# What is the shape of a hanging chain?

You may think that the graph of $\cosh{x}$ looks like a parabola, but it is slightly flatter. It is called a catenary, which is the shape formed by a hanging chain.

You may think that the graph of $\cosh{x}$ looks like a parabola, but it is slightly flatter. It is called a catenary, which is the shape formed by a hanging chain.

You have learnt Poisson distribution in AS Further Statistics but you may still have many questions. Where does that complex probability mass function come from? Why it has the same variance and expectation?

This post will answer all these questions.

PrefaceYou have learned how to prove $\sqrt{2}$ is an irrational number, how about another famous number $\pi$? The number $\pi$ is very useful in our daily life. You need it to calculate the perimeter and area of a circle, you need it in trigonometry and geometry as well. You might have heard that just like $\sqrt{2}$, $\pi$ is an irrational number too. But how can you prove this? Let’s start our journey from an easier question: How to define number $\pi$?

PrefaceI have been complaining about the course design of A Level Mathematics and Further Mathematics for quite a long time. The A Level syllabus is not “friendly enough” for students who are interested in pure mathematics. For example, it teaches you the “algorithm” to work out the inverse of a matrix but never tells you why it is correct; it teaches you how to find the derivative of a function without strictly define the limit of a function.

The smallest even number greater than $2$ that cannot be expressed as the sum of two prime numbers is ____. The maximum prime number which cannot be expressed in the form $n^2+1 (n\in\mathbb{Z})$ is ____. Give any $n (n>2)$ points $A_1, A_2, …, A_n$ on the plane, the distance between any two points is less than 1, than $\max{\min_{1\leq i< j \leq n}{A_iA_j}}=$____. The equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has no positive integer solutions for $x,y,z$, then the minimum value for the positive integer $n$ is ____.

PrefaceThe United Kingdom Mathematical Olympiad for Girls is, well apparently, according to its name, a math competition for girls specifically. Which also means it is a little bit easier than the BMO Round 1, so it is a good opportunity for me to have an attempt of writing a serious solution for such competitions. I may keep posting my own solutions to some BMO or IMO questions, or may not.

The beginning of the story A fun journey to an integer sequence.
Everything started from the Puzzle 137 from www.puzzleoftheweek.com. I was not interested in those kindergarten-ish questions at first, but found this one is quite fun at the end.
StarterCounting the number of the triangles is pretty easy, and I recommend this puzzle to your any 3 year old cousins to “develop their early mathematical thinking”.
There are 27 triangles when the height is 4:

Hi There~Planet Cloud is alive again!
This time, this blog will be more serious. Instead of sharing musics and funny stories, more notes on academic experiments will be posted, as well as some trivial daily thinking will be shared.
Chinese content will not be provided any more.
Hope you enjoy it!!