Maybe they are not burnt out? Maybe they are not getting power?
Heres my thought,
I tried to look at the flexible pcb on them and trace it but it is really hard to do.
But if you look at the wiring diagram here :
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LoginRoll down the page to find a schematic of it inside a picture of the string.
It appears you should be able to cut it anywhere marked giving you a three led segment.
I am thinking ... (this is always dangerous!)
Maybe the power from each section does not connect to the power pads but bypasses them to the inside section of the diode pad. So you do not need a diode at each section because it would have a voltage drop and across a full string you would not want it as it would not work. You would only want one diode on the whole string to protect from hooking the string up backwards.
You can handle one voltage drop but not one for every 3 leds!
So to get power to the section with out a diode you need a diode added to pass the power. With out it you have no power applied. So a test would be to take one of the sections you think you killed and solder a jumper across the diode pads. Make sure to hook the power up the correct direction as you have no protection.
this is what the engineer means by "It might burn up with out it." He means it could be hooked up backwards on the power and be damaged with out a diode to protect it I am thinking.
My issue is my strips need to be cut in the middle for what I am doing and this is at a spot with a diode I believe.
Let me know what you find out as I will be very interested myself.
RJ