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A Sangaku Problem

I will be giving a talk Saturday at the NES-MAA meeting in Bridgewater MA in which I will talk about sangaku, or Japanese temple geometry problems. These problems, created by people from a wide walk of life, were beautifully drawn on wooden tablets which were then placed in Buddhist temples or Shinto shrines. Hundreds have been discovered, and probably thousands existed at one time. I became involved in this when I was challenged to solve one of the difficult sangaku. Here I present one of the easier ones, from the delightful book Sacred Geometry: Japanese Temple Geometry by Fukagawa Hidetoshi and Tony Rothman.

I chose this problem because the diagram is so beautiful, the solution is fairly simple, yet satisfying, and it is one of the few sangaku created by a woman (Okuda Tsume).

In a circle of diameter AB = 2R, draw two arcs of radius R with centers A and B respectively, and 10 inscribed circles, two green circles of diameter R, four red circles of radius t, and four blue circles of radius t'. Show that t = t' = R/6.

In the diagram below, we follow the convention of labeling the center of a circle with the radius of that circle.

E22. A simple probability problem

How many rolls of one fair die does it take, on average, before all six numbers show up? Make a guess and then see if you can figure it out.