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So, here is the question: Is there a home-made "gun" that I can make and shoot the pigeons with that will be a bit more accurate than a slingshot?


Any decent air rifle will do. A lower powered model won't kill a pigeon unless you score a headshot, which is good if you just want to scare them away. If you want something more lethal, get a higher powered air gun, maybe .22. They're all very accurate at least up to 20 yards, which is probably good enough for what you need.