# Monsky's theorem

In geometry, **Monsky's theorem** states that it is not possible to dissect a square into an odd number of triangles of equal area.^{[1]} In other words, a square does not have an odd equidissection.

The problem was posed by Fred Richman in the *American Mathematical Monthly* in 1965, and was proved by Paul Monsky in 1970.^{[2]}^{[3]}^{[4]}

## ProofEdit

Monsky's proof combines combinatorial and algebraic techniques, and in outline is as follows:

- Take the square to be the unit square with vertices at (0,0), (0,1), (1,0) and (1,1). If there is a dissection into
*n*triangles of equal area then the area of each triangle is 1/*n*. - Colour each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates.
- Show that a straight line can contain points of only two colours.
- Use Sperner's lemma to show that every triangulation of the square into triangles meeting edge-to-edge must contain at least one triangle whose vertices have three different colours.
- Conclude from the straight-line property that a tricolored triangle must also exist in every dissection of the square into triangles, not necessarily meeting edge-to-edge.
- Use Cartesian geometry to show that the 2-adic valuation of the area of a triangle whose vertices have three different colours is greater than 1. So every dissection of the square into triangles must contain at least one triangle whose area has a 2-adic valuation greater than 1.
- If
*n*is odd then the 2-adic valuation of 1/*n*is 1, so it is impossible to dissect the square into triangles all of which have area 1/*n*.^{[5]}

## Optimal dissectionsEdit

By Monsky's theorem it is necessary to have triangles with different areas to dissect a square into an odd number of triangles. Lower bounds for the area differences that must occur to dissect a square into an odd numbers of triangles and the optimal dissections have been studied.^{[6]}^{[7]}^{[8]}

## GeneralizationsEdit

The theorem can be generalized to higher dimensions: an *n*-dimensional hypercube can only be divided into simplices of equal volume, if the number of simplices is a multiple of *n*!.^{[2]}

## ReferencesEdit

**^**Aigner, Martin; Ziegler, Günter M. (2010). "One square and an odd number of triangles".*Proofs from The Book*(4th ed.). Berlin: Springer-Verlag. pp. 131–138. doi:10.1007/978-3-642-00856-6_20. ISBN 978-3-642-00855-9.- ^
^{a}^{b}Xu, Moor (April 4, 2012).*Sperner's Lemma*(PDF) (Technical report). University of California, Berkeley. **^**Monsky, P. (1970). "On Dividing a Square into Triangles".*The American Mathematical Monthly*.**77**(2): 161–164. doi:10.2307/2317329. JSTOR 2317329. MR 0252233.**^**Stein, S. (2004). Kleber, M.; Vakil, R. (eds.). "Cutting a Polygon into Triangles of Equal Areas".*The Mathematical Intelligencer*.**26**: 17–21. doi:10.1007/BF02985395.**^**Verrill, H. A. (September 8, 2004). "Dissecting a square into triangles" (PDF). Louisiana State University. Archived from the original (PDF) on August 18, 2010. Retrieved 2010-08-18.**^**Mansow, K. (2003),*Ungerade Triangulierungen eines Quadrats von kleiner Diskrepanz (en. Odd triangulations of a square of small discrepancy)*(Diplomarbeit), Germany: TU Berlin**^**Schulze, Bernd (1 July 2011). "On the area discrepancy of triangulations of squares and trapezoids".*Electronic Journal of Combinatorics*.**18**(1): #P137. Zbl 1222.52017.**^**Labbé, Jean-Philippe; Rote, Günter; M. Ziegler, Günter (2018). "Area Difference Bounds for Dissections of a Square into an Odd Number of Triangles".*Experimental Mathematics*: 1–23. arXiv:1708.02891. doi:10.1080/10586458.2018.1459961.