Tag: even

Posted in Science & Nature

Seven Bridges Of Konigsberg

Posted on by Jinavie

The city of Königsberg (capital of Prussia, now Kaliningrad, Russia) has the Pregel River running through the middle, with islands at the centre of the river connected by seven bridges. Is it possible to cross all of these bridges while only crossing them only once each?

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If you try to solve this problem, you soon discover that it is incredibly difficult not to cross the same bridge twice. But it is difficult to tackle this problem in a brute force manner. To calculate all of the permutations in the order of bridges, you use 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040, meaning that there are 5040 possible arrangements of bridges. Then how can you prove if the problem is solvable or not?

The great mathematician Leonhard Euler, upon being asked to solve the problem, is reported to have said that the problem is impossible to solve on the spot. In 1735, he proved his answer by modelling the seven bridges of Königsberg in a diagram of four dots connected by lines (representing the bridges).

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By using this model, the problem is converted into a “draw in one stroke” problem, which is also called a Euler walk to honour Euler’s contributions. Euler discovered many properties and laws regarding such problems. If a certain point is the starting point, then the line must first leave the point, then even if it comes back to the point, it must leave again. Ergo, the starting point must have an odd number of lines connected to it. The opposite applies to the ending point, where a line must enter the point, and if it leaves the point it must come back to it. Ergo, the ending point must also have an odd number of lines connected to it. In the case of a Euler walk, the starting and ending points are identical, so the number of lines is the sum of two odd numbers, making it an even number. Thus, to find out whether a picture can be drawn using one line, use the following laws:

  1. If there are no points of odd degree (odd number of lines), the starting and ending points are identical.
  2. If there are two points of odd degree, the starting and ending points are different.
  3. If there are one of more than two points of odd degree, it is impossible to draw using one stroke.

Thus, a Euler walk is only possible if there are 0 or 2 points of odd degree. Looking at the seven bridges of Königsberg problem, we can see that A is connected to 5 lines and B, C and D are connected to 3 lines each. As there are four points of odd degree, we have thus proved that it is impossible to draw a path that crosses all the bridges while not crossing any bridge more than once.

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Posted in History & Literature

Communism

Posted on by Jinavie

A professor was lecturing about communism. The students insisted that communism worked since no one would be poor and no one would be rich – a great equalizer. To show the students whether communism worked or not, the professor designed a social experiment. The professor announced to the class that all grades will be averaged and everyone will receive the same grade.

After the first test, the grades were averaged and everyone got a B. The students who had studied hard were upset while the students who had studied very little were happy.
But when the second test came, the students who had studied little studied even less and the ones who had studied hard before decided that since they could not make an A (even if they got 100%, the bottom half of the class would pull the average down), they also studied less. The second test average was a D.
No one was happy. By the third test, the class average had fallen to an F. Now no one was rich, but everyone was equally “poor”.

Communism ultimately fails because it removes one of the major driving forces of economies – incentive. Money is a key (but not the only) incentive that motivates people to work harder as they are rewarded with a better quality of life. But when private wealth is abolished and wealthy is “equally” (note that it is not “equitably”) split among the population, there is no longer motivation to try harder. Because no one wants to work for the benefit of strangers (due to psychological phenomena such as responsibility splitting and the monkeysphere), the economy does not grow and everyone becomes poor.

Communism essentially relies on the goodwill of the people while disregarding all realistic factors of economic growth. It is an extreme ideal that has failed in every instance in history. The only times communism works is in a small society setting such as a small village or tribe where there are less than 150 people (within the monkeysphere, thus people actually care about all of the others). The idea behind it is admirable – that all human beings have equal rights to a certain quality of life. However, it disregards the most important factor of economics; that resources are scarce and we cannot fulfil the infinite needs of everyone. Monetary incentives at least allow us to seek out these resources ourselves, with market economies rewarding such behaviour with money.

Thus, we should not focus on “equality”, which means that everyone should receive the “same”, but “equity”, which means that everyone should receive a “fair share” according to how much they have worked and contributed to the society.