Here are a few ideas for exercises that hopefully lead to "fun" facts in (algebraic) topology. If I remember correctly, most (but not all) of them are contained in Prasolov's book you mentioned.
a) Let them try to solve the three utilities problem. This boils down to embedding the graph $K_{3,3}$ in the plane, which is impossible. b) Explain that this is impossible. If you take Euler's theorem (about the Euler characteristic) for granted, the proof is quite accessible. c) Let them try to solve the same problem on the surface of a mug, which is a torus. I'd expect them to succeed with some trial and error, and possibly some hints (the handle lets you avoid one intersection). d) Discuss the difference between the two cases, and how Euler characteristic works for different surfaces.
Let them discover the Borromean rings by themselves. That is, give them three loops (that can be open and closed at will) and tell them to arrange it so that the three loops together are tangled, but removing one loop makes the other two untangled. The solution is not unique, but there's a good chance they will arrive at this one, because it's the simplest.
a) Consider the following puzzle. There are two nails in the wall, and the goal is to hang a picture on them (using a loop attached to the picture), according to two rules: the picture hangs if both nails are in place, but if either nail is removed, it falls down. b) After they arrive at a solution, discuss why it's the same puzzle as the Borromean rings. In both cases we're studying the same fundamental group: of the space minus two untangled circles (in 1) or of the plane minus two points.
a) Let $A,B,C$ be Borromean rings. Find a surface spanned by $C$ (i.e. a surface with $C$ as its boundary) which doesn't intersect $A$ or $B$. b) Discuss why this surface cannot be a disk (because you could shrink $C$ and remove it without touching $A$ or $B$). c) If possible, introduce Hurewicz's theorem. It explains the difference between spanning a disk (i.e. being trivial in the fundamental group) and spanning a surface (i.e. being trivial in the first homology group).