A few months back we taught a course on curves and surfaces to undergraduates and asked them to slice a bagel into two linked halves as in [here][1]. Of course, you need at least two bagels per student since inevitably most of them end up cutting the first bagel into two unlinked pieces. 

The 15-tile sliding puzzle (may be a bit outdated by now) is also a good way to introduce permutation groups and even permutations in particular. 

And lastly, the game of [Sim][2] (not to be confused with sim city) where two players take turns in drawing edges in red and blue on set of 6 vertices. The rule is that if an edge already exists between a and b then one cannot draw another one. The aim is to avoid a triangle in your own colour. It is known that this game always has a winner. Obvious generalizations to more colours and more vertices lead to Ramsey theory. I actually took this route while lecturing to high school kids and they get into it if you start your talk by playing a few games on the blackboard.


  [1]: http://www.georgehart.com/bagel/bagel.html
  [2]: https://en.wikipedia.org/wiki/Sim_(pencil_game)