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On the mystery quadrant: that is a surprising result.
The ten LEDs in that half-quadrant are connected in series.
So, if one LED were damaged such that it gave no connection (or it was in backwards), I would expect all five to light up when you connect an extra LED across the pins of that one location. You would not be able to light an LED at any of the other locations.
And on the other hand, if one LED had a short circuit (for example, if its two pins were soldered together), I would expect the other four to normally be lit up, but brighter than usual.
Do you still have the same behavior now that you've replaced that LED-- as in, will the other four locations still light up an LED? And, do you have access to a multimeter?