Like this blog?- New webpage at www.vadimkulikov.org March 24, 2017
- (Ir)rational Behavior of Calculation October 18, 2011
- The Product of Topological Spaces Does Not Obey Cancellation October 12, 2011
- Wild Embeddings October 11, 2011
- Progress October 4, 2011
- Class-metric August 24, 2011
- Intellectual Lazyness August 22, 2011
- Time Limit: One Minute! August 17, 2011
- Douglas Hofstadter: I Am a Strange Loop. August 15, 2011
- 0.999999……=1? (continuation) July 10, 2011
- Walks on Planets July 5, 2011
- 0.999999….. = 1? July 3, 2011
- Prisoners’ problem 6 June 29, 2011
- I am back! June 28, 2011
- A non-associative “group” March 21, 2011
- Why I haven’t blogged March 21, 2011
- Doodling with Fractals and Persistent Worms. January 2, 2011
- Five groups November 23, 2010
- Splitting A Rectangle November 17, 2010
- Sphere November 10, 2010
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
Posted in Combinatorics, Logic, Mathematics, Ramsey Theory, Set Theory, Topology, Wednesday Problem
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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
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
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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
Posted in Algebra, Combinatorics, Mathematics, Recreation, Wednesday Problem
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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
Posted in Combinatorics, Games, Mathematics, Recreation, Wednesday Problem
Tagged chess board
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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
Posted in Combinatorics, Mathematics, Philosophy, Recreation
Tagged Prisoners' problems
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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
Posted in Combinatorics, Games, Mathematics, Recreation, Wednesday Problem
Tagged chess board
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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
Posted in Combinatorics, Geometry, Logic, Mathematics, Topology
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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
Posted in Combinatorics, Mathematics, Recreation, Wednesday Problem
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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: [tex]p | \binom{p}{k}\qquad \qquad \qquad (*)[/tex] You … Continue reading
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
Posted in Combinatorics, Mathematics, Recreation, Wednesday Problem
Tagged Prisoners' problems
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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
Posted in Combinatorics, Mathematics, Recreation, Set Theory, Wednesday Problem
Tagged Prisoners' problems
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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
Posted in Combinatorics, Mathematics, Recreation, Wednesday Problem
Tagged Prisoners' problems
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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
Posted in Algebra, Combinatorics, Mathematics, Wednesday Problem
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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, [tex]n\in\mathbb{N}[/tex]. Suppose that [tex]K_1,\dots,K_n,K_{n+1}[/tex] are subsets of K. … Continue reading
Posted in Combinatorics, Mathematics, Set Theory, Wednesday Problem
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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
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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
Games in Topology (Another Proof of Brouwer’s Fixed Point Theorem)
Theorem (Brouwer’s Fixed Point Theorem, BFPT). Suppose that [tex]B\subset \mathbb{R}^n[/tex] is the closed n-dimensional unit ball and [tex]f\colon B\to B[/tex] is a continuous function. Then there exists a point [tex]x\in B[/tex] such that [tex]f(x)=x[/tex]. Theorem (Jordan’s Curve Theorem) Let [tex]f\colon … Continue reading
Posted in Combinatorics, Games, Mathematics, Topology
Tagged Brouwer's Fixed Point Theorem
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Stable Marriages
Suppose n men and n women survived after a spaceship crush on a planet orbiting Alpha Centauri. They happen all to be heterosexuals and all single (or their partners died in the crush). Each man then ranks women in order … Continue reading
Posted in Combinatorics, Mathematics, Recreation
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Brouwer’s Fixed Point Theorem: Many in One Post
In this post I will (1) give a simple proof of Brouwer Fixed Point Theorem (2) fulfill the promise given here (3) present the Wednesday Problem in the form fill in the details in the below text Theorem (Brouwer’s Fixed … Continue reading
Posted in Combinatorics, Mathematics, Topology, Wednesday Problem
Tagged Brouwer's Fixed Point Theorem
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How Old Are The Kids?
Please comment your solutions, questions and remarks.. This is quite famous. But maybe you haven’t heard it yet: A math student goes to a party organized by her supervisor. The student asks: How many daughters do you have? And the … Continue reading
Posted in Combinatorics, Recreation, Wednesday Problem
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MAD families
Please comment your solutions, questions and remarks.. Maximal Almost Disjoint families. This is not so much of a riddle than just a theorem, but the solution is fun, so I would like to place it here. This is like a … Continue reading
Posted in Combinatorics, Mathematics, Set Theory, Wednesday Problem
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The Game Discrete
Inspired by my post about tic-tac-toe I want to introduce a game. It is played on a 3x3x3 grid or equivalently on a 9×3 grid:
Posted in Combinatorics, Games, Mathematics, Recreation
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Handshaking Lemma
Please comment your solutions, questions and remarks.. This one I learned from Sam Hardwick. Show that the number of those, who have shaken their hands with others an odd number of times, is even. Level 1/5 P.S. This lemma has … Continue reading
Posted in Combinatorics, Recreation, Wednesday Problem
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The Chocolate Game
Please comment your solutions, questions and remarks.. I heard this from Lauri Hella, but he doesn’t remember from whom he heard this. The game is played between two players as follows. There is an [tex]n\times m[/tex] bar of chocolate on … Continue reading
Posted in Combinatorics, Games, Mathematics, Recreation, Wednesday Problem
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Tic-tac-toe And A Non-Constructive Proof
Usually the game tic-tac-toe is played on an infinite board (i.e. 30 times 40 or something). The two players draw X:s and 0:s to the squares of a notepad paper. Player 1 (beginner) is the X-drawer and the Player 2 … Continue reading
Posted in Combinatorics, Games, Mathematics, Ramsey Theory, Recreation
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