April 2018 - Solution
There are five different solutions:
1,2,4 1,2,5 1,3,4 1,3,5 1,2,4 1,2,6 1,3,4 1,3,5 1,2,5 1,2,6 1,3,4 1,3,5 1,2,6 1,3,4 1,3,5 2,7 1,2,6 1,3,4 1,3,5 4,5
Here's an example of a correctness proof for the first solution: In all four experiments, we have one balloon appearing four times and four balloons appearing twice.
4*A+2*(B+C+D+E) = 2*(A+B+C+D+E)+2*A >= 2*(1+2+3+4+5)+2*A = 30+2*A.
On the other hand, since 1,2,4 ≠ 1,2,5≤9, and 1,3,4 ≠ 1,3,5 ≤ 9, and 1,2,4 ≠ 1,3,4, and 1,2,5 ≠ 1,3,5 the sum of all four is at most 7+8+8+9, so 30+2*A≤32, so A=1.
"srcmake" wrote a blog post about the solution: www.srcmake.com/home/ponder-this-april-2018 and even prepared a nice video. Thanks for the solution and for keeping it unpublished till the end of the month, so as not to spoil it for other solvers.
Bert Dobbelaere suggested six experiments to fully determine all nine balloons:
2,3,4 1,3,5 1,2,6 1,7 2,7 1,8