I  have used several examples  on non-mathematicians  that seem to have an impact.

**The stable marriage problem**.    Here you can do no better than   use  [Gale & Shapley  exquisite presentation][1]. If you have not read the  paper, maybe you ought to. It is very rewarding.

 **Six degrees of separation problem.**   Each person is at at most 6 n hand shakes away from the President of US.   (There is a version where Kevin Bacon   takes the place of president, though experimental results show that  the average number of handshakes away from Rod Steiger  is  2.87)  What does this have to do with mathematics?      The paper  [*Random graphs models of social networks*][2] by Newman, Watts and Strogatz   may explain why. The stories in the  first part  of this paper  can be used    as a  conversation starter with a non-mathematician, but mathematically curious person.     The book of Durrett  *Random Graph Dynamics*   studies this and other problem  in greater mathematical detail.The story is juicy.  Open these references for inspiration. 

**Teorema Egregium**, the one that says that the Gaussian  curvature of a surface in $\mathbb{R}^3$, which is an *intrinsic* quantity  can be  expressed in terms of the second  fundamental form, which is an *extrinsic*   quantity. The way I tell  the story   to non-mathematicians is with the help of a handkerchief.    It can be wrapped neatly  around  a cylinder (no folds are formed), by try wrapping  it around  your head so no folds are formed.

Then, take a piece of a very elastic material (say a piece of a   surgery glove. You can wrap it around a cylinder, you can stretch it over a  spherical shape such as the top of the human  head. You can do this without forming folds in the elastic surface.   Then as  the audience what is the difference between a cylinder  and a sphere, why can the handkerchief  (which is flexible but inelastic) can sense the  difference, while the elastic material cannot.

 Sneak in there  the word curvature, tell the story of Gauss who   discovered  that our Universe is curved, [tell of Eddington sharper experiment][3]   on the curvature of the  Universe, and end up  Giving Einstein's   explanation as curvature as gravity that bends the straight lines that are supposed to be the trajectories of photons.   usually when I tell this story, people stop asking about the uses of mathematics. They get why it is  fascinating.

**The Radon transform.** I told this story to my dentist. He installed a 3D printer in his   cabinet. He took many shots of a tooth that was missing  a part. These were automatically  fed into a computer and, I kid you not,  the computer  3D-printed the  missing part, matching size, shape, color, all in the span of  less than one our.

I was so impressed about this that I told him about the Radon transform      that happened to capture my imagination at that time. I told him  that I never thought  the technology   had advanced so  much  that it could simulate the Radon inversion formula.  I left him as impressed about the power of mathematics as I was about the advances in technology.


  [1]: http://www.econ.ucsb.edu/~tedb/Courses/Ec100C/galeshapley.pdf
  [2]: http://www.pnas.org/content/99/suppl_1/2566.full.pdf
  [3]: https://en.wikipedia.org/wiki/Arthur_Eddington#Relativity