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1500 questions
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
- 773
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
11 answers
What is a simple example of an unprovable statement?
Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements.
I've seen a simple natural language statement here and elsewhere that's supposed to…
Michael Harris
- 1,075
96
votes
17 answers
What is the most elegant proof of the Pythagorean theorem?
The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield).
What's the most elegant proof?
My favorite is this graphical one:
According to…
Shane
- 111
96
votes
7 answers
Polynomials such that roots=coefficients
Here is my question :
Are there monic polynomials with degree $\geq 5$ such that they have the same real all non zero roots and coefficients ?
Mathematically, prove or disprove the existence of $n \geq 5$ such that $$\exists (z_1,\ldots, z_n) \in…
Gabriel Romon
- 35,428
- 5
- 65
- 157
96
votes
2 answers
Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$
$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$
The equality numerically holds up to at least $10^4$ decimal digits.
Can we prove that the equality is exact?
An…
Vladimir Reshetnikov
- 47,122
96
votes
11 answers
Demystify integration of $\int \frac{1}{x} \mathrm dx$
I've learned in my analysis class, that
$$ \int \frac{1}{x} \mathrm dx = \ln(x). $$
I can live with that, and it's what I use when solving equations like that.
But how can I solve this, without knowing that beforehand.
Assuming the standard rule…
polemon
- 1,041
96
votes
24 answers
100 blue-eyed islanders puzzle: 3 questions
I read the Blue Eyes puzzle here, and the solution which I find quite interesting. My questions:
What is the quantified piece of information that the Guru provides that each person did not already have?
Each person knows, from the beginning, no…
A Googler
- 3,355
96
votes
12 answers
How to convince a math teacher of this simple and obvious fact?
I have in my presence a mathematics teacher, who asserts that
$$ \frac{a}{b} = \frac{c}{d} $$
Implies:
$$ a = c, \space b=d $$
She has been shown in multiple ways why this is not true:
$$ \frac{1}{2} = \frac{4}{8} $$
$$ \frac{0}{5} = \frac{0}{657}…
user86484
- 821
96
votes
8 answers
Zero probability and impossibility
I read a comment under this question:
There are plenty of events that can occur that have zero probability.
This reminds me that I have seen similar saying before elsewhere, and have never been able to make sense out of it. So I was wondering
if…
Tim
- 47,382
96
votes
2 answers
Predicting Real Numbers
Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to see how others might think about it.
$100$ rooms…
Jared
- 31,451
96
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…
deinst
- 5,646
96
votes
4 answers
Do numbers get worse than transcendental?
Mathematicians have come up with many ways of classifying how "exotic" some numbers are. For example, the most ordinary numbers are the natural "counting" number, and the next most exotic numbers are zero and the negative integers. Next are the…
Franklin Pezzuti Dyer
- 39,754
- 9
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- 166
96
votes
2 answers
A goat tied to a corner of a rectangle
A goat is tied to an external corner of a rectangular shed measuring 4 m by 6 m. If the goat’s rope is 8 m long, what is the total area, in square meters, in which the goat can graze?
Well, it seems like the goat can turn a full circle of radius…
space
- 4,561