The reason for the three round limit is exactly why they have it. People with semi autos firing a dozen rounds wildly,as a group of deer run by. When I bought my first AK a MAK 90 it came with three five round mags. That was 1993.
In my state the 5.56 has always been legal for deer, but 22 cal semi autos have been limited to I think, 7 round mags
while hunting deer, I have no clue where they got that number, but I have had several 5 and 10 round AR mags for maybe 30 years. Everybody that hunts deer with ARs has had them for years. For everything else, hogs, coyotes, etc, no limit. I am a vet of both the Army and Air Force and carried them a lot, most people like 30 round mags, I prefer the 20 for just daily use, like coyote hunting. The ten round mags are fine too, they do not get in the way if you are shooting while resting the gun on the hood, coyotes of course.
For the last 30 years I only hunt old deer, do not shoot babies or does. Makes hunting hard, but after several times of having does and little 6 point bucks just stand their curious at 10 yards, I decided there was no skill in killing them and they will grow twice as much meat if you let them live just three years. The last doe I killed was 1991. I leave them for the children and new hunters to shoot or people who desperately need the meat.
All that said to say, I have used the five round mags in ARs and AKs and the excellent Mini 14 and 30, since 1991. I remember the year because I bought the Mini 30 as a gift to myself after Desert Shield and Storm. I used the 5 round mags in the mini 14 since 1978. The mini 14 and 30 come from the factory with 5 round mags.I have not fired but two rounds when hunting with the 'assault rifles'.
There is just something very manly about hunting with an AR that only has 3-5 rounds in it and you know you will come back with one deer and most of that ammo. Not everybody can do that, mostly just us old guys willing to hunt long and never taking an iffy shot. Most younger guys do not have the mental discipline to do that. Just my two cents, lots of people new to hunting may disagree.
Off the subject a bit but mentioned on another thread, I have ARs in 5.56, 300 BLK and 7.62x39. I use color coded mags, so I do not take the wrong ammo. I use desert tan or fde for the 7.62, black for the 5.56 and blue for the 300. The last mag below is from Midway, it has over 200 reviews so I guess it works.
The 350 gets great reviews for a short range deer round, give us some reports. Might just be the perfect round, I dunno.
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