CTK Insights

The Extreme Principle The Extreme Principle is a misnamed problem-solving tactic akin to the Worst-Case Scenario often used in combinatorics and computer science. It does not make any claim (like, say, the Pigeonhole Principle) per se, but only suggests that, for some problems, looking into extreme circumstances or elements within the conditions of the problem [...]

There are mathematical statements that appear counterintuitive. For example, when it comes to infinities, counterintuitive statements are abundant. At the other extreme, there are statements intuitively obvious that are rather hard to prove. Such, for example, is the famous Jordan Curve Theorem. Naturally, mathematics does not lack in statements of any intermediate kind. Here is [...]

Many book authors end their book Introduction expressing the hope that readers will enjoy reading the book as much as the author(s) enjoyed writing it. Persi Diaconis and Ron Graham do not. Nonetheless, their book - Magical Mathematics - oozes their enjoyment at writing it. The authors are master storytellers. Movingly, Martin Gardner wrote Foreword [...]

This is a beautiful pieces by Andy Liu, University of Alberta, from the College Mathematics Journal, Volume 42, Number 5, November 2011, p. 372 Parallel lines are usually defined as lines with no points in common. Parallelism is clearly symmetric. If line 1 has no points in common with line 2, then line 2 also [...]

Parabola has an easily verifiable property. The segment joining points and crosses -axis in point . The equation of the segment is , from which . This may be a curious fact in its own right. What does it say? Taken at a face value, it simply shows a way to obtain the product of [...]

In a recent post, I have implied that Socrates new how to dissect a 2×1 rectangle into a square. There is actually no evidence that he did. However, he certainly knew how to produce a square half the area of a given one. How would he relate the two problems? A sangaku tablet has preserved [...]

What's the task? The task is to combine several 3- and 4-pyramids into larger 3- and 4-pyramids. What's the setup? You'll need 4 tetrahedra and 6 square pyramids. Having 8 tetrahedra and 8 square pyramids will allow to complete 3- and 4-pyramids simultaneously. For your convenience, here are the maps of the pyramids. Just cut, [...]

What is the setup? You'll need a set of cards with numbers 1, ..., N written one per card; N not too small and not too big. Say, for N = 8: How to start? Shuffle the cards somehow and place them in a row: What is the activity? Look at the number on the [...]

Today's activity is based on a problem by Vyacheslav Proizvolov offered at the 1985 All-Union Soviet Math Olympiad. What do you need? Strictly speaking, that activity needs nothing beyond a piece of paper and a pencil. However, it may be convenient to have numbers, say, 1 through 20, written on small paper pieces: What is [...]

The setup Draw 6 dots more or less evenly distributed over a circle: The activity Join the dots by lines of two colors, say, red and blue. Join all pairs of dots. See how many lines of the two colors emanate from each dot. What's to observe At every dot there are at least three [...]