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I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found that I, too, lament the uninspiring quality of my elementary math education.

I want to make a book that discredits the notion that math is merely a series of calculations, and inspires a sense of awe and genuine curiosity in young readers.

However, I myself am mathematically unsophisticated.

What was the first bit of mathematics that made you realize that math is beautiful?

For the purposes of this children's book, accessible answers would be appreciated.

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For me Euclid's proof of the infinitude of primes was the first thing that made me realize the beauty of mathematics. – Manjil P. Saikia Mar 7 '13 at 7:02
Wow. Just last night I had a fierce argument with one of the bartenders of my usual watering hole who is a mechanical engineering student. He insisted that he has a better idea than me of what is mathematics. I am so going to print him a copy of Lockhart's text. Thank you for that link! – Asaf Karagila Mar 7 '13 at 7:59
I can’t remember a time when I didn’t think that mathematics was beautiful and fascinating. – Brian M. Scott Mar 7 '13 at 15:06
Although I don't know if it's what you are looking for, try looking up "vihart" on youtube--Even if it's not helpful, I guarantee you will appreciate it. – Bill K Mar 8 '13 at 2:57
I think it's a shame that this question was voted closed... – Will Mar 10 '13 at 19:50

160 Answers 160

For me, the result that really captured my imagination was the divergence of the harmonic series:

$$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\ldots=\infty $$

It combines some wonderful ideas about the infinite and the infinitesimal, and it seemed (at the time) completely absurd to me that adding infinitely small numbers could result in an infinitely large one.

As an illustration of this idea, say we have a big pile of 1-foot square boards. We stack the first board on the second, hanging half-way (6 inches) over the edge. Then we stack the third on top the second, hanging 1/3 of the way (4 inches) over the edge. The forth is stacked on, hanging 1/4 of the way (3 inches) over the edge. The get the idea. At first glance, one might think that our pile can only extend horizontally a finite distance - we might take bets that it gets at most 2 feet, or maybe 5 or 10 feet horizontally. But it turns out that if we have enough boards (negligibly thin, say), we could build a bridge across any river, any ocean, in fact we could build a bridge across the entire universe this way.

Here is a Wolfram demonstration of this, although their stack is upside-down from how I have described it:

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The stack the way you've described it will collapse, I believe. You need to do it the way the Wolfram demonstration does. – Joe Z. Mar 8 '13 at 20:23
Sure, okay. Hey, I'm a mathematician, not an engineer =) – icurays1 Mar 9 '13 at 6:42

I was in elementary school, drawing 3D shapes in class while bored. I drew cubes by drawing two overlapping squares and connecting the vertices, like the top row in this image:

enter image description here

Then I thought, what if I did the same procedure, but to a cube? So I drew four squares and connected the vertices, like this:

enter image description here

I was struck by the beauty of the resulting image, with its intricate structure of star-like patterns. Here's a static version:

enter image description here

It was years later that I discovered, to much fascination, that this was in fact the four-dimensional analogue of the cube: the hypercube. Hence my username.

Edit: Another thing I remember thinking about when I was younger was that I could not always draw a straight line through three points, but was surprised to find that it would always work for two points.

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This is really amazing. This is my idea of what is math all about. – Adam Sep 25 '13 at 20:45

That the roots of $z^n-1 = 0$ start to form a circle as $n$ increases. It starts out with the simple solution, the quadratic which you've already seen, then the complex plane comes in for $z^3$ and all of a sudden it's like "Hey! Those form circle!"

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Do you really imagine a child reading this and getting amazed? – Integral Mar 7 '13 at 15:40
Well, I was amazed, and 15 years of age - not yet a man, despite my best efforts. – Roberto Tyley Mar 7 '13 at 15:49

Many years ago, before I knew multiplication, I wrote numbers 1 to 10 in a row:

1  2  3  4  5  6  7  8  9 10

Then I wrote a second row, just for the fun of it, starting with 2, increasing each number by 2:

2  4  6  8 10 12 14 16 18 20

And then a next row, starting with 3, with an increment of 3, and so on, until I got:

 1  2  3  4  5  6  7  8  9  10
 2  4  6  8 10 12 14 16 18  20
 3  6  9 12 15 18 21 24 27  30
 9 18 27 36 45 54 63 72 81  90
10 20 30 40 50 60 70 80 90 100   

I showed this to my parents, and they told me it was this thing called the "multiplication table" and explained how it worked. I was amazed.

Still today I'm very proud that I reinvented the multiplication table :)

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Arithmetic series might be interesting: straightforward to explain and amenable to pictorial representation ...and the child might love the fact that they've learnt how to do huge sums that might stump many (non-mathematical) adults.

You could show how $1 + 2 + 3 + \cdots + 100$ could be worked out by pairing numbers from opposite ends of the sum together $(1 + 100) + (2 + 99) + \cdots + (50 + 51) = \underbrace{(101 + 101 + \cdots )}_{\text{50 terms}} = 5050$.

or by adding the series to itself with terms running in ascending and descending order $1 + 2 + \cdots + 99 + 100$

$100 + 99 + \cdots + 2 + 1$

to get $101 + 101 + ... = 101 \times 100$ which is twice the sum.

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The story could even be about a little boy named Carl Freidrich... – Will Mar 7 '13 at 16:31
I also remember a teacher (must have been around 10yo) asking us to calculate the sum of all numbers from 1 to 100, letting us sweat uselessly, then showing us how to pair the numbers. He liked to play tricks on us, but the lessons were always useful. Well, sometimes he played us for fools by being wrong on purpose - once he "calculated" that there were about half of the days in the years that were holidays to show we shouldn't complain about school. Took me years to realize he had counted a lot of days twice (like, 2 months holiday & 52 week-ends.) – Joubarc Mar 8 '13 at 8:55
@Joubarc: That reminds me of a joke. A clerk asks his boss for a raise, and the boss calculates how many days the clerk works there. There were 366 days in that year, and he worked for 8 hours a day, so it became 122. Then, he had to subtract 52 Saturdays, two weeks' vacation, four bank holidays, and 52 Sundays, for a grand total of no days at all. "And you have the nerve to ask me for a raise!" – Joe Z. Mar 8 '13 at 20:10

Thanks to @FacebookAnswers for suggesting Conway's Game of Life, a cellular automaton devised by John Conway in 1970.

A Gosper Glider Gun

Gosper Glider Gun

With its patterns, oscillators, spaceships, glider guns (the minimalist Gosper Gun is shown above), breeders, Turing Machines, and the many derivatives, this "game" has spawnd much thinking and imagining.

A generation $\approx 10^{28}$ Turing Machine in Golly

A 10^28 generation Turing Machine in Golly

Of course it's a challenge to replicate the wonders in a static book, but there's great potential for the CD, ebook, or website.

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Euclidean geometry was the first thing that got me (about grade 9 or 10). That's where I first found out that

1) There is such a thing as mathematical proof (rather than just calculation).

2) Mathematics is not a closed subject: new and interesting results can still be found.

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My favorite was when I was asked:

"If you were to save 1 penny on day one, and double your money for a month every day after that, how much money would you have?"

One I realized the answer was $10,737,418.24 I was flabbergasted. That was when I was able to understand that there is a mathematical model/equation for just about everything in this world; now that's beautiful.

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My answer was going to be about that fable of the grains of rice on a chess board.. Its very similar to your answer though. – gordatron Mar 8 '13 at 15:48

The following riddle blew my mind when I was a kid.

Three men went into a hotel. At the front desk they were told that the room would be \$30, so they each gave \$10.

After the men went to their room the manager realized they booked a room that was only \$25, so he gave the bell boy \$5 in ones to take back to the men.

On his way, he thought, "5 can not be evenly divided by 3 men", so he pocketed two and gave the other three to the men, one to each.

So, effectively each man paid \$9 for the room, a total of \$27. Remember, the bell had \$2 in his pocket. \$27 the men paid + \$2 the bell kept = \$29. Where did the extra dollar they paid go?

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I love this one. Thank you. – Stu Mar 7 '13 at 19:35
What the hell?! Ah, I got it. Crafty.. – Thomas Mar 8 '13 at 3:35
Where did the extra dollar they paid go? Taxes! – BeniBela Apr 27 '13 at 20:54

For me, it was topology, and beautiful Klein bottle and Möbius strip.

enter image description here

Related to this was the realisation that a coffee cup is topologically identical to a doughnut:

Coffe cup to doughnut

This still fascinates me to this day despite not being involved in advanced maths at all.

Coincidentally, I learnt about this from a maths book for children written ~30 years ago :)

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Wait, is that a Klein bottle in a Klein bottle in a Klein bottle? – Joe Z. Mar 12 '13 at 13:09
@JoeZ. What does "in" mean, when you're talking about a Klein bottle? – Andreas Blass Jul 20 '13 at 18:59
You tell me. :P – Joe Z. Jul 21 '13 at 22:38

This was my favourite equation. I was 16 or so, when my father showed it to me. I was amazed, and I programmed an application which drew this:

The equation of love.

The interval should be <-6;6> maybe. I made it a looong time ago after all ;)

orig white black

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The fact that Gabriel's horn has infinite surface area, but finite volume, hence you can "fill it with paint, but you can never cover the whole surface".

Gabriel's horn (also called Torricelli's trumpet) is the graph of $y =1/x$ for $x\geq 1$ rotated around the $x$ axis.

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In elementary school, my math teacher taught us this trick for the 9 multiplication values:

enter image description here

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That's neat! ${}{}$ – Elmar Zander Mar 15 '13 at 9:44

Two instances where I thought math was amazing:

  1. In like 4th grade or whenever you learn areas of rectangles, one of the exercises in my book was to estimate the area of some squiggly shape overlaid on a rectangular grid. I thought this was pretty cool, and reasoned that if you could make the grid "smaller" (higher resolution), you could be more accurate. I mentioned this to my mom, who proceeded to tell me that was basically how Calculus 2 worked. :) That was very fun for me.
  2. Deriving the quadratic formula in Algebra 1. That was fun--it showed that some totally un-intuitive formula could be easily found using other previously found results.
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Your first sentence is incorrect. :) I had the "grid" epiphany, too! – Akiva Weinberger Nov 13 '14 at 1:23

For me, I suppose it was Pascal's triangle. I was first formally introduced to it in one of my high school math classes, where my teacher explained Pascal's triangle, and challenged us to find as many patterns as we could in it. We spent a decent chunk of time doing so, and I was amazed by how a simple rule to generate a simple pattern of numbers could yield so many interesting patterns and properties.

I also found it cool how Pascal's triangle could be used to solve a variety of patterns from binomial distribution to the problem where you try and find the total number of paths on a grid assuming you can only travel in two directions, and demonstrated to me how mathematics is a lot more interconnected then I thought.

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When I was maybe 8 or 9, the following trick was showed to me as a sanity check for calculation by hand.

  1. Take two numbers, let's say $358$ and $77$.
  2. Sum up the digits until you get a single digit number. $$ \begin{array}{r} 358 \to 16 \to 7\\ 77 \to 14 \to 5\\ \end{array} $$
  3. Do the same with the two sums $$ \begin{gather} 7 + 5 = 12 \to 3\quad\text{and}\\ 358+77 = 435 \to 12 \to 3 \end{gather} $$
  4. You get the same result? Try with other numbers. Be Amazed!
  5. Best of all? It also works with products: $$ \begin{gather} 7 \times 5 = 35 \to 8\quad\text{and}\\ 358 \times 77 = 27566 \to 26 \to 8 \end{gather} $$

I could not believe this always worked, it looked at the same time so beautiful and magic! A few years later, I was finally able to prove it by myself. I was so happy!

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This method is called "casting out nines", and works due to modular arithmetic. – Joe Z. Mar 8 '13 at 1:00

It is really difficult to remember my days as an elementary student. I just remember how beautiful I found math to be: the connections I saw between everything I was learning, the beauty of the patterns, the sense-making, the sheer marvel of it all. I cannot pinpoint one "fact" I learned, or one particular "ah-ha!" moment (there were so many), but I attribute my love for math as much to the freedom I was given to actively inquire and explore mathematics, as much as to the many marvels I discovered in this way.

I recall being encouraged (by remarkable teachers) to explore, ask questions, and try to find answers to those questions. I was given a lot a lee-way, apart from classroom lessons, to pursue the connections and patterns I saw, to conjecture, and confirm conjectures, or find counterexamples. Given this encouragement and flexibility, I found mathematics to be akin to solving mysteries. I wondered about what I was learning, and was able to anticipate what this would lead to, before receiving formal instruction. And this was terribly satisfying: the wonder, the pursuit, the discovery, and even "invention" (for myself) of things I would later find to be true.

So in a sense, I discovered as much about math as I learned through formal instruction, and didn't get trapped into the mechanistic learn-a-rule/apply-the-rule/produce-an-answer mode which so many students come to define as "doing math."

So it wasn't so much a matter of the facts I learned that drew me to, and keeps me enamored by, math: it was/is the activities of mathematics: the process of questioning why certain relationships hold, conjecturing, exploring, testing, discovering and chasing down implications, constructing an understanding, and defending or rejecting my hypotheses, and on and on...

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First of all I must say that I really appreciate the idea of such a book. I wish I was exposed to such a book when I was younger as it was relatively late in my life (high school)I started appreciating mathematics. Anyway here is something I consider to be beautiful and simple, that you might find of interest:

The Pigeon hole principle and it's applications. The pigeon hole principle goes something like this:

Assume that you have some pigeons and some holes, and you want to put your pigeons into the holes, then if you have more pigeons than holes at least one of the holes must contain more than one pigeon. For example if I have 3 pigeons, but only two holes then one of the holes must contain at least two pigeons. The more mathematical way to state this is that if you have a set $X$ consisting of $n$ elements and another set $Y$ consisting of $m$ elements and $n > m$ then there cannot exist an injective function from $X$ to $Y$.

Now this statement is fairly obvious and I am sure most people can understand this. But this statement shows up a lot in various disguises.

Here is an example I think is pretty cool: Suppose a group of people are at a party. Each person may introduce himself/herself and shake hands with someone else at the party. I claim that there will always be at least two persons who have shaken the same amount of hands. Here is a proof of that statement:

Suppose there are $n$ people at the party. Then a given person can either shake $0, \;1 \ldots n-1$ different peoples hands. That is $n$ different possibilities, however if there is a person who shakes $0$ hands (that is he doesn't shake hands with anyone) then there can't be a person who shakes hands with $n-1$ persons (that is he shakes hands with everyone except himself), and conversely if a person shakes hands with everyone, then it is not possible that someone else doesn't shake hands with anyone. So there are only $n-1$ possibilities but there are $n$ people, so thinking of the people as pigeons and the possibilities as holes we see that we have $n$ pigeons and $n-1$ holes so at least two pigeons must go into the same hole, that is at least two people must shake the same amount of hands at the party.

You can read more about the pigeonhole principle here:

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I remember that when I was five, I made this reasoning "if I can write the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \space$ then I will be able to write all the numbers".

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For reasonable values of all. – Chris Cudmore Aug 9 '13 at 14:31

I guess this is not as special as the other ones, but this is how mathematics amazed me for the very first time:

I just turned 4 years old (I still vividly remember this), and my mother bought 4 cartons of eggs, a dozen per carton. My mother, trying to challenge me, asked how many eggs we bought in total, and after a short while I said 48 (I've always had a knack for arithmetic and I guess intuitively I knew it was $12 \times 4$). My mother was amazed, and she asked me how I did it. At this point I wasn't formally introduced to any mathematics (no multiplication and division, just basic addition and subtraction using our hands). So when I tried to show using my fingers how I got to 48 by taking 12 four times, it took me a lot longer, and my mother decided to teach me multiplication right then and there. This was the beginning of my interest.

The more I think about this story, the more beautiful it gets. I implement the lesson I learn everytime someone says mathematics is useless!! Ask them to do $12+12+12+12$ with their fingers.

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@AsafKaragila I agree, but it's the part of math everyone will get in school, and the part that everyone, no matter how 'uneducated', uses and finds useful. Of course, if one makes such a statement, you're probably not going to convince them showing beautiful mandelbrot sets, because those are fairly useless to a layman. – OmnipresentAbsence Mar 7 '13 at 18:14
@AsafKaragila No I don't, because music and poetry are barely ever mandatory, while mathematics is (everybody has to learn some mathematics, while music and poetry are hobbies). So convincing people poetry and music are useful is not needed. Of course, you can let people think what they think, but I think it's more beneficial to us all if people know how 'mathy' or world is. Comparing mathematics to music/poetry is like comparing biology to dancing, or physics to acting. – OmnipresentAbsence Mar 7 '13 at 18:20
If you've read the lament linked to in the question, you'll know that this sort of argument is exactly what kills mathematics education in schools. – Joe Z. Mar 8 '13 at 0:53

1) Modular arithmetic fascinated me. I could not believe that with just a few tools, I could find the remainder left when $3^{100}$ is divided by 8. ($3^2\equiv 1$ mod $8$ and hence the result.)

2) Euclid's proof of the infinitude of primes. (Let the number of primes be finite. Let them be $P=\{p_1,p_2,\dots,p_r\}$. Take $k=p_1p_2\dots p_r+1$. None of the primes in $P$ divides $k$, hence $k$ is a prime or divisible by a prime not in $P$, and so we have a contradiction.)

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Solving for an unknown. 2x = 4 so x = 2. Beautiful.

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It becomes hideous when forced to solve endless amount of them. – Phonics The Hedgehog Jun 2 '13 at 2:44

I remember my own observation about Pythagorean triples. I already knew that $3^2+4^2=5^2$ and $5^2+12^2=13^2$, and realized that the same trick can be done starting with any odd number $n$, and the other two will be serial numbers that add up to $n^2$.

For example, starting with $n=7$, we get $24+25=7^2$, and finally $7^2+24^2=25^2$.

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Hilbert's infinite hotel, the realization that $\mathbb{Z}$ is equinumerous with $\mathbb{N}$, and the uncountability of the set of all functions $\mathbb{N} \rightarrow \{0,1\}$. Basically: if it involves infinity, it's interesting.

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When I was young I found a riddle:

  • Think about a number
  • Multiply it by 3
  • Add 1
  • Multiply it by 3
  • Add the number you thought at the beginning
  • Tell me the result:)

The number you thought about is your result without the digit 3 at the end, so i.e. if your result is 53, then you thought about 5.

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This kind of joke was very commom im my school ^^ – Integral Mar 7 '13 at 15:43
@drjimbob: You're missing the "add the number you thought at the beginning" line, which makes the result be $(9x+3) + x = 10x + 3$. – ShreevatsaR Mar 24 '13 at 11:43

When I saw my first list of mathematical axioms (algebraic in this case). This was when I was 11.

  1. If the same quantity or equal quantities be added to equal quantities, their sums will be equal.
  2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal.
  3. If equal quantities be multiplied into the same, or equal quantities, the products will be equal.
  4. If equal quantities be divided by the same or equal quantities, the quotients will be equal.
  5. If the same quantity be both added to and subtracted from another, the value of the latter will not be altered.
  6. If a quantity be both multiplied and divided by another, the value of the former will not be altered.
  7. If to unequal quantities, equals be added, the greater will give the greater sum.
  8. If from unequal quantities, equals be subtracted, the greater will give the greater remainder.
  9. If unequal quantities be multiplied by equals, the greater will give the greater product.
  10. If unequal quantities be divided by equals, the greater will give the greater quotient.
  11. Quantities which are respectively equal to any other quantity are equal to each other.
  12. The whole of a quantity is greater than a part.

It was an almost religious experience, as in "if you take these on faith, the rest can be proven". I compared these to axioms of religious faith. There was, is, nor will there ever be any comparison.

In short, seeing this list sold me on rationality forever.

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"In short, seeing this list sold me on rationality forever." But what about those pesky Cauchy sequences that never converge? – Joe Z. Mar 8 '13 at 20:17
@Joe: SHUSH!!!!! – Stu Mar 12 '13 at 17:42

The possibilities of abstraction. This liberated me. Until I was about 13, I always had trouble with solving problems involving proportionality and inverse proportionality. Until I learned about variables. When I realized that you could just put a symbol instead of the number you don't know and just perform computations with it until everything simplifies in a way you can find back the number, I had a feeling of unstoppability.

The power of abstraction is so great that I'm very saddened by our current educational system in which it has nearly disappeared. All the students I get are struggling with symbols there were I have always seen them as my friends.

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When I was a kid I realized that $$0^2 + 1\ (\text{the first odd number}) = 1^2$$ $$1^2 + 3\ (\text{the second odd number}) = 2^2$$ $$2^2 + 5\ (\text{the third odd number}) = 3^2$$ and so on...

I checked it for A LOT of numbers :D

Years passed before someone taught me the basics of multiplication of polynomial and hence that $$(x + 1)^2 = x^2 + 2x + 1.$$ I know that this may sound stupid, but I was very young, and I had a great time filling pages with numbers to check my conjecture!!!

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One of the things I really like in math is the probability. One of the best examples that I like is on the film 21.

You are in a program show and you have three doors:

One: With the prize, and the other two with monsters;

The presenter tells you to pick up a door. When you finally choose a door, he asks "Are sure about it?

Then for some reason he decides to open one of the wrong doors and asks you: "Are you going to stay with your door or change it?" and he says "But remember I know where the prize is".

So what should have you do?

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Goats aren't monsters! D: – Joe Z. Mar 7 '13 at 21:33
Of course, switch. The first time I have 1/3 chance of being right. The other two doors 2/3. Now, one wrong door is opened, meaning that the remaining door has the unified 2/3 chance of being right. I am flappergasped how many people do not see this. This is where I demand them to envision this example with 1 million doors, you chose one at random. Then the moderator opens 999.998 other doors leaving you with your first choice, and the one that was not openened. Do you really think not switching would be smart? – k0pernikus Mar 8 '13 at 13:07
Isn't this called the Monty Hall Problem? – funkymushroom Mar 8 '13 at 16:28
I remember my teacher telling us this problem, but she had trouble explaining why it was that way. Most of my classmates didn't believe it was better to switch. But I wrote a Basic program on my Commodore 128 running 1000 iterations and found out that it really resulted in 50% chance of winning when switching doors. Several years later I stumbled upon a good explanation of why it is better to switch. – Anlo Mar 14 '14 at 17:51

$$\sqrt{\sqrt{\dotsb\sqrt{x}}} = 1$$

(or its more precise version: $lim_{n \rightarrow \infty} \sqrt[n]{x}$, for x positive)

As a kid I would always type in a number in my calculator and then keep hitting the square root key until the display went to 1. I would also do this with other keys on the calculator to see what would happen (some would blow up past the capacity of the floating point storage and some would go to 0, some to 1).

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Actually $\sqrt[n]{x}\to 1$, as $n\to\infty$. – Asaf Karagila Mar 8 '13 at 21:15

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