CTK Insights

The three angle bisectors of a triangle meet at incenter of the triangle. Reversing the problem we may ask a relevant question: Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the angle bisectors. If so, construct the triangle. Solution Given three concurrent lines: α, β, [...]

The medians of a triangle meet at a point known at the center of the triangle. Reversing the problem we may ask a relevant question: Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the medians. If so, construct the triangle. Solution Given three concurrent lines: [...]

Imagine a pyramid with no symmetries or regularities whatsoever. To construct a pyramid like that, pick a plane, four arbitrary points in the plane and one point outside. The lines (or rays) joining the latter to the four points in the plane serve as the edges of a slanted and likely irregular pyramid. However, the [...]

It's hard to overestimate the influence Alcuin of York (c. 732-804) had on Western civilization. He also left the earliest known European collection of puzzles, Propositiones alcuini doctoris caroli magni imperatoris ad acuendos juvenes - Propositions by Alcuin Teacher to the Great Emperor Charles to Sharpen up the Young. The collection consists of 53 problems [...]

A problem for the innocent minds There is a well known problem of finding two points of the same color in the plane all points of which have been colored either red or blue distance 1 unit apart. It is simple but not trivial. This said what about finding two points of different colors under [...]

The Bottleneck Principle The Bottleneck Principle is a problem-solving strategy according to which it may be useful to look into the circumstances in which the conditions of a problem at hand are either hardly or not at all satisfied. It is different from the Worst-Case Scenario in that the latter looks at the problem as [...]

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 [...]