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Brian asked a question yesterday about estimating the height of a tree using a stretched arm and a pencil and measuring surface distances, which turned out to be quite easy to solve. It reminded me of similar problems I studied in the past. The most famous set is from the book 海岛算经 (Hai Dao Suan Jing or Sea Island Mathematical Manual) by the third century Chinese mathematician 刘徽(Liu Hui), who is famous for calculating π to 3.1416.

I reread the book again, and find its problems still refreshing to ponder. Here is the first problem, translated into English, posed to you as today's math quiz (can you, with your computers and calculators, out smart someone from AD 263?):

Q: Now one surveys a sea island. He erects two poles, both three zhangs tall, and one thousand bus apart. He lets the two poles and the sea island line up. From the forward pole he walks back one hundred and twenty-three bus, puts his eye to the ground and sees that the tip of the pole coincides with the top of the island. From the rare pole he walks back one hundred and twenty-seven bus, puts his eye to the ground and sees that the tip of the pole coincides with the top of the island. What is the height of the island and the distance from the island to the poles?

(In ancient Chinese measurement, 1 zhang(丈) = 10 chi(尺), and 1 bu(步) = 6 chi(尺).)