708
votes
25answers
113k views

How long will it take Marie to saw another board into 3 pieces?

So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long ...
697
votes
53answers
417k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...
616
votes
160answers
38k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

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 ...
564
votes
16answers
52k views

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
500
votes
25answers
58k views

Splitting a sandwich and not feeling deceived

This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an ...
490
votes
27answers
116k views

How to study math to really understand it and have a healthy lifestyle with free time?

Here's my problem. I'm studying math and when I really work hard, I think I understand things very good, but that comes at a big cost: in the last few years, I've had practically zero physical ...
418
votes
17answers
40k views

Is value of $\pi = 4$?

What is wrong with this? SOURCE
418
votes
6answers
70k views

Why can you turn clothing right-side-out?

My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the ...
398
votes
30answers
46k views

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
394
votes
10answers
402k views

Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
394
votes
20answers
65k views

Mathematical difference between white and black notes in a piano

The division of the chromatic scale in $7$ natural notes (white keys in a piano) and $5$ accidental ones (black) seems a bit arbitrary to me. Apparently, adjacent notes in a piano (including white or ...
377
votes
6answers
26k views

Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?

So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}} $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as ...
376
votes
12answers
103k views

Does Pi contain all possible number combinations?

I came across the following image, which states: $\pi$ Pi Pi is an infinite, nonrepeating (sic) decimal - meaning that every possible number combination exists somewhere in pi. Converted ...
373
votes
21answers
65k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
358
votes
11answers
23k views

My son's Sum of Some is beautiful! But what is the proof or explanation?

My youngest son is in $6$th grade. He likes to play with numbers. Today he showed me his latest finding. I call it his "Sum of Some" because he adds up some selected numbers from a series of numbers, ...
357
votes
33answers
22k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
352
votes
10answers
55k views

What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
350
votes
7answers
13k views

“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
345
votes
35answers
41k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
303
votes
0answers
14k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
288
votes
23answers
38k views

Proofs that every mathematician should know? [closed]

There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge etc. So I'd like to know what ...
288
votes
29answers
44k views

My sister absolutely refuses to learn math [closed]

My 13-year-old sister has a problem which, given the way math is currently taught, I doubt is anything but all too common. She has a low grade in her math course and only ever attempts to memorize ...
275
votes
21answers
15k views

On “familiarity” (or How to avoid “going down the Math Rabbit Hole”?)

Anyone trying to learn mathematics on his/her own has had the experience of "going down the Math Rabbit Hole". For example, suppose you come across the novel term vector space, and want to learn more ...
269
votes
33answers
34k views

Pedagogy: How to cure students of the “law of universal linearity”?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} ...
254
votes
14answers
30k views

Find five positive integers whose reciprocals sum to $1$

Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was ...
249
votes
7answers
78k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
236
votes
6answers
48k views

In Russian roulette, is it best to go first?

Assume that we are playing a game of Russian roulette (6 chambers). Assume that there is no shuffling after the shot is fired. I was wondering if you have an advantage in going first? If so, how big ...
235
votes
23answers
8k views

Why don't we define “imaginary” numbers for every “impossibility”?

Before the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had, so we said $i$ to be a new type of number. How come we don't do the ...
234
votes
29answers
18k views

A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
231
votes
15answers
26k views

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
229
votes
7answers
67k views

Multiple-choice question about the probability of a random answer to itself being correct

I found this math "problem" on the internet, and I'm wondering if it has an answer: Question: If you choose an answer to this question at random, what is the probability that you will be correct? ...
228
votes
8answers
13k views

The length of toilet roll

Fun with Math time. My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
226
votes
4answers
23k views

The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
223
votes
13answers
194k views

Fourier transform for dummies

A vague question of Kevin Lin which didn't quite fit at Mathoverflow: So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)? ...
220
votes
7answers
14k views

Best Sets of Lecture Notes and Articles [closed]

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
219
votes
15answers
10k views

Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantage/disadvantages of proving ...
216
votes
11answers
20k views

Is '10' a magical number or I am missing something?

It's a hilarious witty joke that points out how every base is '10' in its base. Like, 2 = 10 (base 2) 8 = 10 (base 8) My question is if whoever invented the ...
215
votes
41answers
30k views

Riddle: 1 question to know if the number is 1, 2 or 3

I've recently heard a riddle, which looks quite simple, but I can't solve it :/ The girl thinks of a number which is 1, 2, or 3 The boy then asks just 1 question about the number The girl can only ...
213
votes
29answers
20k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
213
votes
13answers
28k views

Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?

Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet." I have a (not very mathematically intelligent) friend who ...
212
votes
64answers
50k views

'Obvious' theorems that are actually false

It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true". Now, I know a fair few examples of things that are obviously true and that can be proved ...
207
votes
92answers
18k views

Surprising identities / equations [closed]

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...
205
votes
23answers
17k views

Zero to the zero power - is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 \cdot 0^x = 1 \cdot ...
203
votes
65answers
14k views

Funny identities [closed]

Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?
197
votes
32answers
11k views

Can't argue with success? Looking for “bad math” that “gets away with it”

I'm looking for cases of invalid math operations producing (in spite of it all) correct results (aka "every math teacher's nightmare"). One example would be "cancelling" the 6s in $$\frac{64}{16}.$$ ...
197
votes
4answers
84k views

What is the intuitive relationship between SVD and PCA

Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional dataset into fewer dimensions while retaining important ...
194
votes
16answers
8k views

Optimizing response times of an ambulance corp: short-term versus average

Background: I work for an Ambulance service. We are one of the largest ambulance services in the world. We have a dispatch system that will always send the closest ambulance to any emergency call. ...
192
votes
28answers
27k views

Too old to start math [closed]

I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some ...
192
votes
5answers
20k views

Is $7$ the only prime followed by a cube?

I discovered this site which claims that "$7$ is the only prime followed by a cube". I find this statement rather surprising. Is this true? Where might I find a proof that shows this? In my ...
189
votes
24answers
15k views

Is mathematics one big tautology?

Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties ...

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