Category Archives: Algebra

(Ir)rational Behavior of Calculation

Posted on October 18, 2011 by Vadim Kulikov

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

A non-associative “group”

Posted on March 22, 2011 by Vadim Kulikov

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

A Year Problem

Posted on November 3, 2010 by Vadim Kulikov

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

Using Abstract Algebra To Understand Basic Combinatorics

Posted on October 11, 2010 by Vadim Kulikov

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 →

Posted in Algebra, Combinatorics, Mathematics | Tagged | 5 Comments

Euler’s Riddle

Posted on June 16, 2010 by Vadim Kulikov

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 Many Good Colorings?

Posted on March 31, 2010 by Vadim Kulikov

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