We don't have mountain lions/cougars here in Southern Illinois. At least not according to the DNR. Despite that, I've spoken first hand with at least half a dozen locals who've seen one over the past couple years, and a lady I work with claims a family member has a picture of one taken with one of those game cameras (but I have not seen this picture to verify it.) I generally don't "worry" about them; just try to take prudent precautions like I would with anything else. Since I never step into the woods without some form of adequate armament anyway, this didn't require much change. I just look up at the trees a little more...
The only time I've really been concerned was when we took our little girl out (1 1/2 yrs old then) for a walk in the woods. Our hired hand had seen a cougar jump into the adjacent corn field the day before. At one point I was carrying my daughter and she started patting her thigh & calling "kitty kitty", like she does to call the house cats. My heart was racing for a few seconds as I strained to look for whatever cat she was calling. I never did see anything, but I guess that doesn't mean nothing was there.
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Some general thoughts about subjects that we've touched on in this thread, many of which you guys probably already know-
Pistols can be about the best compromise weapon to carry, but their "stopping power" is seriously lacking, and it's very difficult to hit moving targets with them beyond very close ranges. Some tips to increase their effectiveness-
Practice. (duh!

) Really, I'm not talking about shooting tight groups on paper here. Try hitting moving targets. Take a big coffee can or milk jug, give it a swift kick, and try to put a couple holes in it before it hits the ground. It's pretty humbling & frustrating at first. Be honest with yourself here. If you can't reliably hit this target from only a few feet away, I doubt the vitals of a running critter will be any easier. Try the same with a shotgun, and the result is a destroyed target nearly every time.
Head shots are not the same thing as brain shots.
If you can make the bullet travel the full length of the animal's body (ie., shoot straight on into the chest rather than broadside) this seems to greatly improve effectiveness.
If you're that close, press the muzzle directly to the critter's flesh. The fireball of gasses entering the body will increase a pistol's killing power by a HUGE amount. (you may also get splattered, but it's worth the tradeoff)
Don't worry about magic bullets. No hollowpoint, cast lead, whatever, can turn a pistol into anything more than a pistol. I've noticed that I generally get cleaner kills on smaller tough critters (raccoons mostly) with decent hollowpoints, but the differences are not dramatic, and would be hard to tell in any one given situation.
With blades, I've always found that a serious chopping blade has far more "stopping power" than pistols, given a decent blow. (at least with smaller critters) The problem of course is range.
If you're worried about hitting something you don't want to up by the critter's mouth, then by all means give it a good blow to the middle of the back or upper hindquarters. The spinal cord is actually less protected in the middle, because the spinal bones get taller on the neck. The hindquarters are the largest muscle groups in the body and are well supplied with blood. This can result in a bleedout just as fast as an arrow through the heart. In my experience the critter will then forget about whatever it was doing and turn to fight, which they can still do adequately even on 2 legs. But this is far preferable to reducing your swing's power in order to make a more precise cut in the neck or head region, or thrusting.
IF your blade has adequate cutting power, then cuts will always be
preferable to thrusts. I don't know what the minimum would be for cougar. A good 15"-18" blade (not overall) can easily do some serious damage on a deer carcass or smaller critters though. This also gives you some reach, so you don't have to be as close to their weapons.
Your blade must be
quick and sharp. My khuk wasn't near quick enough for my liking, so I reground it. I've found polished edges to work best on skin & flesh with a khukri, since it delivers a straight shearing cut. With straighter blades where you can add a bit of drawing motion to the cut, a coarser edge will be more aggressive & work noticeably better against things covered in fur.
Guess that's enough rambling for now...