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Do you like the summer heat? Our lead article makes August/September even hotter as Erwan Brugallé and Kristin Shaw introduce us to a new but growing area of geometry in their paper "A Bit of Tropical Geometry." Ever wonder what angles in a right triangle can be trisected with a ruler and compass? If the sides have integer length, then you will find the answer in Wen D. Chang and Russell A. Gordon's note "Trisecting Angles in Pythagorean Triangles." If you like our Problem Section, then you will love this month's longer than usual version. We close with Michael Barr's review of Noson Yanoksy's *The Outer Limits of Reason*. Stay tuned for the October issue when we get a lesson in additive combinatorics from Persi Diaconis, Xuancheng Shao, and Kannan Soundararajan.—*Scott Chapman*

Vol. 121, No. 7, pp.563-660.

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Erwan Brugallé and Kristin Shaw

This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro’s patchworking. Each definition is explained with concrete examples and illustrations. The text is a modification of a translation from a French text by the first author. There is also a newly-added section highlighting new developments and perspectives on tropical geometry. In addition, the final section provides an extensive list of references on the subject.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.563

Marcelo Brafman

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.589

Bryan Brown, Michael Dairyko, Stephan Ramon Garcia, Bob Lutz, and Michael Someck

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the *Monthly*, despite its intense coverage of quotient sets over the years.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.590

Vinícius G. Pereira de Sá and Celina M. H. de Figueiredo

The toss of a coin is usually regarded as the epitome of randomness, and has been used for ages as a means to resolve disputes in a simple, fair way. Perhaps as ancient as consulting objects such as coins and dice is the art of maliciously biasing them in order to unbalance their outcomes. However, it is possible to employ a biased device to produce equiprobable results in a number of ways, the most famous of which is the method suggested by von Neumann back in 1951. This paper addresses how to extract uniformly distributed bits of information from a nonuniform source. We study some probabilities related to biased dice and coins, culminating in an interesting variation of von Neumann’s mechanism that can be employed in a more restricted setting where the actual results of the coin tosses are not known to the contestants.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.600

Richard Katz, Mike Krebs, and Anthony Shaheen

We prove that if a real-valued function of the plane sums to zero on the four vertices of every unit square, then it must be the zero function. This fact implies a lower bound in a “coloring of the plane” problem similar to the famous Hadwiger–Nelson problem, which asks for the smallest number of colors needed to assign every point in the plane a color so that no two points of unit distance apart have the same color.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.610

D. Zagier

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.618

Rajendra Bhatia and Rajesh Sharma

The farther a normal matrix is from being a scalar, the more dispersed its eigenvalues should be. There are several inequalities in matrix analysis that render this principle more precise. Here it is shown how positive unital linear maps can be used to derive many of these inequalities.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.619

Wen D. Chang and Russell A. Gordon

Suppose that *a*^{2} + *b*^{2} = *c*^{2}, where *a*, *b*, and *c *are relatively prime positive integers, and consider the right triangle *T *with sides *a*, *b*, and *c*. We prove that both of the acute angles in *T *can be trisected with a compass and unmarked straightedge if and only if *c *is a perfect cube.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.625

Diego Marques

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.631

Rom Pinchasi

Given an odd number of axis-aligned unit squares in the plane, it is known that the area of the set whose points in the plane that belong to an odd number of unit squares cannot exceed the area of one unit square, that is, 1. In this paper, we consider the same problem for other shapes. Let *T *be a fixed triangle and consider an odd number of translated copies of *T *in the plane. We show that the set of points in the plane that belong to an odd number of triangles has an area of at least half of the area of *T *. This result is best possible. We resolve also the more general case of a trapezoid and discuss related problems.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.632

Meinolf Geck

We present a shortcut to the familiar characterizations of finite Galois extensions, based on an idea from an earlier MONTHLY note by Sonn and Zassenhaus.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.637

Samuel G. Moreno and Esther M. García-Caballero

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.639

F. Dreher and T. Samuel

The Hausdorff–Alexandroff Theorem states that any compact metric space is the continuous image of Cantor’s ternary set *C*. It is well known that there are compact Hausdorff spaces of cardinality equal to that of *C *that are not continuous images of Cantor’s ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of *C*. Here, we present a compact countably infinite non-Hausdorff space that is not the continuous image of Cantor’s ternary set.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.640

Robert A. Beezer

Associated with any matrix, there are four fundamental subspaces: the column space, row space, (right) null space, and left null space. We describe a single computation that makes readily apparent bases for all four of these subspaces. Proofs of these results rely only on matrix algebra, not properties of dimension. A corollary is the equality of column rank and row rank.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.644

Problems 11789-117895

Solutions 11655, 11656, 11657, 11659, 11660, 11664, 11667

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.648

*The Outer Limits of Reason* By Noson Yanofsky

Reviewed by Michael Barr

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.658