution solution (spoiler) | Comment 1 of 7
Start with an example:

 60 83 85 87 89 100 102 112 125 129
 
These add up to 972. As each number is counted four times, the total of the five numbers is 972/4 = 243.

The largest of the ten is 129 and the smallest is 60. These are the sums of the largest two and the smallest two of the five, respectively, so subtracting the total of these two, representing four of the five, from the total of the five, 243, will give the value of the middle of the five: 243 - (129+60) = 54.

The third largest of the ten will be the middle plus the largest (of the five). So in this case 112 = 54 + 58, making 58 the second largest number.  Then 129 - 58 gives you the largest number, 71.

Symmetrically with the preceding paragraph, the third smallest of the ten will be the middle plus the smallest of the five. In this case 85 = 54 + 31, making 31 the second smallest of the five. Then 60 - 31 gives you 29, the smallest of the five.

In this case we have figured out
 
 29 31 54 58 71

  Posted by Charlie on 2018-07-30 11:03:24
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