Generating a list of numbers whose combinations are mathematically unique I'm trying to design a puzzle for a game I'm making, and the puzzle is fairly similar to the one from the power plant in Myst. The idea is that I have 9 buttons, and each button supplies an amount of power to the "system". The solution to the puzzle is to press three specific buttons (order doesn't matter) to reach a specific power threshold.
Obviously, I have to be careful about how I choose how much power each button provides since if a number can be reached by more than a single combination, that can destroy the complexity of the puzzle somewhat. My first naive idea was to use the first 9 prime numbers, but a quick check through a program showed that that wasn't going to fly. It had occurred to me that I could just choose a number that only occurs once in the list of results, but I would prefer it if I could choose which 3 buttons to press at random.
How do I go about generating this list of 9 numbers such that adding any 3 of the numbers together results in a unique value? Is there some statistics/probability thing that I can use or do I just have to trial-and-error it?

Partial answer.
Just starting a brute force greedy algorithm, the sequence
$$
1,2,4,6,10,16
$$
works as far as it goes. The pattern continues adding $8$, $10$ and $12$ 
$$
24, 34, 46
$$
to give you nine numbers with distinct sums of threes. The totals range from $1+2+4=7$ to $24+34+46=104$.
You should check this.
I wonder if this greedy solution is optimal in some sense. I wonder if there's a systematic way to generalize.
