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1500 questions
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votes
11 answers

Expected number of unpecked chicks - NYT article

In this article, the winner of the math competition answered this question correctly: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of…
AAC
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89
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7 answers

If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$

Let $a$ and $b$ be positive numbers such that $a+b=1$. Prove that: $$a^{4b^2}+b^{4a^2}\leq1$$ I think this inequality is very interesting because the equality "occurs" for $a=b=\frac{1}{2}$ and also for $a\rightarrow0$ and $b\rightarrow1$. I tried…
Michael Rozenberg
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89
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7 answers

Number of ways to write n as a sum of k nonnegative integers

How many ways can I write a positive integer $n$ as a sum of $k$ nonnegative integers up to commutativity? For example, I can write $4$ as $0+0+4$, $0+1+3$, $0+2+2$, and $1+1+2$. I know how to find the number of noncommutative ways to form the sum:…
Yellow
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89
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5 answers

How to solve these two simultaneous "divisibilities" : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I cannot progress anymore. Thanks !
user14479
89
votes
7 answers

In what sense are math axioms true?

Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers. The kid asks: why? Well, it's an axiom. It's called commutativity (which is not even true for most groups). How do I "prove" the axioms? I can say, look, there are $3$…
user4951
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89
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2 answers

Determining answers to a true/false test by guessing optimally ($k2^{k-1}$ questions, $2^k$ attempts)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the real exam. After each mock exam the teacher tells…
Sergio
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88
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31 answers

What are some conceptualizations that work in mathematics but are not strictly true?

I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to iteratively learn and unlearn along the way. I'm…
MGA
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88
votes
3 answers

Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by cubing both sides and using $x = \sqrt[3]{2}$ for…
Paramanand Singh
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88
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1 answer

Does $X\times S^1\cong Y\times S^1$ imply that $X\times\mathbb R\cong Y\times\mathbb R$?

This question came up in a recent video series of lectures by Mike Freedman available through Max Planck Institut's website. He proves the "difficult" converse direction, that $X\times \mathbb R\cong Y\times \mathbb R$ implies $X\times S^1\cong…
Cheerful Parsnip
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88
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
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88
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1 answer

Can someone explain this integration trick for log-sine integrals?

I was working on this rather challenging log-sine integral: $$ \int_{0}^{2\pi}x^{2}\ln^{2}\left(2\sin\left(x \over 2\right)\right)\,{\rm d}x = {13\pi^{5} \over 45} $$ The upper limit is a waiver from the norm of $\frac{\pi}{2}$. Anyway, when…
Cody
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88
votes
5 answers

Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top

Having a bit of a problem calculating the volume of a take-away box: I originally wanted to use integration to measure it by rotating around the x-axiz, but realised that when folded the top becomes a square, and the whole thing becomes rather…
Nemui
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88
votes
9 answers

Terence Tao–type books in other fields?

I have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that there is no big list of exercises at the end of…
Ovi
  • 23,737
88
votes
6 answers

Any rectangular shape on a calculator numpad when divided by 11 gives an integer. Why?

I have come across this fact a very long time ago, but I still can't tell why is this happening. Given the standard calculators numpad: 7 8 9 4 5 6 1 2 3 if you dial any rectangular shape, going only in right angles and each shape consisting of 4…
noncom
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88
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12 answers

Why determinant of a 2 by 2 matrix is the area of a parallelogram?

Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$. How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$? Are the areas of the following parallelograms the same? $(1)$ parallelogram with…
user
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