# Colouring Proofs

Today i will introduce colouring proofs. For this i will give the classic example.

Problem:
Let a chessboard be given. The two black corners are cut off. Is it possible to completely cover the board with domino bricks.

Sollution:
No. A domino brick on the board always  covers one black and one white tile. Since there are more white tiles than black tiles, it is impossible to completely cover the board with dominoes.

# Huygens Problem

A few months ago, I stumbled a pond a new problem solving idea. It is called the telescope method. I did know about it but I just used it subconsciously. I want to present a rather easy problem that can be solved with this method. In 1672 Huygens asked Leibnitz to solve this problem, which he did using the same method.

The problem is this: What is the sum off all reciprocals off the triangle numbers.

First we find a formula to express a triangle number. it is well known that this formula is:        (n*(n+1))/2 . Let x be the solution to the problem. Let dn denote the nth triangle number.

Now there is a nice trick that you can use:

This has a great benefit for infinite sums. We add this into the equation above and get:

Lets se what happens to the first few n. 1/1-1/2+1/2-1/3+1/3-14+1/4-1/5. As you can see, everything except the 1/1 cancels itself out. 1/1+(-1/2+1/2)+(-1/3+1/3)+(-14+1/4)+(-1/5…  =1/1+0+0+0+0….

We have 1=1/2x. Simple algebra gives us x=2, which is the solution to the problem.

# The Farmer Problem

The problem goes like this:

There are 2 points on a plane, which are not the same. One represents a farmer and the other point represents a pig. Then there is a straight line on the plane, which does not go through either of the points. The line represents a river. The farmer wants to give the pig water. To do his, he has to first go to the river and then to the pig. How can you find the fastest way for him to do so.

The answer is astonishingly easy. You have to reflect the (pig) point at the river as in the picture (a=b) :

It is known, that the shortest path between two points on a plane is a straight line. Now we connect the farmer with the reflected point is a straight line. Then we call the point where the line cuts the river c. Now we just have to connect the farmer with c and the pig with c. This is the shortest path, since the left side of the river was just reflected to the right side and we had the shortest path before we reflected back..

# Life or Death problem

There are 2 doors in a room. Behind one of them there is hell and behind the other one is heaven. There is a person in front of each of the doors. They know where heaven is and where hell is. One of them always lies and the other one always says the truth. Can you choose the door to heaven if you are only allowed to ask one question?

Yes. You have to ask one of the persons, if the other person would say that heaven is behind door 1.

Proof:

Lets assume door 1 is heaven. If you ask the liar he would answer no since the other person would say yes to the question if door 1 is heaven. If you ask the person that says the truth, he would also say no since the other person would say no to the question if door 1 is heaven.     Lets assume door 1 is hell. If you ask the liar he would answer yes since the other person would say no to the question if door 1 is heaven. If you ask the person that says the truth, he would also say yes since the other person would say yes to the question if door 1 is heaven.     To conclude, door one is heaven if the answer to the question is no otherwise it is hell.

# A Math Problem

Let ABCE be a square. Let D be a point, so that CED is right angle and ED has the same length as CD. Use 2 straight lines to cut the pentagon into three pieces and arrange these, so that they form a square.