I suspect that you are getting serial and parallel connection mixed up. Which is it? Serial is +Ve to -Ve chains, parallel is all +Ve together and all -Ve together.Do these LEDs have resistors? If no then they should if you want them to last more than a few minutes in parallel.In series you can get away with one resistor to set the current for the whole chain.
The original PSU with this is definitely 6 volts, not 20 (stated on the transformer). If I touch a single + wire to the + supply the single light lights as expected.Each LED originally had two wires fed to it + and -. I have had a meter on the supply in addition and it reads 6 volts.If I block connect all 10 of the + lines to the supply the all light correctly.