# CTK Insights

• ## Pages

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

21 May

### Area and Perimeter Splitters in a Triangle

It is then meaningful to ask whether a line could be simultaneously an area and a perimeter splitter, and seek a characterization of such lines if they exist. As a matter of fact, they do, and there may be 1, 2, or 3such lines. This was shown by A. Todd a senior at university. In truth, the problem has an interesting history.

20 May

### Probabilities in a Painted Cube

Now generalize: cut the cube into nxnxn smaller cubes and ask the same question. The problem is not awfully difficult but needs some figuring out. Following the foregoing pattern, we eventually arrive at 1/n. But here is a delightful shortcut

15 May

### Fowl Photos for Subitizing

Subitizing is the ability to discern the number of object on a group without actually counting the objects. Even babies and animals do subitizing with small groups. The photos below present an opportunity for subitizing in increasing order of difficulty.

15 May

### Reflections captured in photo II

When shooting in proximity to a puddle, the reflection of an object comes out sharp is the focus taken is as if you were shooting the object itself. This becomes quite clear when you compare the two pictures below. In the first one focus is on the reflection, in the second it's on the surface of the puddle.

07 May

### Applying Daniel Dennett's New Book

TweetLast week I got ahold of the new book by Daniel Dennett - Intuition Pumps And Other Tools for Thinking. I knew of Dennett - a prolific author and a noted philosopher at Tufts University - from his earlier books, Brainstorms, The Mind's I, Consciousness Explained.   I've been recently spending time in physical therapy […]