The LCDMOD project is working on software patches to NES light gun games, that might allow them to be compatible with flat-screen TV’s. I’ve been corresponding with the creator Aleksey and his goal is for people to use the original, unmodifed Zapper and only require a ROM patch! He’s just opened the public beta for the Duck Hunt ROM and is asking for feedback: http://neslcdmod.com/roms/
To use the ROM, simply load it on a ROM cart and boot it normally (controller in port 1, zapper in port 2). My suggestion is to try it in “auto” mode at first and if you don’t have any luck, press Start on the controller, the use the up and down arrows to manually toggle the delay.
I had mixed results with my TV – I’m using an Analogue NT Mini and while it did work, it was only a fraction of the time. Check out my quick demo video below for more info:
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