Prove that there are two lists of 16 integers that produce the same list of pairwise sums Just a little mathematical curiosity of mine. 
Given a list of n integers (numbers in the list can be negative or positive, and do not have to be distinct), we calculate the list of pairwise sums by computing all 1/2(n)(n-1) sums of two pairs of numbers from the list and arrange them in order from smallest to largest
For example: given the list (1, 2, 5, 9) the list of pairwise sums would be (1+2, 1+5, 1+9, 2+5, 2+9, 5+9) = (3, 6, 10, 7, 11, 14) and organizing by order we get (3, 6, 7, 10, 11, 14). So, the list of pairwise sums of (1, 2, 5, 9) is (3, 6, 7, 10, 11, 14)
How can I prove that there are two lists of 16 integers that produce the same list of pairwise sums
I tried inducting on the length of the lists, but I failed. Then I tried to directly construct the lists using numbers with properties that I thought would give nice results (like powers of 2), but I failed again.
Any help would be greatly appreciated. 
 A: You were essentially one step away from the solution. Indeed, the lists are better constructed inductively (sort of). Also, the number 16 is suggestive of some scenario with duplications.
Say we take two arrays you already know to have the same lists of sums:
$$(1,7,13,15)\quad (3,5,11,17)$$
Let's add $100$ to one of the arrays and join them. Depending of the choice of that one array, this can be done in two ways:
$$(1, 7, 13, 15, 103, 105, 111, 117)\quad (3, 5, 11, 17, 101, 107, 113, 115)$$
Guess what? These two have similar sums as well!
Now let's add $1000$ to one of them... well, I think you got the idea.
A: It's not possible to have two lists with the same pairwise sum, having at least 3 entries
Let's start out with two distinct lists, each in ascending order
$$\{x_1, x_2...., x_n\}, \{y_1, y_2....., y_n\}$$
Let us assume $x_1 \neq y_1$
Now, looking at the first three elements, we get the following equations
$$x_1 + x_2 = y_1 + y_2$$
$$x_1 + x_3 = y_1 + y_3$$
$$x_2+x_3 = y_2+y_3$$
Solving these, you get $x_1 = y_1$
Hence, if the two lists had the same pairwise sum, and we remove the same element from them, the resulting lists should still satisfy the property. Do you see the problem with this? We could technically keep removing pairs till the lists become identical. Hence the largest lists that can have equal pairwise sums is 2
