Category Archives: Wednesday Problem

(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 $$A,B,C$$ such that $$A$$ is not homeomorphic to $$B$$, but $$A\times C$$ is homeomorphic to $$B\times C$$. Exercise 2. Find path connected metric topological spaces $$A,B,C$$ such that $$A$$ is not homeomorphic to $$B$$, … Continue reading

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

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

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

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

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

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

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

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

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

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

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

If You Beg For Money, You Might End Up Doing Math

Yesterday I was walking around with my friend in a market hall. Two girls of age 15 asked us whether we could give them 40 euro cents. It is about \$0.6. We asked them what could we gain in response. … Continue reading

Posted in Education, Meta, Recreation, Wednesday Problem | 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

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 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

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

Back To College

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

Posted in Calculus, Mathematics, Wednesday Problem | 2 Comments

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

Intermediate Value Functions

Please comment your solutions, questions and remarks.. A continuous function $$f\colon\mathbb{R}\to\mathbb{R}$$ 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

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

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 <$$\pi/2$$)? For example … Continue reading

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

The Coin Placing Game.

See the Wednesday Problem’s vague rules here. Some games again. According to Sam, from whom I heard this riddle, it divides people into cathegories i) those who realize the answer immidiately and ii) those who think quite long about it. … Continue reading

Posted in Games, Recreation, Wednesday Problem | 6 Comments

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 | 2 Comments

The Monk

Please comment your solutions, questions and remarks.. The posts have been a bit advanced lately. Let us lighten the atmosphere by this riddle which admits a simple solution, though mathematicians tend to use calculus in solving it: An Indian monk … Continue reading

Posted in Calculus, Recreation, Wednesday Problem | 2 Comments

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

Happy Transition 2009-2010!

Please comment your solutions, questions and remarks.. Which one is bigger: $$\sqrt{2009 + \sqrt{2010}} + \sqrt{2010 + \sqrt{2009}}$$ or $$\sqrt{2009 + \sqrt{2009}} + \sqrt{2010 + \sqrt{2010}}$$? (If you use a calculator, show that it does not lie) 2/5 … Continue reading

Joulupukki Is Fair: Your Christmas Riddle

Please comment your solutions, questions and remarks.. Joulupukki came to a kindergarten. He had some number of candies to give to the children. He saw that there are more boys than girls and that he could divide the candies evenly … Continue reading

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 | 2 Comments

Ants

Please comment your solutions, questions and remarks.. This funny riddle I heard from Marcin Sabok. There are n ants on the unit interval, [0,1]. Each has a direction, left or right. In the picture above there are three ants and … Continue reading

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 $$n\times m$$ bar of chocolate on … Continue reading