Give participants some paper, some conductive tape, some LEDs, and maybe a little bit of information about how electric circuits work, and then you have all you need for a paper circuits program. The potential variations on this are impressive, through the use of conductive paint pens, different types of LEDs, or different goals for construction. Switches can be added into the design, either at the battery or in the circuit itself, parallel and series circuits can be used (which can lead to learning about forward voltages, resistance calculations, and best practices for making sure your circuits don't let the magic smoke out when the projects start getting sufficiently complex. (Most beginning programs won't need those calculations, as the necessary resistance can be provided in other ways, but they might find out that certain combinations of mixed-color parallel circuits don't work because of their different forward voltage requirements and the tendency of electricity to take the pathway of least resistance.) There's enough challenge in getting it right the first few times, but once that's done, the sky is the limit for imagination.
Plenty of problem-solving techniques are possible when you start engineering things to try and work the same way that human appendages do, or when you want to try and produce something that is suitable for a specific goal purpose in mind. A deep dive into the design process and mapping out what the requirements are for success and how someone plans on going about it makes for really good STEAM learning, even, and sometimes especially, if there are failures along the way that require reworking and iterating on the design. Making approximations of muscles and skeletal structures and then manipulating them really helps learners understand their own systems and their requirements.
Even things that look like they're going to be very simple can produce complex learning and designs once you start getting into optimization questions. A straw rocket is pretty easy to construct, once the basics of making sure the seals are tight in all the correct places so the rocket will fly. Building a rocket that can fly the farthest, the fastest, or the steadiest involves refinement of ideas and figuring out what does and doesn't work as modifications to the original design. Or possibly trying a completely new and radical design to see if anything good happens with it.
A thing we have to keep remembering with regard to the Maker Mentality and programs like these that may have a suggested goal is that what qualifies as success for the participants is different than what qualifies as success for the facilitator. If the participant's goal is to build a rocket that does a loop-the-loop after being launched, and they get that, it's a success, regardless of whether it does well in any other category. It's surprisingly easy to become directive, especially with participants who are experiencing frustration, but while we can guide and suggest and experiment together, ultimately the participant is the one who is directing the activity and what they want to know.