Lafazar: 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.

WARNING: SPOILERS!

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.