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The Broken Bamboo Problem

Triangle geometry through broken bamboo

The broken bamboo problem involves the application of the Pythagorean theorem. It first appeared as problem 13 in the Chiu Chang Suan Shu (The Nine Chapters on the Mathematical Art), likely at the time of the Han Dynasty. The accompanying figure first appeared in Yang Hui's 1261 work A Detailed Analysis of the Mathematical Methods in the Nine Chapters.

The Broken Bamboo Problem

The broken bamboo problem considers an upright bamboo stalk 10 chi high that is fixed at the base and "broken" partway up its height. While the rooted portion below the "break" is left in place, the upper fragment of stalk is bent over at the break point until it touches the ground 3 chi to the right of the base. The Pythagorean theorem (known as the gou-go relation) is then applied to find the height of the portion of the stalk that remains erect.

Artifact format

Unknown (lost)

Artifact origin



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Additional Reading

  • Dauben, J. W. "Chinese Mathematics." Ch. 3 in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 284 and 287–288, 2007.
  • Joseph, G. G. The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed. Princeton, NJ: Princeton University Press, pp. 257–258, 2011.
  • Lam, L. Y. "On the Existing Fragments of Yang Hui's Hsiang Chieh Suan Fa." Archive for History of Exact Sciences, Vol. 6, pp. 82–88, 1969.
  • Lam, L. Y. A Critical Study of the Yang Hi Suan Fa, a Thirtheenth-Century Chinese Mathematical Treatise. Singapore: Singapore University Press, 1977.
  • Maor, E. The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press, pp. 64–66, 2007.
  • Mankiewicz, R. The Story of Mathematics. Princeton, NJ: Princeton University Press, pp. 34–36, 2004.
  • Martzloff, J.-C. A History of Chinese Mathematics (Trans. S. S. Wilson). New York: Springer-Verlag, pp. 300–301, 1997.
  • Needham, J. Science and Civilization in China, Vol. 3. Cambridge, England: Cambridge University Press, 1959.