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The 15-tile sliding puzzle (may be a bit outdated by now) is also a good way to introduce permutation groups and even permutations in particular.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