Invariant of 4 congruent triangles

I'm back to the usual content, everyone! Well, not really usual, since this problem is more of a puzzle than a problem, but you get the point. No more psychology articles for a while. Anyways, this puzzle comes from the Moscow Maths Olympiad (or the Mexican Maths Olympiad; I don't really know what MMO stands for when it comes to Maths) from 1995, and it goes something like this:

Suppose you have 4 congruent right-angled triangles. After each step, we cut one of the triangles along its height. Prove that, after any number of steps, there are still at least 2 congruent triangles.

Solution:

Let the division of the longer leg of the triangle over the smaller leg be equal to \(x\). Thus, when we cut a triangle along the height, we get two smaller triangles with areas proportional to \(x\) and \(1-x\). Let us presume that the initial area of the triangles is 1. At the first step, we have 4 triangles, which have \(x\) and \(1-x\) 0 times in the formula of their areas, so we write that 

$$S(0) = (0,0) (0,0) (0,0) (0,0).$$

After the first step, one triangle is split in two parts, one of area \(x\) and one of area \(1-x\), so 

$$S(1) = (1,0) (0,1) (0,0) (0,0) (0,0)$$ because \(x\) and \(1-x\) appear only once in the formula of the area of two triangles. Continuing this, we get that, at the p-th step, 

$$S(p) = (m_1, n_1) (m_2, n_2) ... (m_{p+4}, n_{p+4})$$ 

You may see where I'm going with this. We can represent all these pairs as positive lattice points. For two triangles to be congruent, their respective lattice points must be the same. Now, let us presume that, at the start of the game, we put 4 coins of value 1 on the lattice point (0,0). After the first step, we move half a coin on (1,0) and half a coin on (0,1), and, after each step, this process continues. In order for two triangles to be congruent, there must be 2 coins on at least one lattice point. We shall prove that, if there is one coin in all the lattice points, then the sum of values is smaller than 4, which should conclude the proof. 

The number of coins on each lattice point is dependent on the position of the lattice point. The coins in the point (m,n) have value \(2^{-m-n}\). Thus, we have to prove that 

$$\sum_{n∈ℕ} \frac {n+1}{2^n}<4$$

This series, as shown by Wolfram Alpha (https://www.wolframalpha.com/input/?i=sum+n%3D0...+infinity+(n%2B1)%2F2%5En), converges to 4, and is strictly increasing, so the value of the coins is always smaller than 4, which leads to a contradiction. The series could have been proven to be smaller than 4 by hand, but I preferred using Wolfram Alpha for a quicker result. 

Since the sum of values in all positive lattice points is 4, this means that at least a lattice point has 2 coins on it, which concludes the proof.

I found this problem really interesting, mainly because of the ingenuity required to solve it. I found it in Arthur Engel's excellent book 'Problem Solving Strategies' (did you really think I'd go as far as to search the Mexican Maths Olympiad from 1995?) while preparing for my interview at Oxford. Apologies for not posting anything for the past two weeks, I was in Oxford for about a week, and then I had to catch up with my homework, so it has been a pretty rough period. Tune in next time for the first post under the 'Theory' tag, which has been a long time coming! 

Cris.

A Cheerful Introduction to Jungian Functions

I haven't released an article in about a week... oh boy. I've been advertising this article for a while now, so I'll do all my best to make it count. Next article might be about an inequality, but, until then, the number of Maths posts on this blog remains the same.

In this article, I will introduce you (if you weren't already introduced) to Jungian Psychology and its successor, the Myers-Briggs Type Indicator, MBTI for short. Perhaps I will discuss Socionics in a sequel to this article (if there will be a sequel to this article). It isn't a pleasant thought to think that I am writing an article on personality types when I can't even decide what my type is...

Carl Jung, in his book 'Psychologial Types', discussed the problem of types in different contexts and observed several patterns. Other than historical contexts, Carl Jung observed these patterns in his work as a psychiatrist. Other than the classical 'Extrovert' vs 'Introvert' options, he also discovered something he called 'cognitive functions', which are basically ways of perceiving/judging the world. Each person has a personality formed by a permutation of the 8 possible cognitive functions, which I'll discuss shortly.

Each function is composed of two letters: the first letter is the preferred type of perceiving/judging, while the second letter explains how the user of the function uses it, whether he uses it in an introverted manner or an extroverted manner.

The perceiving functions do literally what the name says: they perceive information. A person can perceive information either by sensing or by intuition: the intuitive person will read between the lines, they process information through possibilities, hidden meanings and impressions, while the sensing person will perceive information through the body senses. The perceiving functions are:

• Ne: Extroverted intuition. This one is the impersonation of every mad scientist in movies. People with Ne as one of their first two functions are highly creative individuals. They are characterized by their need for debate and meaningful discussion, and also by their hatred for rigid environments. They are free-spirited individuals who enjoy life. 
• Ni: Introverted intuition. This one is the deepest of the cognitive functions, as it is intimately linked with the subconscious. Individuals with this cognitive function as one of their first two functions in the stack can use most of the brain in order to predict how events turn out, it basically works as a glimpse into the future, making these people look like prophets. People with this personality function are also the main antagonists in movies. Such humans have incredible mental landscapes, making Ni one of the most creative functions.
• Se: Extroverted Sensation. This one is the most adventure-seeking function. These people thrive on intense experiences, and people with this function as one of their first two functions in the stack are some of the most thrill-seeking people. This function basically lives in the present, and can notice details about recent events quite nicely.
• Si: Introverted Sensation. This one is the most calm of them all. People like this perceive new information through a prism of old experiences and sensations. This function is the most melancholic of them all, and also the most common one, as about 30-40% of the population has this function as either their primary or secondary function. This function is also correlated with good memory.

The judging functions, just like the perceiving ones, do literally what their name says. A person can judge given information through the lens of either feelings and values or logical reasoning. The judging functions are:

• Fe: Extroverted Feeling. This is widely considered as the most extroverted function, as it involves connecting on a personal level with most of the people a person knows. People with Fe among their first two cognitive functions are very popular and great talkers, they are empaths with a great desire to help people. However, people like this may be also manipulative, as they have an intuitive knowledge of what other people want. This function usually involves sync-ing in with other people's feelings and with the social norms, and it requires a great deal of socializing for it to flourish.
• Fi: Introverted Feeling. While the Fe function adheres to the social norms, this function draws values from within. People with Fi as the first are very idealistic and romantic, they are easily influenced by people, but they are mostly trying to do good.
• Te: Extroverted Thinking. This function is the one most concerned with efficiency. People with this function among their first 2 functions are obsessed by the efficiency of their actions, and all of their judgements are objective and impersonal.
• Ti: Introverted Thinking. This function is the one obsessed with problem solving (aka the part of me making this blog). It is derived from an internal logical system and thrives on solving problems from all possible subjects. 

These are, more or less, the cognitive functions. Each individual possesses, in some measure, all of the cognitive functions above. Jung believed that each individual has a primary and auxiliary function (hence why I talked about the first two functions), and these functions reflect the individual's attitude the most. In the next psychology article, I will discuss about the MBTI model of describing a personality, which is basically a consequence of the Jungian model. Keep in mind that, while there isn't much evidence to back up this theory, it resembles reality the most, which is why I decided to write about it. Another thing to mention is that no two individuals who have the same cognitive functions are the same. I've heard many arguments that categorizing people is wrong, which, in some cases, is true. However, in the case of psychological types, it's not wrong to categorize people, and it's a fairly logical thing to do. The cognitive functions don't describe WHAT a person thinks, but rather HOW a person thinks, which, in my opinion at least, makes the cognitive functions model almost foolproof to such arguments.

Cris.