# Category Archives: Combinatorics

## Class-metric

In two weeks I start lecturing Topology I at our University. I am excited. This course was my favourite among the basic undergraduate courses. The course concentrates on metric topology and its goal is to prove simple results about complete … Continue reading

## Prisoners’ problem 6

Three prisoners meet a guard. The guard says: I have hats with me, two of which are black and the others are white. The prisoners ask: “How many hats there are all together?” He says: “It’s a secret!”. The guard … Continue reading

Posted in Combinatorics, Recreation, Wednesday Problem | Tagged | 2 Comments

## Doodling with Fractals and Persistent Worms.

Suppose we have a rubber line of length 1 m and a worm at the other end. The worm moves 10 cm in a minute and its goal is to reach the other end moving along the rubber line. However … Continue reading

Posted in Combinatorics, Geometry, Mathematics, Recreation | 1 Comment

## A Year Problem

There are more year problems than years. But since I have been pondering on this particular one, I will present it here. You are allowed to use +, -, / and * (plus, minus, division and multiplication) signs and bracketing. … Continue reading

## More Chessboard

You are free to comment your solutions, questions and remarks.. You have two kinds of allowed moves. One move is to jump with a piece over another piece in a horizontal or vertical direction like this: and to jump with … Continue reading

## Prisoners’ Problem 5

There is a famous class of problems concerning public announcements. There is a philosophical (sociological) appeal in these problems. Can a public announcement change the behaviour of the citizens? Of course, if it is for example a news article about … Continue reading

## Incomplete Chess Board

Please comment your solutions, questions and remarks.. I learned this puzzle from Juha Oikkonen, but it is probably quite famous anyway. You have left your chess board on the table in the summer house and when you came back you … Continue reading

## Map Colouring Problem And Compactness

Suppose G is a planar graph embedded into the plane. The graph divides the plane into regions. Let us say that two regions are adjoint if they have a common edge. Question: (Q) Is it possible to colour the regions … Continue reading

## Poisoned Bunnies

Please comment your solutions, questions and remarks.. Imagine that you have one thousand bottles in front of you. You know that one of them is filled with poison and others with water, but you do not know which one is … Continue reading

## Using Abstract Algebra To Understand Basic Combinatorics

Pascal’s triangle has many fascinating properties. One of them is for any given prime number p, the number of k-element subsets (0< k < p) of a p-element set is divisible by p: $$p | \binom{p}{k}\qquad \qquad \qquad (*)$$ You … Continue reading

Posted in Algebra, Combinatorics, Mathematics | Tagged | 5 Comments

## Prisoners’ problem 4

The rules are as in the previous problem, but this time there are only 100 prisoners. How should they do so that 50 of them certainly survives? 3/5

## Prisoner’s problem 3

Please comment your solutions, questions and remarks.. There are infinitely many prisoners in a prison. The prisoners are taken to the yard and everyone is given a hat which is either black or white. No one knows the colour of … Continue reading

## Prisoner’s problem 1

Please comment your solutions, questions and remarks.. We are going to have several prisoner problems! Better solve them before you end up in jail. There are 2010 prisoners in the prison. One day the chair of the prison gathers the … Continue reading

## How Many Good Colorings?

Please comment your solutions, questions and remarks.. Suppose G is a graph in which each vertex has three neighbours, for example: Suppose we have three colors, red green and blue, with which we want to color the edges of the … Continue reading

## Nasty Finite Combinatorics

Please comment your solutions, questions and remarks.. I was blamed for having too easy Wednesday problems. So beware! The level is coming up! Suppose that K is a set of n elements, $$n\in\mathbb{N}$$. Suppose that $$K_1,\dots,K_n,K_{n+1}$$ are subsets of K. … Continue reading

## Andre’s Reflection Method

I promised to post something about random walks. I learned the content of this post while listening to Vilppu Tarvainen’s bachelor’s degree talk and I consider it is quite cute. Let us consider a one dimensional random walk: formally it … Continue reading

Posted in Combinatorics, Mathematics, Probability, Recreation | 1 Comment

## Even More On BFPT

I wouldn’t mind writing this unless I haven’t already posted two different proofs of Brouwer’s Fixed Point Theorem. Namely a friend of mine and a reader of this blog, exposed me to another way of proving it. This proof has … Continue reading