How to entertain a crowd with mathematics? [closed]

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."

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.

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.