If we number the pins like this:
1 2 3
4 5 6
7 8 9
You can start by finding out the value for pins 7 and 9. You just need to realize that you can put the wire around only a single pin which identifies their value completely (For pin 7: 19-x=17, where 19 is the number of lights on top and 17 is the number of lights on bottom, thus pin 7 has a value of 2). Note that all values are negative, sort of like a resistance (even though real resistors would not work like that), so I am going to omit the minus signs from now on.
Then wrap around pins 4+7 and 6+9 to find out the values for 4 and 6. (for pin 4: 19-x-2=11, thus pin 4 must have a value of 6)
Then 1+4+7 to find out the value of 1
3+6+9 to find out 3 will not work because their sum equals or exceeds 19 (can't tell which).
Then 7+8 or 9+8 to find out 8.
And so on, always making new paths with exactly one unknown in them.
Solution for the value lost at each pin, note that each digit only appears once:
3 5 4
6 1 8
2 9 7
So a path that equals 9 would be around pins number 4,5,7 (values: 6+1+2=9).
That's good work. I spent over an hour on that thing and started to figure out what you've described here, but was missing a fundamental part of the logic; that is, subtracting the 17 to get the exact number. Very strong work.