I just bought the Ruger Mark III a month ago. The above video is not the same as my gun. You have to do exactly as the instructions say. With my gun, you have to make sure you put the magazine in and take it out EXACTLY with the instructions.
I missed one time doing the magazine routine, and it took me 1/2 hour to figure out what was wrong.
Also, the first 2 times I attempted this puzzle, I had to really smack the barrel to get it apart. It has gotten easier the more I do it. But I still have the instructions right in front of me every time.
Good Luck, it gets easier!