Posted in Philosophy

## Truth And Lies

You are walking along the road when you reach a fork, where you see two guards chatting with each other. You approach them to ask which way you should go, but you notice a peculiar sign that says: “One of us always tells the truth, one of us always lies”. There is no way for you to know who tells the truth and who tells lies. If you could only ask one question to help you walk down the right path, what would you ask the guards?

Posted in Science & Nature

## Seven Bridges Of Konigsberg

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?

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).

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.

Posted in Science & Nature

Anyone who has studied mathematics to some degree will know about algebraic equations. An algebraic equation is an equation that can be solved to find the unknown value of x. A quadratic equation is an algebraic equation with , or in other words has two valid solutions to x. Generally speaking, a quadratic equation can be expressed in the following fashion: ax² + bx + c = 0. a, b and c are constants and the equation can be solved to find x. A quadratic equation is definitely more complicated to solve compared to a linear equation and it can be solved using various means and applications such as factorisation. As these methods are learnt in school and this Encyclopaedia is technically not a mathematics textbook, such methods will not be delved into.

If you have not learnt it already, there is a shortcut method to solving quadratic equations: the quadratic formula. This formula can easily find x if you simply substitute in the values for a, b and c. Of course this formula only works if the solutions are real numbers. The quadratic formula is as follows:

As you can see, because of the ± sign, the formula can be used to find both solutions to a quadratic equation. Even without factorising, it can find the answer as long as you substitute numbers into it on a calculator, making maths class very easy. However, as mentioned above the Encyclopaedia of Absolute and Relative Knowledge is not a mathematics textbook and one should instead learn properly from their teacher, not using the formula until they have been taught it properly.

Posted in Science & Nature

## Fermat’s Last Theorem

In the 17th century, a lawyer called Pierre de Fermat conjectured many theorems while reading a mathematics textbook called Arithmetica, written by an ancient Greek mathematician called Diophantus. He wrote his theorems on the margins of the books. After his death, a version of the Arithmetica with Fermat’s theorems was published and many mathematicians checked over Fermat’s proofs. However, there was one theorem that could not be solved. Fermat wrote on the theorem: “I found an amazing proof but it is too large to fit in this margin”.

Fermat’s last theorem is as follows:

No three positive integers x, y, and z can satisfy the equation
xⁿ + yⁿ = zⁿ for any integer value of n greater than two.

For example, x² + y² = z² can be solved using Pythaogorean triplets (e.g. 3, 4, 5) but there are no values for x, y and z that solves x³ + y³ = z³. This theorem remained unsolved for 357 years until Andrew Wiles finally found the proof in 1995.

There are many stories surrounding Fermat’s last theorem, but by far the most interesting is related to suicide. In 1908, a German mathematician called Paul Wolfskehl decided to kill himself after being cold-heartedly rejected by the woman he loved so much. He decided to shoot himself at midnight and in the remaining time started reading some mathematics texts until he found a flaw in Kummer’s theory, which disproved Cauchy and Lamé’s solution (the leading solution at the time. After Kummer’s essay, most mathematicians of the time gave up on Fermat’s last theorem). After researching Kummer’s essay, Wolfskehl found that it was far past midnight and he felt great pride in reinforcing Kummer’s solution. His depression was gone and through mathematics he found new meaning in his life. Wolfskehl, who believed that the theorem saved his life, made a resolution to donate his wealth to whoever solved Fermat’s last theorem, putting up 100,000 marks as a prize. This prize was claimed by Wiles in 1996 (then worth \$50,000).

Posted in Life & Happiness

## Why And How

When an obstacle blocks the way, the first response a person shows is thinking “Why did this happen? Whose fault is it?”. A person looks for the person responsible and ponders what appropriate punishment should be given.
In an identical scenario, an ant first thinks “How, and with whose help, can I solve this problem?”.
In the ant world, there is no concept of “crime”.

It is obvious that there is a great gap between people who ask themselves “Why didn’t this work?” and those that ask “How can I make it work?”.
In modern times, the world is dominated by people who ask “why”. However, in the future a day will come when the world is ruled by those that ask “how”.

(from The Encyclopaedia of Relative and Absolute Knowledge by Bernard Werber)

Posted in History & Literature

## Three Daughters

A man asked how old a man’s three daughters were. The father replied with the following statement.
“The product of their ages is 36.”
“It’s hard to determine their ages from just that.” the man asking replied.
“The sum of their ages is same as the number of my house.”
“I still can’t figure out the answer!” the man replied again.
“My eldest daughter is blonde.” the father said, and the man, now smiling, replied.
“Oh, is that so? Then I can figure out how old your daughters are.”

How old is each daughter? And how did the man figure it out?
A computer cannot solve this problem, as it can only be solved using human logic.

Posted in Life & Happiness

## Feynman Problem Solving Algorithm

Richard Feynman is a world-renowned genius physicist, famous for his ability of solving some of the most difficult problems in physics. He said that his intelligence was all thanks to his unique yet “normal” problem solving method, which he used to solve most of his problems. Here is the algorithm:

1.  Write down the problem.
2.  Think very deeply.