# CTK Insights

• ## Pages

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 […]

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

01 Feb

### Mathematical Circle Diaries, Year 1

The country of FarAwaynia is composed of several states and also has several political parties. Once, a group of FarAwaynian politicians got together for a dinner. It is known that the group contained people from at least to different states and from at least two different parties. Prove that there were two politicians at the dinner that represented different states and belonged to different parties.

14 Jan

### Inspiration, Perspiration and Satisfaction

Still, there is great satisfaction in having solved a problem - even a simple one, and extra satisfaction in being able to appreciate an elegant proof; this kind of satisfaction is multiplied manifold after you devised a solution on your own. Yes, it all may start with inspiration, but to keep the flame burning involves hard work. The upside is that eventually it all comes together when, with evolving habit of solving problems, one begins to realize how right is the paraphrase of the well known statement by Thomas A. Edison: Solving problems is one percent inspiration and ninety-nine percent perspiration. The more you sweat the greater is the satisfaction.

13 Jan

### A Pair of Probability Games for Beginners

The second problem is subtler and sounds rather deceptive: of $6$ possible outcomes of one throw of a single die, $2$ or $5$ come up with the probability of $\displaystyle\frac{1}{3} \lt \frac{1}{2}$. However, the situations changes drastically when two dice are tossed

24 Oct

### Highway Musings

On long drives I made it a habit to talk math, so I prepared a few problems to discuss from the night before. But this time I had to rack my brain all by myself, if only to fight away drowsiness. The warm-up problem was simple: The product of 22 integers is equal to 1. Show that their sum cannot be zero. I would modify it in several ways

11 Sep

### Solving Puzzles with Socrates

This remark helps solved the following problem: in the diagram below, sum the areas of the circles in the two squares; which is larger: the sum of the two areas on the left or that of the four circles on the right? Or, may they per chance be equal?

07 Aug

### Evolution of a problem and an answer II

The figure below depicts a deep circular lake, 300 yards in diameter, with a small island at the center. The two small spots are trees. A man who cannot swim has a rope a few yards longer than 300 yards. How does he use it as a means of getting to the island?

12 Jul

### Mathematics and Calculations

I posed the problem to my (future) 9th grader son whose math teacher I am not happy with. It seems to me that what he does is mostly memorizing formulas and putting in numbers to get numeric results. But the boy proved to still have his wits about him. Without batting an eyelid he gave his answer: "It's simple. It is faster to start moving right away. There is no advantage in losing any time at the outset."

16 Mar

### Pinocchio as Epimenides

TweetThis is a short note for the record. I've been preparing a page on the indivisibles (not ready yet), when my eighth grader boy came up to share his solutions to a couple of olympiad problems. We talked a few minutes about the olympiad and then, sensing his mood, I decided to catch the moment […]