One week of mathematics, #4

it was thursday, the last whole day on math course. It was snowing all the time if I remember correctly. Again we had two morning lectures and then one evening.

The first one was one of the best there, it was about combinatorics and random walk across square grid. I mean, really cool, lets see:

Take a grid.
Now imagine you want to get from top left point to right down corner at the least number of steps. In those steps you always want to move to right or down, so every time you get closer. You can only move across black lines, now lets look how many ways there are.

OK, look at the next picture. There are clearly too many ways to get there, coin-gridwe need some bette way to do this.coin-grid2

Lets number those crossroads.

You start at “1”. There are only two options to go. You can go vertically or horizontaly. So there is only one way to get to point 2 and 8. When you want to get to point 9 you could either go through 2 or 8 so there are two ways to get there, as this we can fill in whole grid.coin-grid3

You can see that the numbers always add up and there are 924 ways to get to other side. Also the grid is symmtrical. I could also say that you can not step on some point. Then you will add up from left top to right bottom but the number on that step would be zero because no way would come through the spot. Or also you could say that I have to go through one spot. Well I would make two rectangle/squares from this table and first take the ways to get to the spot and then from the spot to the end, then I would multiply both numbers to get the number of ways, one can get there.

Such a random walks can also be extended to higher dimensions, it gets kind of mess but it would be fun to do it in giantic 5 dimensional cube, I mean the ways would increase extremely, like a lot.


After this amazing stuff we had lecture which at the start seemed to be pretty good. It was about usage of excel in mathematics, particularly matrices.

The guy probably thought that we know a lot more and I do not think that there were many (maybe any) people who understood him. When we asked him something, he just said to us that we should read English Wikipedia that he wont waste time there. He was asshole.

During the rest of the day I felt like walking zombie. I had headache, I was sick a lot and there was one more lecture in the evening about conics, but it was too difficult for me to follow it and I was way too tired and went to sleep at around 20:30.