Maths does not need colours

If the world would be in black and white maths was exactly the same that it is.

There is indeed no need of colours in maths at all. The basic fact that justifies this is that we can code maths with only zeros and ones, or in other words,  in black and white.  In this sense, maths is a science that do not “see in colours”.

In practical terms, if we could create a mathematical machine to simulate a human being it should be unable to see colours, however it could distinguish among them. Let me explain this. It is easy to write a program in order to distinguish colours, and surely it will work well in most of the cases. However, distinguish is different from see. If we take a look to what the machine really sees whenever it watches a colour, we should realise that it is indeed an image in black and white of what a colour is. The reason is that the machine understands in mathematical terms. This makes a huge difference between our perception of the word and the one of a mathematically made machine.

Paradoxically, it is usually said that teaching mathematics needs colours. This is because we are not mathematically programmed machines, moreover, colours allows us to incorporate, understand, remember and differentiate mathematical objects (surely a phycologist could say more about this). More in general, we are used to perceive our surrounding using colours, so, it is not very surprising that representing maths with colours do help us to understand them better (this is an interesting fact, that many maths professors could exemplify vasty).

It is a bit sad that a colour is indeed a not definable object in mathematics, so a proper definition of something so trivial is far beyond what we (mathematicians) can define. Defining a colour is a problem that more successfully than we, physicists, artists, phycologists, physicians and writers have accomplished, however, its fully comprehension looks at least intriguing.

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Why maths and life are two different things?

Of course I do not intend to even try to answer this question, rather, I will explain an example that happened to me yesterday.

Imagine there are three lifts on the same wall.  You are in a rush (the stairs are far away not to be considered) and you need to stand up somewhere in order to take the lift as quick as possible. Where should you stand?

 |Lift|________|Lift|_____________________|Lift|
|
|
|                          0H   <-YOU
|
|

Mathematically the answer is simple if you consider that each lift has the same probability of arriving. In this case you should locate in front of the lift in the middle.

In the real life, however, this is far from being optimal. First of all, it is not very polite. Secondly, you may not been able to see if one of the other lifts arrive before. So a better answer is to consider the smallest distance from the lift in the middle that allows you to see the other two lifts. This answer uses the mathematical answer, but it improves it accordingly the real word.

This example clearly shows that maths and life are two different things, however, maths can help you to improve a decision in real life if you use it carefully.

Bingo winning strategy

I will write about a problem by my friend Edgardo Roldán Pensado. He told me it long time ago. If you are a mathematician the solution may be trivial, however, I personally admire the way he faced the situation and came out with this nice problem.

I won’t bore you with a longer introduction, let start the problem.

Suppose you bought two days ago (in amazon.com for example) a Royal Bingo Supplies Wooden Bingo Game (like the one in the picture)

Bingo

  • Wooden Bingo Game Set with instructions.
  • Perfect for old-fashioned fun with a nostalgic twist.
  • Includes with 18 Bingo cards, 150 Bingo chips, a Bingo board, brass cage and 75 wooden balls.
  • Great for parties, barbeques or family game nights.
  • Recommended for ages 3 and up.

Today you received your game and you invite seven other friends to play together. Each player takes a bingo card and you start playing. Five hours later, when everybody is already tired of playing, you decide to count the times each player has won. Everybody is very  surprised how lucky a single player was, who won many more times than anybody else.

Is there a Bingo winning strategy? Or do you have a very lucky friend?

Well…Everybody has a very lucky friend, so no discussion about this. However, there is also a Bingo winning strategy, so, there is a chance the winner was a Bingo´s tactician (with a bit of luck).

How does the Bingo´s tactician play?

This guy chose his bingo card at the end, after analysing the bingo cards of the other player. Each card was (hopefully, I do not know if it is true) created with a uniform random distribution (for each square you choose a number from 1 to 75 with probability 1/75.). He considered a metric on the set of bingo cards, for example, given two bingo cards a=(x_1,\ldots,x_{24}) and b=(y_1,\ldots,y_{24}), d(a,b)=\#\{i:x_i\neq y_i\}, with the convention that \# \emptyset :=0. He was a bit lucky enough to be able to find a bingo card that maximises the d distance with respect to the bingo cards of the other player.

Why is this a winning strategy?

Suppose that the 7 bingo cards chosen by the opponents of the winner were very close with respect to the distance d, moreover, suppose that all the 7 bingo cards were exactly the same. On the other hand, the “lucky one” chose a different bingo card. So there is \frac{1}{2} probability one of the other seven players (and then all) wins, and \frac{1}{2} probability the “lucky one” wins. Now, suppose, the 7 bingo cards chosen by the opponents are very close (with respect to d), but all different. Then the “lucky one” player wins with probability close to \frac{1}{2}, whilst  and the other players with probability close to \frac{1}{14}.