# Category Archives: Mathematics

## 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

## TopoLogic

The Continuum Hypothesis trilogy will continue later. Today I’ll recall some discussion I made a year ago or so in the student seminar. I guess someone would say that it is surprising, how closely related these two branches of mathematics … Continue reading

Posted in Logic, Mathematics, Topology | 2 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

## Continuum Hypothesis III

Suppose you pick randomly a real number. What is the probability that it equals to 1? The probability is zero. Suppose $$X\subset [0,1]$$ is a countable subset of the unit interval. What is the probability that a randomly picked real … Continue reading

## 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

## Continuum Hypothesis II

Differentiability of Space Filling Curves A Peano curve is a surjective (onto) function $$f\colon\mathbb{R}\to\mathbb{R}^2$$. Apparently such an f cannot be smooth. To see this consider the restrictions of this function to closed intervals $$f\restriction [n,n+1]$$. By smoothness and compactness the … Continue reading

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

## Continuum Hypothesis I

This is the first part of the forthcoming trilogy in four parts. The trilogy will be devoted to the understanding of the Continuum Hypothesis by looking at some statements that are equivalent to it. In this post I just expose … 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

## My Brain Is Open, the mathematical journeys of Paul Erdös: A Book Review

During my stay at the Mittag-Leffler Institute I read this book written by Bruce Schechter. The book is written in accessible English, so that I had no problems reading it though English is not my mother tongue. It tells the … 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

## 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:

## A Pizza Pun

$$\textrm{Suppose we have a pizza with radius }z$$ $$\textrm{ and thickness }a\textrm{. Then the volume is}$$ $$\pi \cdot z \cdot z \cdot a.$$

## 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

## Where Do Random Walks Lead

Rami Luisto asked the following question: Pick randomly n unit vectors from the plane, $$x_1,\dots,x_n\in \mathbb{R}^2$$. What is the probability that the sum of these vectors has magnitude less than or equal to one, $$|x_1+\cdots+x_n|\leqslant 1$$? (magnitude = norm = … Continue reading

Posted in Calculus, Mathematics, Probability | 3 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

## The Monkey and the Balls

I heard this story here at Mittag-Leffler institute from someone who heard it from someone else, who probably also heard it from someone else. (And so on.) What is the probability that I heard it from someone else? Just kidding. … Continue reading

Posted in Mathematics, Philosophy, Probability | 2 Comments

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

## 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

## Erdös and the individual property of dimensions 1 and 2.

Some time ago I read a biography of Paul Erdös titled ‘my brain is open’ written by Bruce Schechter. The book contains some popular exposition of math. As a biography the book is good, but much of the mathematics and … Continue reading

Usually one thinks of real numbers represented by a sequence (possibly infinite) of numbers between 0 and 9. For example $$34,140414\dots$$ One can use various bases like binary, hexadesimal, but the most common is decimal, although with the growing trend … Continue reading