Sangaku

In my exploration of the Internet in search of non-Western approaches to mathematics, I was delighted to discover a tradition in medieval Japan known as Sangaku, based on a form of traditional Japanese mathematics known as wa-san.

As I discovered from a number of sources, including here at Wikipedia , because Japan was cut off from Western societies during the period known as the Edo Period, the Japanese developed their own system to deal with the analysis of shapes and numbers. Citizens ranging from samurai, farmers and laborers would devise difficult mathematical problems and represent them in colorful wood block cuttings that they would hang from the temple walls. These served as both offerings to the gods, as well as challenges to other temple goers. An example is the picture above from 1847 appearing in the city of Uchiko.

There is apparently an excellent book, called the Sacred Geometry of Japanese Temples, which discusses the history of Sangaku and traditional Japanese mathematics in greater detail. I found this excellent set of geometry problems here (but note that the links to the answers no longer work!).

Let's start with the most straightforward one:

Q1) In the diagram below, three orange squares are drawn inside of a green triangle. If the radius of the blue circle in the upper left of the triangle is equal to 3, what are the radii of the other two blue circles?

Q2) In the diagram below, the three circles are tangent to each other and to the line beneath them. If the radius of the orange circle is 10, and the radius of the blue circle is 6, what is the radius of the red circle?