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1500 questions
97
votes
6 answers
How was the normal distribution derived?
Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as make sure that the area under the curve was…
Cisplatin
- 4,855
97
votes
6 answers
Can you make a sphere out of a plane?
I had this idea to build a model of Earth in Minecraft. In this game, everything is built on a 2D plane of infinite length and width. But, I wanted to make a world such that someone exploring it could think that they could possibly be walking on a…
986
- 1,438
97
votes
5 answers
Cutting sticks puzzle
This was asked on sci.math ages ago, and never got a satisfactory answer. Given a number of sticks of integral length $ \ge n$ whose lengths add to $n(n+1)/2$. Can these always be broken (by cuts) into sticks of lengths $1,2,3, \ldots ,n$? You are…
deinst
- 5,736
97
votes
2 answers
Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing
Prove without calculus that the sequence $$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$ is strictly decreasing.
user 1591719
- 45,325
97
votes
2 answers
When does a sequence of rotated-and-circumscribed rectangles converge to a square?
Recently I came up with an algebra problem with a nice geometric representation. Basically, I would like to know what happens if we repeatedly circumscribe a rectangle by another rectangle which is rotated by $\alpha \in \left( 0, \frac {\pi}…
samgiz
- 774
97
votes
5 answers
Reversing an integer's digits is multiplicative for small digits
So my 7 year old son pointed out to me something neat about the number 12: if you multiply it by itself, the result is the same as if you took 12 backwards multiplied by itself, then flipped the result backwards. In other words: $$12 × 12 =…
BCA
- 813
97
votes
8 answers
Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
I'm having trouble computing the integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$ I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as $u=\sin(x)$ and $u=\cos(x)$, but it was not very…
Michael Li
- 2,291
97
votes
3 answers
Modelling the "Moving Sofa"
I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here . In this question I am going to rotate the L shaped hall instead of moving a sofa around the corner. By rotating the hall…
newzad
- 4,963
97
votes
4 answers
The "pepperoni pizza problem"
This problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely randomly. All pepperoni have the same radius $r$. A…
Ben S.
- 505
97
votes
7 answers
Why are all the interesting constants so small?
A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is there some kind of normalization going on, or some…
Greg L
- 1,865
97
votes
1 answer
Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc... & Representation Theory of Special Functions
Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question here . The gist of the theory is as follows: The…
bolbteppa
- 4,669
97
votes
8 answers
A map is continuous if and only if for every set, the image of closure is contained in the closure of image
As a part of self study, I am trying to prove the following statement: Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$, where…
Holdsworth88
- 8,988
97
votes
7 answers
What is special about the numbers 9801, 998001, 99980001 ..?
I just saw this post , and realized that 1/9801 =…
Lazer
- 1,635
96
votes
6 answers
$x^y = y^x$ for integers $x$ and $y$
We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
Paulo Argolo
- 4,400
96
votes
5 answers
Intuition behind using complementary CDF to compute expectation for nonnegative random variables
I've read the proof for why $\int_0^\infty P(X >x)dx=E[X]$ for nonnegative random variables (located here ) and understand its mechanics, but I'm having trouble understanding the intuition behind this formula or why it should be the case at all.…
bouma
- 1,183