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 [...]

This Is Just Plain Counting I and many others think it's a good idea to start a math class with a simple non-traditional problem to get the students into the right mood for the class. My main source for the problems below is a Russian booklet by E. G. Kozlova intended for early and middle [...]

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 [...]

Nowadays, finding the area of curvilinear shapes falls in the purview of calculus. But the problem of finding areas draw much interest in antiquity and preoccupied mathematicians ever since. One of the acknowledged results by Hippocrates of Chios (470-410 B.C.) is the Squaring of a Lune. The problem of squaring a shape refers to a [...]

At the beginning of his career, Doug Rohrer - presently Professor of Psychology at the University of South Florida - was a math teacher. As such, he was used to begin his mathematics classes with thought provokers, the kind of puzzles that are intrinsically provocative and whose solution - often surprising - does not require [...]

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, [...]

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 [...]

The setup A cube whose faces have been partitioned by the midlines into four squares. The activity Color the squares in as many colors as you wish, with a single caveat: no two adjacent squares, i.e., squares that share an edge, may be of the same color. The task Determine the maximum possible number of [...]