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- 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

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# Category Archives: Mathematics

## New webpage at www.vadimkulikov.org

Dear readers, If you want to follow my new blog, please visit www.vadimkulikov.org ! A lot of new and unexpected stuff there ! Not only a blog, but also a podcast and much much more.

Posted in Mathematics
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## (Ir)rational Behavior of Calculation

We had the following set of exercises with a group of secondary school students (in Ressun lukio). Assume that a and b are positive real numbers. Then assume one of the following: (i) a and b are both rational (ii) … Continue reading

Posted in Algebra, Calculus, Education, Mathematics, Wednesday Problem
1 Comment

## The Product of Topological Spaces Does Not Obey Cancellation

Exercise 1. Find metric topological spaces [tex]A,B,C[/tex] such that [tex]A[/tex] is not homeomorphic to [tex]B[/tex], but [tex]A\times C[/tex] is homeomorphic to [tex]B\times C[/tex]. Exercise 2. Find path connected metric topological spaces [tex]A,B,C[/tex] such that [tex]A[/tex] is not homeomorphic to [tex]B[/tex], … Continue reading

Posted in Mathematics, Topology, Wednesday Problem
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## Wild Embeddings

Yesterday in a discussion in Komero we concluded that if f is an injective (one-to-one) map from the unit interval [0,1] to the Euclidean plane (or [tex]\mathbb{R}^n[/tex]), then the interval is homeomorphic with its image. The proof is as follows: … Continue reading

Posted in Algebraic Topology, Mathematics, Topology
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## 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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## Time Limit: One Minute!

I found this lovely geometrical riddle in Martin Gardner’s book “My Best Mathematical and Logic Puzzles”. The most lovely thing about it is that Gardner gives a time limit: one minute! So, now draw out your stopwatch and read the … Continue reading

Posted in Geometry, Mathematics, Popular, Recreation, Wednesday Problem
11 Comments

## Douglas Hofstadter: I Am a Strange Loop.

Two weeks ago I finished reading Douglas Hofstadter‘s 2007-book I Am a Strange Loop. I haven’t read other Hofstadter’s books and I want to write this review before I read any. My next Hofstadter-book is going to be Gödel Escher … Continue reading

Posted in Book & Article Reviews, Logic, Philosophy, Popular
2 Comments

## 0.999999……=1? (continuation)

Earlier, I posted a pretending-to-be-funny discussion on this subject and now continue in a more formal fassion. Namely the dialogue in the previous post didn’t actually arrive to any conclusion as to whether 0.9999… is 1 or not. When you … Continue reading

Posted in Calculus, Mathematics
2 Comments

## Walks on Planets

My gps-equipped radiowave controllable robot Lori is somewhere on the surface of Earth. I order her to drive 100 miles South and she obeys. Then I order her to drive 100 miles West and she obeys. Then I order her … Continue reading

Posted in Geometry, Mathematics, Popular, Recreation, Wednesday Problem
7 Comments

## 0.999999….. = 1?

The question in the title is constantly discussed on web forums and at our department’s corridors; sometimes even by professional mathematicians and is in fact already quite seen and banal. In our department this question is often raised by the … Continue reading

Posted in Calculus, Education, Mathematics, Philosophy, Popular
5 Comments

## 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

## A non-associative “group”

Here is what I’ve been doing for the past hour: proving that associativity does not follow from other group axioms including that the left and right inverses are the same and the neutral element is unique. Let us define the … Continue reading

Posted in Algebra, Mathematics
1 Comment

## 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

## Splitting A Rectangle

Find all ways to cut a rectangle into two connected pieces of equal area so that they can be rearranged to get a new rectangle with different dimensions (side lengths) than the orgininal one. I already know countably many ways. … Continue reading

Posted in Geometry, Mathematics, Recreation, Wednesday Problem
2 Comments

## Sphere

Pick randomly three points on the unit sphere. The geodesic triangle with these points as vertices divides the sphere into two parts. What is the probability that the north pole is contained in the smaller of these parts? For instance … Continue reading

Posted in Geometry, Mathematics, Probability, Wednesday Problem
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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
12 Comments

## 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
3 Comments

## 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
2 Comments

## 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
7 Comments

## 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

## Irrational Rectangles

Please comment your solutions, questions and remarks.. This riddle is not easy. I solved it only after I got a hint. I got the hint without wanting it though, and I was already thinking in the right direction; so with … Continue reading

Posted in Geometry, Mathematics, Wednesday Problem
9 Comments

## Uncountable Borel Sets

Here I promised post some tough stuff. Well, I meant some fruits of our recent research by Hyttinen, Friedman and I. I will not go into the subject here, just post a link to the forthcoming paper. There is an … Continue reading

Posted in Logic, Mathematics, Set Theory
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## Ambient Isotopy

Note: One is able to do puzzles 2 and 3 without reading or understanding the text before them. In knot theory, people define equivalence of knots using the concept of ambient isotopy. Two knots (embeddings of the unit circle into … Continue reading

## Around Jordans Curve Theorem I

There are more or less three theorems that are often called Jordan Curve Theorems, while there is a distinction between them. Let us denote by E the Euclidean plane, [tex]E=\mathbb{R}^2[/tex] and by [tex]E^n[/tex] the Euclidean n-dimensional space, [tex]E^n=\mathbb{R}^n[/tex]. The Jordan … Continue reading

Posted in Mathematics, Topology
Tagged Brouwer's Fixed Point Theorem, Jodan's Curve Theorem
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## Euler’s Riddle

Please comment your solutions, questions and remarks.. I found in a recreational book the following riddle attributed to Leonard Euler. The book gives a clumsy solution involving solutions of quadratic equations, a lot of fractions and half a page of … Continue reading

Posted in Algebra, Mathematics, Recreation, Wednesday Problem
2 Comments

## How To Find Your Way Out of Woods

Please comment your solutions, questions and remarks.. A lumberjack got lost in a forest. He knows that the area of the forest is S and that there are no meadows in the forest. Show that he can get out from … Continue reading

Posted in Geometry, Mathematics, Recreation, Wednesday Problem
3 Comments

## The Most Important Relation In Mathematics 1

Part 1: Basics In this and few following posts I will argue that not only in mathematics, but in all human reasoning, the role of equivalence relations is crucial. The notion of an equivalence relation is the key to any … Continue reading

Posted in Education, Foundations, Mathematics, Philosophy
3 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

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
2 Comments

## Prisoner’s problem 2

Please comment your solutions, questions and remarks.. Consider a prison with several millions prisoners. The head of the prison arranges the following challenge to the prisoners. One prisoner at a time will be called to a room with a lamp. … Continue reading

Posted in Mathematics, Probability, Recreation, Wednesday Problem
Tagged Prisoners' problems
2 Comments

## 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
5 Comments

## 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
1 Comment

## Back To College

Please comment your solutions, questions and remarks.. Show that the following inequality holds for all [tex]k>0[/tex] and [tex]x\in \mathbb{R}[/tex]: [tex]e^x>kx-k\ln k.[/tex] Level 1/5

Posted in Calculus, Mathematics, Wednesday Problem
2 Comments

## 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

## How Many Are True?

Please comment your solutions, questions and remarks.. At most 1 statement of this post is true. At most 2 statements of this post are true. At most 3 statements of this post are true. At most 4 statements of this … Continue reading

Posted in Logic, Mathematics, Recreation, Wednesday Problem
1 Comment

## Solution To The Previous Problem

Exceptionally I will present a solution to the previous Wednesday Problem. Only not shall I present any simplest solution but a cool one. The question was the following. Suppose a function [tex]f\colon \mathbb{R}\to\mathbb{R}[/tex] has the property that given any reals … Continue reading

Posted in Calculus, Mathematics
3 Comments

## Intermediate Value Functions

Please comment your solutions, questions and remarks.. A continuous function [tex]f\colon\mathbb{R}\to\mathbb{R}[/tex] on the real numbers has the following property: Given any two reals a<b, the function f gets all possible values between f(a) and f(b) on the interval [a,b] (i.e. … Continue reading

Posted in Calculus, Mathematics, Wednesday Problem
1 Comment

## Sorry I Am Late

Please comment your solutions, questions and remarks.. How many of the statements below the line are true? 2/5 ———— This post was not published on Wednesday 24. February 2010 There are at least 2 true statements in this post (after … Continue reading

Posted in Logic, Recreation, Wednesday Problem
3 Comments

## 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
2 Comments

## Pinocchio Is Omnipotent

Please visit my new website at www.vadimkulikov.org to find more about mathematics, science, cognitive science and art!

Posted in Logic, Mathematics, Philosophy, Recreation
15 Comments

## More Tricks With Triangulations

Please comment your solutions, questions and remarks.. I learned this riddle from Sergei Chmutov. It is also a problem concerning triangulation and sharp angled triangles. Suppose we have a triangulated regular polygon with odd number of edges. The vertices of … Continue reading

Posted in Geometry, Mathematics, Recreation, Wednesday Problem
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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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## Triangulation

Please comment your solutions, questions and remarks.. In the last week’s many in one post I explained what is a triangulation. Can you triangulate a square using only sharp angled triangles (i.e. triangles whose all angles are <[tex]\pi/2[/tex])? For example … Continue reading

Posted in Geometry, Mathematics, Recreation, Wednesday Problem
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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
2 Comments