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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?

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closed as off topic by DonAntonio, Arkamis, Lord_Farin, Joe, user17762 Apr 25 '13 at 22:36

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I heard on a math podcast that you only need $\pi$ accurate to 39 decimal places to "construct" a perfect ring around our known universe... –  Eleven-Eleven Apr 25 '13 at 19:52
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This is off topic...and perhaps some historical issue: tell them about FLT and how it contributed to form and develop amazing areas in mathematics, or tell them about the invention of differential & integral calculus and how that battle between England and Germany isn't over yet, or tell them how the great Archimedes helped to defend his city against the conquering romans... –  DonAntonio Apr 25 '13 at 19:52
    
@ChristopherErnst I heard that too, that we would need somewhat more than 30 decimals of pi to calculate the circumference of the universe to 1 atom accurate. But that's just a factoid –  MathisFun Apr 25 '13 at 20:00
    
@DonAntonio Oh, I didn't know that, I saw some other subjective questions on this site too, so I thought this would be allowed. –  MathisFun Apr 25 '13 at 20:05
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I hope that you don't literally want to see heads pop. –  treble Apr 25 '13 at 23:20

14 Answers 14

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."

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I hadn't heard of the first one before, that's a nice one! –  TMM Apr 25 '13 at 20:16
    
Yes, I know of this one. This was the first problem I encountered during my analytic geometry study. It took me a while to crack! –  MathisFun Apr 25 '13 at 22:21

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.

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With a very large layman audience, probably the most convincing argument is to actually play the game and keep track of results... –  N. S. Apr 25 '13 at 20:35

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.

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My teacher once gave out a test (for fun) with 10 questions, all of them unintuitive, one was almost a carbon-copy of the example you give at 2, except with HIV. Needless to say everyone got terrible grades. –  MathisFun Apr 25 '13 at 22:38

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.

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My god Newcomb's paradox blew my mind too, definitely a very good candidate! –  MathisFun Apr 25 '13 at 20:08

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.

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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.

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Yes, these are awesome. I previously heard about sphere eversion and hairy ball theorem, but I always thought those were deep inside the realm of topology –  MathisFun Apr 25 '13 at 20:03
    
Yes, but you do not need to be very formal of course. The problem is sphere eversion e.g. may be time-constraint-wise problematic. –  Lord Soth Apr 25 '13 at 20:04

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/

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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:

  1. Tell a story, not lessons.
  2. Involve the audience.
  3. Keep it simple. You want audiences to be able to follow what you are doing.
  4. Use daily objects that the audience can relate to.
  5. Use visuals.
  6. Keep an element of surprise.
  7. 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.
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I'm not sure this what you're after, but here it is anyway.

Do you know about modular arithmetic?

Clock arithmetic is a more descriptive term for the aforementioned.

What time is $11:00 + 17 \text{ hours}$? The computation goes as follows: $11+17=28$. Now from $28$ you subtract $24$ as many times as you need until you get a number between $1$ and $24$. So $28-24=4$. Another way of doing this is dividing $28$ by $24$ and keeping the remainder. The remained is what you're after. You basically just need to know the euclidean algorithm to present this.

Now the real presentation (which is an application of modular arithmetic).

What I'll present below is a way of computing the day of the week of any given day of the month of any year. There are several of these. You can find'em here.

I like the following one better because it doesn't require you to memorize too many things. In case it looks complicated, don't give up. It's easy and anyone can do it provided they can do a simple calculations in their heads. With practice you'll nail it and do it easily. Also it actually helps you on day-to-day life.

Firstly we give each month a number, which I'll call month code or MC for short.

The codes go as follows: $$\begin{align} &\text{January: } &1\, &\text{April: } &0\, &\text{July: } &0\, &\text{October: }&1\\ &\text{February: }&4\, &\text{May: }&2\, &\text{August: }&3\, &\text{November: }&4\\ &\text{March: }&4\, &\text{June: }&5\, &\text{September: }&6 \,&\text{December: }&6 \end{align}$$

There's an easy way to memorize this. The first column is $144=12^2$, the second is $25=5^2$ and the third is $6^2=36$. The fourth column you'll just have to remember.

We'll also need a code for each year and a code for each day of the week. The days of the week will be numbered $0,1,2,3,4,5,6$, respectively to Saturday, ..., Friday. Saturday can also be $7$. You can use both $0$ and $7$ because in our arithmetic $0=7$. These are the day codes (DC). Each day of the month will have it's own code DMC and it will be itself, for instance, today is the $25^{th}$ of April, therefore today's DMC is $25$.

The year codes (YC) are computed and not memorized.

To compute the day week of any date, just do DMC+MC+YC, then divide by $7$ and save the remainder. The remainder will be the DC of the day you're after.

For instance, we're in $2013$. This year's code is $1$ (trust me for now).

Let us find which day of the week it is today: $25+0+1=26=7\cdot 3+\color{green}5$ , therefore today is Thursday. And xmas will be on $25+6+1=32=7\cdot 4+\color{green}4$ therefore this year xmas will be on a Wednesday.

Now to find the YC. We take the year $1900$ and give it the YC $0$.

Instead of describing how to get the YC I will exemplify it:

Let's compute the YC of $2013$:

  1. $2013-1900=113$
  2. Find the quotient of the above divided by $4$ (because of leap years), $113=\color{green}{28}\cdot 4+1$. Save $28$.
  3. Add $113$ and $28$: $113+28=141$.
  4. Find the remainder of the above when divided by $7$: $141=20\cdot 7 +\color{green}1$. This is your YC.

Another example, for instance $1996$ (which with any luck is the year you were born):

  1. $1900-1996=96$
  2. $96=\color{green}{24}\cdot 4$
  3. $96+24=120$
  4. $120=17\cdot 7 + \color{green}1$

Crap, it's the same YC. Doesn't matter, you will get how it's done. Note that if you were indeed born in $1996$, then the day of the week of your birthday this year is the day of the week in which you were born.

Important: When you're computing the YC, check if it is a leap year, for if it is the YC will be what you get with the algorithm above only after the $29^{th}$ of February. For January and February you want to take whateverYC you get and subtract $1$. If it happens to be $0$, then $0-1=7-1=6$, becaus $0=7$.

Note: After having the YC of a certain year, it's easy to get the YCs for years on a relatively small neighborhood. You've already found that the YC of $2013$ is $1$. To get the YC for $2014$ just add $1$. To get the year code for $2015$ add $2$. To get the year code for $2016$ ($\color{red}{\text{danger - leap year}}$), add $3+1$ for March onwards and add $3+1-1$ for January and February.

Help: If anyone can prove why this algorithm works or tell me where to find a proof, I'd appreciate it.

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Yup, I read quite some thing about modular arithmetic quite some time ago. I remember going through Gauss' proof of quadratic reciprocity; pretty interesting stuff. By the way, I was indeed born in 1996 and this is very interesting, this can be very useful in day-to-day life! –  MathisFun Apr 25 '13 at 21:42
    
@MathisFun It's good for impressing girls too. You're sure to make them smile when you guess the day of the week in which they were born. –  Git Gud Apr 25 '13 at 21:44
    
I'm certainly going to use that when I going to France next week! –  MathisFun Apr 25 '13 at 21:53

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. :)

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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.

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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 :)

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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

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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.

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