Problem Solve Out Loud
If you want your children to learn to problem solve, model it! Children are mimics, one of their main ways of learning is through watching others.
I model failure and problem solving out loud. I never realized how much minor problem-solving I do in a day until I made an effort to describe it as I did it. I think adults have a bad habit of showing children only the good and forgetting to walk kids through processing the hard.
For instance, if I'm having trouble opening a jar lid I'll talk it out. "Ugh! I can't get this lid open. I'm going to try again. It's not working!" I'll stomp my foot or let some of my true frustration into my tone "This is so frustrating. I'm going to try again. Ok, maybe I need a new plan. Maybe I need to use a dishtowel." And so on and so forth until I solve the problem, talking out what I'm doing. As adults we handle minor frustrations so easily we might not even feel frustrated but those moments are big for kids and they won't be able to handle hard things down the road without working through the small things and laying the groundwork for handling big emotions. There is no need to put on a big show or to overact, but don't hide your emotions and subsequent calming actions from children.
Ask for help! Kids are surprisingly capable. Ask them to brainstorm ideas to help you. Kids have creative minds and unique, often funny approaches to situations. Give them the opportunity to help you too.