Whiteboard

Welcome to the whiteboard!

In this page you will find listed all ideas I have for future posts! You can leave a comment with new ideas or suggesting that I focus on finishing a specific post.
By giving me your input, I will understand better what are the things you enjoy reading the most. The ideas I present are separated into some categories and, inside each category, the ideas are ordered from most recent to oldest.

Please understand that this is just a sneak peek into my brain and some of my ideas. Not every post will go through this listing and (unfortunately) it is likely that not every idea in these lists will become a post.

Problems

The list that follows has an overview of the problems I am thinking of including. If you want to check all problems I have published already, you can do so here.
  • Present a problem based on one of MathGurl's videos.
  • A weird hotel with $N$ rooms has an even weirder habit that involves all guests in all rooms. The problem has to do with number theory and asks us to count how many rooms in the hotel will satisfy a given property.
  • A riddle with pirates, a monkey, some coconuts and a desert island. Using tools from number theory we have to find the least natural number satisfying a property deduced from the problem statement.
  • A logic riddle on how to determine the size of a very special, circular train.
  • Split the set of the first $2^n$ non-negative integers into two sets that satisfy some nice properties.

Twitter proofs

The following list has the themes for all the twitter proofs I already thought of. You can read all the published twitter proofs here .
  • a proof that the interpolating polynomial (of degree $n $) of a set $\{(x_i, y_i)\}_{i=0}^{n} $ of points is unique;
  • by using elementary properties of real polynomials, show that all polynomials of odd degree have at least one real root;

Tutorials

These are the ideas for programming and mathematics tutorials I have thought of writing. Here you find all the published tutorials.
  • As a continuation to this post, write about simple properties of groups and how they work, with a special focus on the group of permutations on 3 symbols, $S_3$.
  • An introduction to the mathematical tool that is induction, already used in some problems (just check the label "induction" here) with a couple of practical examples.

Pocket maths

The posts on pocket maths are supposed to be short, simple, and about the mathematics that is around us. These are all the pocket maths posts in the blog.
  • Showing how one can create a simple magic trick with a grid of numbers.
  • A post on an excellent trick to become really fast to calculate cubic roots of some numbers.
  • Folding a piece of paper in half is easy to do with some rigour, as opposed to folding a piece of paper into thirds (it requires you to be a bit more handy). This post will show why it is impossible to crease a sheet of paper at 1/3 of its length by using successive "halving folds".
  • The method I use to compute averages mentally.

Misc

Whatever isn't included in any of the previous categories ends up listed here! I will try to be as explicit as possible, when describing my idea for the post.
  • Sharing with you a spatial memory game I created with Python and pygame.
  • Creating a 3D version of the well-known snake game. And by 3D, I mean that the snake can also move up and down.
  • Using the Thue-Morse sequence to generate some interesting turtle graphics.
  • Writing a little bit about the algorithm used in here to generate mazes and implement it.
  • Giving a formal definition of what is P and NP in the famous P vs NP problem.
  • Share a local version of the online game agar.io
  • Writing about the immensely weird Conway base-13 function.

Popular posts from this blog

Pocket maths: how to compute averages in your head

Pt En Being able to do basic arithmetic calculations in your head is a great skill. Not because it is sexy but because it is useful in your daily life: it can help you check the change you are given when shopping, it can help you know if you will have enough money to pay for your groceries, it can help you estimate how much things cost after the discounts, etc... This often reduces to being able to sum and subtract decently; sometimes you need to make a couple of small multiplications, but that is it. More likely than not, you don't need to compute averages every day. But sometimes you just want the scoring average of your team for the past few games, or the average price per person of a given meal, or the average time you spent stuck in traffic this past week... And averages may appear nastier than simply adding or subtracting, because averages also require you to perform a division: in fact, you have to add all the numbers you want and then divide the total by ho...
Read more

Tutorial on programming a memory card game

Image
Pt En Este post vai ser um tutorial, não muito detalhado, sobre como fazer um jogo de memória com Python e pygame. O jogo que vamos implementar é um jogo comum: viramos uma série de cartas para baixo e temos que as virar duas a duas, tentando encontrar os pares. Claro que quando viramos duas cartas que não são um par, temos de as voltar de novo para baixo. Quando estou a criar um jogo, gosto de o ir desenvolvendo por etapas funcionais: partir o processo em várias fases que representem pontos nos quais eu tenho algo que posso testar. Deste modo, não só o processo se torna muito mais interessante, como posso ir controlando o aspeto do que estou a produzir. Deixo de seguida uma lista das etapas que eu pensei para este projeto; cada ponto da lista descreve a funcionalidade que o jogo já suporta: Criar um ecrã onde mostro todas as cartas dispostas, face para baixo; Clicar em cima de uma carta faz com que ela se vire para cima; Clicar na segunda carta verifica se encontrei...
Read more

Markov Decision Processes 02: how the discount factor works

Image
Pt En In this previous post I defined a Markov Decision Process and explained all of its components; now, we will be exploring what the discount factor $\gamma$ really is and how it influences the MDP. Let us start with the complete example of last post: In this MDP the states are Hungry and Thirsty (which we will represent with $H$ and $T$) and the actions are Eat and Drink (which we will represent with $E$ and $D$). The transition probabilities are specified by the numbers on top of the arrows. In the previous post we put forward that the best policy for this MDP was defined as $$\begin{cases} \pi(H) = E\\ \pi(T) = D\end{cases}$$ but I didn't really prove that. I will do that in a second, but first what are all the other possible policies? Well, recall that the policy $\pi$ is the "best strategy" to be followed, and $\pi$ is formally seen as a function from the states to the actions, i.e. $\pi: S \to A$. Because of that, we must know what $\pi(H)...
Read more