We thought of 5 numbers. Adding them in pairs, we got the following numbers: 0, 2, 4, 4, 6, 8, 9, 11, $13,15$. Determine the numbers.
It is known that ten pairs can be formed from five numbers. Since there are no three equal sums among the given sums, all five numbers must be different. Let's denote the five numbers in order of magnitude by $a ; b ; c ; d ; e$.

By adding the 10 sums, we get 72. Each of the five numbers appears in four pairs, so $a+b+c+d+e=72: 4=18$. The smallest of the ten sums is clearly $a+b$, and the largest is $d+e$. Thus, $a+b=0, d+e=15$, so $c=3$.

The second of the ten sums in order is $2=a+c$, from which $a=-1$, and using $a+b=0$, we get $b=1$.

The ninth of the sums in order is $13=e+c$, from which $e=10$, and using $d+e=15$, we get $d=5$.

Therefore, the sought numbers are: $a=-1, b=1, c=3, d=5, e=10$.

It can be verified that the pairwise sums of these numbers also give the other values.