MIDI IN

Today's question is  I’ve thought of five numbers.  If I three of them together in each possible combination, the totals I get are 10, 14, 15, 16, 17, 17, 18, 21, 22 and 24.   What are the five numbers? I'd love to see an answer involving algebra. I started with the assumption (correct, I believe) that the lowest three of the five numbers will add to 10, and the highest three will make the 24.  So we've got A+B+C = 10 and C+D+E = 24 (if A-E go lowest to highest). We can get something from this (D+E)-(A+B)=14 and 2C+A+B+D+E = 34 It's interesting that two combinations make 17 but that's all I've got in that direction. [Future me: I did a bit more work on the algebra and thought I was getting somewhere when I boiled those things down to 2C+2A+2B=20, then realised that's the A+B+C=10 that we started with! D'oh!] So I did my usual thing and wrote a program to "try all of the numbers". As I worked I realised that we don't know whether all of the...