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
Author Archives: Vadim Kulikov
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
Leave a comment
(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
Leave a comment
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
Leave a comment
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
Leave a comment
Intellectual Lazyness
My friend gives private mathematics lessons to young pupils, mostly high school students who need some help in their curriculum math. Here is a story about a student of him. My friend asks him: how much is 5+6? The guy … Continue reading
Posted in Education, Philosophy, Popular
4 Comments
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
Leave a comment
Can Mathematics Solve Philosophical Problems?
This post is not an attempt to answer the question of the subject, but merely to discuss related topics. The idea dates back to the time when I first learned what is the likelihood function in probability theory. When one … Continue reading
Posted in Philosophy
2 Comments
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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
Leave a comment
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