# CTK Insights

• ## Pages

04 Apr

### Weekly report, the week of March 28, 2016

The present collection supplies a perfect illustration to the fact that even the most simplest of the problems may be looked at from various angles. I wish that the authors of school textbooks that often include only answers or "solutions to the odd-numbered problems" paid more attention to the possibility of alternative view points.

27 Mar

### Weekly report, March 21-27, 2016

TweetThis week a discussion on tweeter, brought to mind a quote by Underwood Dudley I used years ago Can you recall why you fell in love with mathematics? It was not, I think, because of its usefulness in controlling inventories. Was it not because of the delight, the feeling of power and satisfaction it gave; […]

26 May

### The Jeweler’s Observation, a look back

TweetPaul Brown, an Australian math teacher and author of Proof, a book that I may characterize as a well-written guided introduction into that most fundamental activity, has brought to my attention a recent post at the Futility Closet blog, The Jeweler’s Observation, which I fully reproduce below: Prove that every convex polyhedron has at least […]

01 May

### A wrapping surprise

As you may surmise, the path will behave - if I may say so - in a more rational way. Given the incommensurate dimensions of the box it was rational to expect an endless path. This is what you get on the second attempt. But there remains a question to ponder: Why was the first path so short? Jim Henle leaves to his readers to find the answer.

01 Aug

### Distance to the Horizon on the Fourth of July

TweetI had the luck to celebrate the past 4th of July with our friends in their newly acquired home just above the marina in Atlantic Highlands, NJ. The view from their backyard was absolutely breathtaking. The ambient light that appeared to blur the background made the view even more enchanting. Here is a map that […]

20 Dec

### Beautiful Geometry

And this is how it goes: 51 chapters that combine pedagogically meaningful artwork together with informative, and often eye opening, text. The book ends with a short Appendix which lays foundations for several mathematical concepts mentioned in the text.

This is truly an enjoyable, simple book that meets if not exceeds the author's expectations. It's a good seasonal present, too.

19 Jun

### Climbing Pyramidal Slopes

It is not very steep and may be even tedious, but - at the end - the answer (summit) proves to be somewhat simpler than the climb that led there.

I slipped once and got an answer that included the golden ratio. Since the latter commonly pops up in unexpected situations, I was not at all surprised. However, I noticed in time that the golden ratio would lead the path downhill.

11 Jun

### Parallel Chords in Conics

Points A,B,C,D lie on ellipse such that AB||CD. AC and BD meet in H. AD and BC meet in G. Prove that GH crosses AB and CD in their midpoints, E and F. Why this is true? A short answer is truly cryptic: Because this is true for circles

07 Jun

### Dynamic Software as Serendipity Enhancement

Checking the "Extra" box will suggested a few more properties: angle MAN is not the only angle in the diagram that equals 45 degrees (e.g., angle ADN and angle CMD; angle NLM=90 degrees; some intersections (N,D,L,E,M) are concyclic; there are several similar triangles (e.g., ALN and NLD).)

There are probably other properties. Should you find any, please let me know

23 May

### A square in parallel lines

The problem is credited to V. V. Proizvolov and may serve an example where using dynamical software to sketch a diagram proves to be a distraction. I put together a GeoGebra applet and looked at the possible properties of the configuration with several elements added. Meanwhile, Hubert Shutrick pointed out its obvious, salient feature. The solution is a one liner