i shoot them here on the farm because they build dams which flood some of my timber. i set up a ladder stand near a dam, dig a 2-3ft. hole in said dam around first light, and let the water drain all day. before sundown i set up a spotlight attached to a car battery, shining in front of the hole i made that morning. beaver often show up about 30 minutes to 1 hr after full dark. they aren't usually afraid of the light as long as it isn't moving. i shoot them using a 12ga and #1 buck. have often killed 2-4 in one sitting. another place on the farm has a large ditch from an adjoining farm. about 4 years ago i shot 13 coming up the ditch in 3 hours. local warden has given me a depredation permit so i shoot them usually at night. 17hmr, 22lr, 223 have all worked for me but 12ga works best for me.
