## Top new questions this week:

### Why do differentiation rules work? What's the intuition behind them? (Not asking for proofs)

Differentiation rules have been bugging me ever since I took Basic Calculus. I thought I'd develop some intuitive understanding of them eventually, but so far all my other math courses (including ...

(calculus) (real-analysis) (derivatives)

### Is it possible to represent every irrational number as a (limit of) an infinite sum of rational numbers?

For instance, we can certainly represent π in this fashion. $$\frac{\pi}{4} \;=\; \sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} .\!$$ $\ln(2)$ is also irrational. And even that can be represented as an ...

(irrational-numbers)

### Expected number of people to not get shot?

Suppose $n$ gangsters are randomly positioned in a square room such that the positions of any three gangsters do not form an isosceles triangle. At midnight, each gangster shoots the person that is ...

(probability) (recreational-mathematics) (puzzle) (average) (geometric-probability)

### Have there been (successful) attempts to use something other than spheres for homotopy groups?

Homotopy groups are famous invariants in algebraic topology. They have a myriad of wonderful properties: For $n \ge 1$, $\pi_n(X,*)$ is a group; for $n \ge 2$, this group is abelian. $\pi_n$ defines ...

(algebraic-topology) (homotopy-theory)

### why $\rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$

Let $n\ge 7$ be positive integers,show that $$f(n)=\rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$$ Anyone know this problem background?or maybe have best proof or best result?

(number-theory) (inequality) (prime-numbers) (approximation) (least-common-multiple)

### Can we recover a space from its continuous functions?

Let $X$ be a topological space and let $\mathcal{F}(X,\Bbb R)$ be the set of continuous function from $X$ to $\Bbb R$. Can we recover the topology of $X$ by only the knowledge of ...

(general-topology)

### Group with two generators of order 3 is finite

A group $G$ is generated by two elements $a$ and $b$ such that for any $g\in G:$ $g^3=e$. Show that $G$ is finite. I don't understand how this is so. I don't know if $G$ is abelian, so I can ...

(abstract-algebra) (finite-groups)

(riemann-zeta)