In this paper, we show that a lattice balanced polygon of odd area be cut into an odd number of triangles of equal areas. The first result of this type was obtained by Paul Monsky in 1970. He proved that a square cannot be cut into an odd number of triangles of equal areas. In 2000, Sherman Stein conjectured that the same holds for any balanced polygon. We also show between the equidissection problem and tropical geometry. Bibliography: 9 titles.
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B. M. Bekker and N. Yu. Netsvetaev, “Generalized Sperner lemma and subdivisions into simplices of equal volume,” J. Math. Sci., 91, No. 6, 3492–3498 (1998).
S. Lang, Algebra, Addison-Wesley, Reading, Massachusetts (1965).
P. Monsky, “On dividing a square into triangles.” Amer. Math. Monthly, 77, 161–164 (1970).
P. Monsky, “A conjecture of Stein on plane dissections,” Math. Z., 205, 583–592 (1990).
I. Praton, “Cutting polyominos into equal-area triangles,” Amer. Math. Monthly, 109, 818–826 (2002).
S. Stein, “A generalized conjecture about cutting a polygon into triangles of equal areas.” Discrete Comput. Geom., 24, 141–145 (2000).
S. Stein, “Cutting a polygon into triangles of equal areas,” Math. Intelligence, 26 (2004), No. 1, 17–21.
S. Stein, “Cutting a polyomino into triangles of equal areas,” Amer. Math. Monthly, 106, 255–257 (1999)
A. Hales and E. Straus, “Projective colorings,” Pacific J. Math., 99, No. 1, 31–43 (1982).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 403, 2012, pp. 142–157.
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Rudenko, D. On equidissection of balanced polygons. J Math Sci 190, 486–495 (2013). https://doi.org/10.1007/s10958-013-1265-1
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DOI: https://doi.org/10.1007/s10958-013-1265-1