How to entertain a crowd with mathematics? I am a high school student who follows a university level curriculum, and recently my teacher asked me to hold a short lecture to a crowd of about 100 people (mostly parents of my classmates and such, I'm not the only one to do something, other kids will sing and play the piano and such). I actually self-studied linear algebra and basic differential equations, but I felt that would be too boring (and the non-boring parts would be too difficult) to explain. I then decided to try and explain Euler's identity, since it looks so counter-intuitive and almost everyone knows $e$, $\pi$ and $i$. But I decided that it's undoable in like 10 minutes; only a handful of people would be able to follow it if I rush through it. 
So I guess my question is; Is there any mathematical 'thing' which is easy to follow and will blow the minds of the parents, literally? I literally want to see some heads pop. It doesn't need to be related to Linear Algebra/Calculus, but I need some interesting problems/theorems/formulas etc. which is understandable to the layman. I think this question might be useful to some other people on this site who are in a similair situation, and want to show math can be mindblowing. 
At the moment the best idea I've come up with is the birthday problem, I think my introduction would be pretty epic:
Me: Hello audience! Let me ask you a question: How big does a group of people have to be minimally if you want the chance of a pair sharing a birthday to be 100%?
Crowd: 367!
Me: Very good crowd! And approximately how many people for it to be 99%?
Crowd: Around 360?!
Me: Nope, 57.
Crowd: POP!
And I would continue to explain why, which is not hard at all.
Are the similair mathematical results which will blow the mind of a layman?
 A: The Monty Hall problem is nice for such purposes. It's probably even more counterintuitive than the birthday paradox. 
One way I tried to convince the crowd "switching" is really better is to use a generalization with $10$ doors, and opening $8$ of the remaining doors when the contestant makes his initial choice. For some this was convincing enough that switching may be better with $3$ doors as well, but some will be left confused even after your explanation.
A: When giving a talk on mathematics, the only thing that matters is the level of your audience. To me, it sounds like your audience is probably on average one that has been heavily removed from any kind of mathematics beyond simple algebra and geometry. If your goal is to entertain people then your talk should be essentially devoid of any derivations that go beyond basic intuition or simple manipulations. To get people interested in anything, you need to make them form some kind of emotional connection with what you are trying to convey. The only way you'll accomplish this is by going very slowly and starting with something anyone can quickly grasp. If something is surprising, then it should be so immediately because it's something that anyone can think is surprising. That's why things like $e^{i\pi}=-1$, while potentially very interesting to a motivated high school student, will essentially be out of grasp for most of the audience which probably does not even remember what $e$ is, nor has any emotional attachment to it's role in mathematics. An excellent example of communicating scientific ideas to the general audience would be something like TED talks or the movie Between the Folds which is about mathematics and origami (it's available on Netflix by the way). 
If you want some ideas:
1) The Greeks, Eratosthenes in particular, was able to estimate the circumference of the the entire earth to within 2% of the actual size. To appreciate this, please consider that this was done over 2000 years ago by someone who had only been to Greece and Egypt using nothing more than a glorified protractor and some basic geometry. By the way, this dispels the commonly held false fact that people thought the earth was flat back then. 
2) People do not understand conditional probability. An example from the link is the following: 8/1000 women have breast cancer. There's a 90% chance that a woman with breast cancer has a positive mammogram. There's a 7% chance that a woman without breast cancer has a positive mammogram (a false positive). You went to the doctor and have a positive mammogram. What's the actual probability you have breast cancer? Try guesstimating the answer first and then working it out by hand. These kinds of issues abound in all walks of life, for example in famous court trials and security. You'll find many more examples and explanation of why people have trouble with this stuff in Kahneman's book Thinking Fast and Slow. The correct answer to the breast cancer problem is 9%. If you only have ten minutes, give people some of the estimates that the NY Times article above suggests for some of these problems. You don't even have to introduce Bayes rule or anything, even if you can convince people that $A\cap B$ is different from $A|B$ you've gone a far way. 
To summarize: create an emotional attachment between your topic and the audience by making people relate to it. Anyone can understand folding origami or trying to work out basic arithmetic. Your focus should be on conveying ideas and not on derivations.
A: In my experience, the best way to entertain and engage a largely indifferent audience with math is to minimize the technical details and maximize the sense of conflict.  Ideally, tell them about a conflict within mathematics, and how there isn't one "right answer" and they can choose their own answer and it's legitimate.  For math undergraduates, a favorite of mine is the axiom of choice, but it's too advanced for the crowd you have.
My specific suggestion for you is Newcomb's paradox, a topic from game theory, but really philosophy.  You can present the problem, let the audience think for themselves how they would act, and give the surprising answer that both answers are legitimate in some sense.  There's even a great postscript, namely that your answer reveals whether you believe in free will or not.
A: Some suggestions:
1) Cardinality of sets. The basics are not technically involved at all and you can tell the story of Hilbert's Hotel. 
2) Monty Hall's Paradox. 
3) The two envelop paradox. 
4) The liar paradox. 
5) The sentence written bellow is false. 
6) The sentence written above is true. 
7) There exist two antipodal points on earth where the temp is the same. 
8) If you stir a cup of coffee, when the liquid comes to rest at least one molecule will return to its original position. (without proving Brouwer's fixed point theorem). 
9) Proving Brouwer's fixed point theorem by means of Sperner's Lemma. 
10) Three disks are in a bag. One is blue on both sides, one is red on both sides, and one is red on one side, blue on the other. You reach into the bag, pull one disk out and look at one of its sides to see that it is blue. What is the probability that the other side is also blue?
11) The non-existence of a uniform distribution on the natural numbers. 
12) The equality (with proofs) 0.999.......=1.0000.......
13) The uncountability of the real numbers. 
14) Two real numbers between $0$ and $1$ are chosen by some non-atomic distribution (that just means that the probability that both numbers are equal is $0$). You do not know the distribution. I show you one number and you have to guess if the other number is bigger or smaller. Find a strategy to maximize you chances of success (hint: you can do better than 50%). 
15) Vitalli's example of an unmeasurable set (most of the technicalities are simple enough, but this could be too much for some). 
16) Euclide's proof of the infinitude of primes + a discussion of related open problems in number theory. 
17) Fun with the Moebius streep (cutting in up in various ways). And then comes the Klein bottle. 
A: Bottema's Theorem 
You've found a treasure map.  Two large rocks and a tree made a triangle, and the lines between the trees and rocks were used to make two big square plots.  The treasure was buried between the two opposite corners of the square plots.
(show the picture)
You get to the site, and you find the two big rocks.  But the tree and the plots are long gone.  How can you find the treasure?
(pause)  Now move the position of the tree around.  The treasure is always in the same place.
There are many other interesting interactive math demonstrations there, such as
1. Pick's Theorem
2. Minimally Squared Rectangles
3. Densest Tetrahedral Packing
4. Pentagon Tilings
5. The Circles of Descartes
6. The Bomb Problem
7. Random Chord Paradox
8. The Penrose Unilluminable Room
9. Drilling a Square Hole
With 3, talk up that mathematicians have answered this wrong for 4000 years. With 4, mention that a housewife solved this problem when all the mathematicians got it wrong. With 8, mention that it was a teenager that solved the problem. 
I've given various math entertainment lectures -- and of the hundreds of quick pieces of fun math, it's Bottema's theorem that always seems to work the best, as the tree gets moved.
Another good one -- the Homicidal chauffeur problem.  
To finish you can appeal to the people that still don't like math if they'll give up all the items they have that have mathematically generated curves.  Then explain Guilloché Patterns, which are on all money.  "You can just lay the money on the table if you still don't like math -- otherwise, my work is done."
A: First of all, good luck with entertaining people with math. Three things that you may want to talk about:
Idea 1: I think you may talk about sphere eversion, i.e. how to turn a sphere inside out. You may use visual aids for that too, there are also videos on the web that explain how and why this is doable. People like drama, so your story will go like this: You ask the audience, is it possible to turn a sphere inside out? They will say (hopefully if they understood what you are talking about) no. And then you say, that is what Smale's advisor told him when he found a non-constructive proof that it is indeed possible. And then, who found the actual method to do it? One of the first was Morin, who happens to be blind. This is a great and amazing story.
Idea 2: Another possible show might be Basel's experiment, but you will need some help to perform it. Maybe you may call some of your friends, and while they do Basel's experiment, you talk about something else.
Idea 3: Hairy ball theorem. If you have a cat, this becomes easier to explain.
A: A lot of ideas can be found in the T.V. series Numb3rs:
http://en.wikipedia.org/wiki/Numb3rs
You might want to check the companion Wolfram site too:
http://numb3rs.wolfram.com/616/
A: You might want to "steal" a couple topic ideas from ViHart or Numberphile.
Fractals is always interesting and easy to entertain since they are visual and you can use real-life objects (leafs, pineapples, etc) that people can easily relate to.
Another topics suitable for entertaining crowds would be mathematic magic. You may instruct your audience to pick any random number, do some simple calculation and guess that whatever the number they picked, it all led to 42 (or some sort of that). Another common magic trick is the Number Guessing Game. Ask a member of the audience to pick a secret number, then ask if their number is in some cards containing a list of numbers, then guess what their secret number is. If you then follow these magic tricks by explaining how and why the trick works, it could be educational as well as entertaining.
Another possible topic is the Seven Bridges of Königsberg for Graph Theory. You can start by saying you want to play a game, involve your audience by asking them to try solving the bridges with maps that are actually solvable (you may want to offer them a small prize for answering), then after the audience felt comfortable with the game, present them with an unsolvable bridges. Follow up with explaining of why it's unsolvable.
Keep these in mind when entertaining an audience:

*

*Tell a story, not lessons.

*Involve the audience.

*Keep it simple. You want audiences to be able to follow what you are doing.

*Use daily objects that the audience can relate to.

*Use visuals.

*Keep an element of surprise.

*Rehearse and practice. A lot. You have to know your topic inside out, more than just what you're planning to talk about. Practice with a test audience if necessary.

A: If its ok to go a little into topology, I think the Möbius band characteristics are perfect for the job. e.g. what happens when you cut it in half one, and then two times. At least the first time I knew about that i was baffled and simply couldn't believe it happened by just twisting a paper strip.
A: Simple ballistics.
Find yourself a reliable but safe cannon - perhaps one that shoots tennis balls or something similar. Find yourself a sufficiently impressively small target.
Do the maths - hit the target first time - pop!
Loads of variants - controlling the cannon with a robot with laser sights would be cool. :)
A: There is that thing with the ring around the earth.
Suppose that you have a ring perfectly fit around the Earth's equator. For this purpose, even suppose that this ring is at sea level and it carves through anything which isn't. This ring has circumference $x$ feet. Suppose that you add $1$ foot to this ring. How high would the ring be over the sea level?
Initially people would think it would raise the ring by less than an inch or so, because the circumference of our planet is so large that one meter is insignificant.
BUT! As it turns out the circumference is increased by enough to allow a cat to pass underneath. That is mind blowing mathematics! And it only require people to know the formula for circumference of a circle.
Read this link for more information.
A: First, I'd like to say that I'd love to have you in my mathematics classes at the university level :)
In a short talk I gave at our Phi Beta Kappa induction ceremony last December, I stole a question one of my graduating seniors was asked on job interviews at financial institutions. If you can get the audience debating a bit, it works out quite well. A number of the parents (and a few of the students) told me afterwards how interesting they'd found it. Here's the question. I hope most of our math geeks here will get it quickly :) 
MathIsFun bicycles precisely 10 miles in an hour. Along the way, he hits red lights, stops to chat with a friend, stops for a coffee, etc. The question is this: Is there some 30-minute interval in which he bicycles precisely 5 miles? 
It's interesting how wrong some people's intuition can be, and perhaps this is, in the end, some good propaganda for learning a bit of math :)
A: Personally, I think you're aiming too high. Something easier to grasp and follow along with would be better.
Something with some humour is always good, but if not, perhaps something fun and/or useful that people can more easily follow along with.
Have a look at
http://ed.ted.com/lessons/the-magic-of-vedic-math-gaurav-tekriwal 
