Interesting questions (with answers) about concepts in topology for an amateur audience I have been asked to hold an introductory math quiz for the Freshmen batch in my college. It entails interesting questions about different areas of mathematics presented in such a way so that it seems it has nothing to do with that area of mathematics. An example of such a problem is the Futurama Theorem.
These questions should not be in a language which involves terms from topology (like topological spaces, homeomorphism, etc) considering the amateur audience for whom this is being presented. Personally I haven't been able to find any such questions except a few which involves showing equivalence of different knots.
 A: Here is a list of suggestions:


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*Take a disk of paper. Crumpled it and place the crumple paper over the place of the initial disk. One point of the crumpled paper will be at the vertical of its initial position in the disk. This is the fixed point theorem.

*Create a Möbius strip. An example of a surface with only one side.

*Describe the construction of the Peano curve. A curve filling a square.

*How to paint an infinite surface with a finite amount of painting. The painter paradox.

*The Schwarz lantern. Or how to approximate a surface of finite area with triangles whose areas are infinite.
You may find other examples on topology in my mathcounterexamples.net web site.
A: A nice (and classic) problem would be the Bridges of Königsberg (https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg).
A: I always liked the: hang a picture on two nails, such that if you remove one the picture falls down .
While the solution to this can be found with a bit luck and without the knowledge of fundamental groups the more complicated ones (hanging it on $n$ nails) is probably impossible without any mathematical advanced ideas. 
A: Here's just a line of thought. I think it would be fruitful to communicate some of the more geometric ideas of algebraic topology or just generally cool results from more general topology. Here are some examples of questions I would think would be insightful:


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*Given a square sheet of paper, how can you make a torus out of it? (In algebraic topology this leads to the idea of quotient spaces)

*How can you recover a plane from a sphere? (In topology a sphere of dimension $n$ minus a point is homeomophic to to $\mathbb{R}^n$)

*Given a torus, can we deform any loop on it to a point? (In algebraic topology this is the result that the fundamental group of a torus is non trivial)
All of these examples convey the sort of geometric problems one asks in algebraic topology, without needing to develop the necessary terminology to explain it rigorously. I think that's good in this case because you want to convey an idea of what math can be like without going into all the hairy details of how it works.
A: As you meantioned, knots and braids are a good source of these kinds of problems. Here are a few that you may not have encountered.
For instance, you could demonstrate that the braid groups on the sphere have torsion elements and then relate this to the Dirac belt trick/plate trick - normally this is explained by using the fact that $SU(2)$ is simply connected, but it can equally well be explained by the fact that the 2-string braid group on the sphere $B_2(S^2)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$.
Some simple invariants can be demonstrated with very little prior knowledge as well, such as the colourability of knots - so you could show how to distinguish one knot from another by showing that one is tricolourable and one is not.
In my experience, people are always rather intrigued by the existence of Brunnian links (and braids) and there's certainly plenty of material here to work with.
The existence of Seifert surfaces via the Seifert algorithm is very pictorial and is pretty fun to play with on paper.
A: Give out paper, scissors and glue.
Get them to make a Möbius strip. Explain how it is a 2D object, but needs the 3rd dimension to fully realise the shape.
For homework, get them to make a Klein bottle. :)
A: The three utilities problem is a disguised version of asking whether $K_{3,3}$ is a planar graph.  It's not, but $K_{3,3}$ is regularly embeddable in the torus -- if given one "handle" attached to the plane, you can attach the last house/utility pair.  The general question is "how many handles do you need" for various numbers of utilities and houses.  This is Turán's brick factory problem and is open (upper and lower bounds are known).
Brouwer's fixed point theorem tells me that if I start with a flat piece of paper, crumple it up, and drop it back where it started, at least one crumpled point projects (perpendicularly to the original sheet) back to its initial position.  (The linked version of this example attributed to Brouwer is better stated.)  Similarly, the Borsuk-Ulam theorem leads to the non-technical result that at every instant, there is a pair of antipodal points of Earth with identical temperatures and barometric pressures (or any other pair of continuous scalar properties you like).
The hairy ball theorem states (very informally) you can't comb a hairy ball without a cowlick.  But you can comb a hairy torus without one.  So there is a boring automated way to paint a torus, but not so much for a sphere.
There's always More Knot Theory...
You can demonstrate tying a knot in string without letting go of the ends during the tying (description and illustration).  This gives an overhand knot, which is a chiral knot.  You can tie its reflection by reversing the helicity of the starting state of the arms.  Interestingly, not all knots are chiral; there are amphichiral knots.
A: Try an exhibition on knots: http://groupoids.org.uk/popmath/cpm/exhib/knotexhib.html
The aim was to use knots to present some basic methods in the development of mathematics. Knots were used  because of their familiarity and visual aspect. 
See also http://groupoids.org.uk/outofline/out-home.html
