Mel has four weights. He weighs them two at a time in all possible pairs and finds that his pairs of weights total 6, 8, 10, 12, 14, and 16 pounds. How much do they each weigh individually?

Note: There is not one unique answer to this problem, but there is a finite number of solutions.

### Solution

There are exactly two possible answers: Mel's weights can be 1, 5, 7, and 9 pounds, or they can be 2, 4, 6, and 10 pounds. No other combinations are possible.

### Explanation

Let the weights be a, b, c, and d, sorted such that a < b < c < d. We can chain inequalities to get a + b < a + c < a + d, b + c < b + d < c + d. Thus, a + b = 6, a + c = 8, b + d = 14, and c + d = 16. But we don't know if a + d = 10 and b + c = 12 or the other way around. This is how we get two solutions. If a + d = 10, we get 1, 5, 7, and 9; if b + c = 10, we get 2, 4, 6, and 10.

### More on the Problem

Where this problem really gets weird is that the number of solutions depends on the number of weights. For example, if Mel has three weights and knows the weight of all possible pairs, then there is only one possible solution for the individual weights. The same is true if he has five weights.

But now suppose that Mel has eight weights, and the sums of pairs are 8, 10, 12, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 36, 38, and 40. Now what are the individual weights?

This time, there are three solutions:

- 1, 7, 9, 11, 13, 15, 17, 23
- 2, 6, 8, 10, 14, 16, 18, 22
- 3, 5, 7, 11, 13, 17, 19, 21