I have been asked to teach mathematics/physics to a few 8th grade/9th grade kids for a summer camp. I have been thinking about it and I realized that I could go about it in two ways: One of the ways is to give them random recreational thought provoking puzzles to encourage problem solving nature (This is the way I started my journey into mathematics). The other way is to give them radical ideas of mathematics disguised as thought provoking puzzles, in order to give them a feel of what mathematical theories are like. The latter is an attempt to bridge the divide between theoretical mathematics and fun problem solving. I finally decided to follow an hybrid of the two and teach a mixed bag of olympiad-like problems, puzzles and theoretical material.
I have coached students for math olympiads before and I have a good source of olympiad problems from my books, internet and so on, so I am not looking for olympiad problems.
I have seen ideas here, but as I said I am looking for foundational ideas disguised as problems. Since there are a lot of experts who visit this site, I am requesting to know simple non-trivial problems that motivate theories and those problems should be explainable to 8th/9th grade kids. So, how can I introduce Calculus, Geometry, Combinatorics and Algebra with a fun problem? Even if there exists a theorem with an involved proof, suggesting a special case that is easy to illustrate will do.
Finally, I have not seen anybody attempt to encourage mathematics in my locality, and this is a new idea for me too. So if you think this is a bad idea ("8th/9th grade is too early for deep mathematics" e.t.c), I would love to hear your criticisms.
P.S: From the feedback that I have collected and the way mathematics is taught according to the curriculum, the talented ones believe math is boring but an important tool and others are mostly scared of mathematics. Nobody seems to be enjoying it as much and many of them have an uninformed view that graduate/undergraduate math is boring. I basically want them to see the fun side of interesting graduate mathematics and then make an educated choice for their future. Of course 8th/9th grade might be too early to expose kids to undergraduate mathematics. However, I want to do the best I can. So, please help me.