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One square and an odd number of triangles

  • Martin Aigner
  • Günter M. Ziegler

Abstract

Suppose we want to dissect a square into n triangles of equal area. When n is even, this is easily accomplished. For example, you could divide the horizontal sides into \( \frac{n}{2} \) segments of equal length and draw a diagonal in each of the \( \frac{n}{2} \) rectangles. But now assume n is odd. Already for n = 3 this causes problems, and after some experimentation you will probably come to think that it might not be possible.

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References

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    N. Jacobson: Lectures in Abstract Algebra, Part III: Theory of Fields and Galois Theory, Graduate Texts in Mathematics 32. Springer, New York 1975.Google Scholar
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    P. Monsky: On dividing a square into triangles, Amer. Math. Monthly 77 (1970), 161-164.MathSciNetzbMATHCrossRefGoogle Scholar
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    F. Richman & J. Thomas: Problem 5471, Amer. Math. Monthly 74 (1967), 329.MathSciNetCrossRefGoogle Scholar
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    S. K. Stein & S. Szabó: Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Math. Monographs 25, MAA, Washington DC 1994.Google Scholar
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    J. Thomas: A dissection problem, Math. Magazine 41 (1968), 187-190.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 1
  1. 1.FU BerlinBerlinGermany

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