The 10 most-cited articles in American Mathematical Monthly, 2023

A classical combinatorial number theory problem is to determine the maximum size of a Sidon set of {1,2,…,n}, where a subset of integers is Sidon if all its pairwise sums are different. For this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by 0.2% for infinitely many values of n. We show that the maximum size of a Sidon set of {1,2,…,n} is at most n+0.998n1/4 for n sufficiently large.

A parametric version of Brouwer’s fixed point theorem, called Browder’s theorem, states that for every continuous mapping f:[0,1]×X→X, where X is a nonempty, compact, and convex set in a Euclidean space, the set of fixed points of f, namely, the set {(t,x)∈[0,1]×X:f(t,x)=x}, has a connected component whose projection onto the first coordinate is [0,1]. Browder’s original proof relies on the theory of the fixed point index. We provide an alternative proof that uses Brouwer’s fixed point theorem and is valid whenever X is a nonempty, compact, and convex subset of a Hausdorff topological vector space.

We consider a combinatorial settlement model on a rectangular grid where each house must be exposed to sunlight from east, south, or west. We are interested in maximal configurations, where no additional houses can be added. Once the settlement is completely built, it seems natural to consider the building density of the obtained maximal configuration. In this article we consider two different random models which produce maximal configurations and, using simulations, we plot an estimate of the distribution of the building density (actually, the occupancy—the total number of houses built) and we conjecture that the means of these distributions converge to a certain limit as the grid dimensions grow to infinity.

We present a simple inductive proof of the Lagrange Inversion Formula.

The telescoping method is skillfully employed to evaluate double series expressions for ζ(2) and ζ(3), as well as the Catalan constant G.

Cake cutting is a classic fair division problem, with the cake serving as a metaphor for a heterogeneous divisible resource. Recently, it was shown that for any number of players with arbitrary preferences over a cake, it is possible to partition the players into groups of any desired size and divide the cake among the groups so that each group receives a single contiguous piece and every player is envy-free. For two groups, we characterize the group sizes for which such an assignment can be computed by a finite algorithm, showing that the task is possible exactly when one of the groups is a singleton. We also establish an analogous existence result for chore division, and show that the result does not hold for a mixed cake.

Political candidates often shift their positions opportunistically in hopes of capturing more votes. When there are only two candidates, the best strategy for each of them is often to move toward the other. This eventually results in two centrists with coalescing views. However, the strategy of moving toward the other candidate is no longer optimal when enough voters abstain instead of voting for a centrist who does not represent their views. These observations, formalized in various ways, have been made many times. Our own formalization is based on differential equations. The surprise and main result derived from these equations is that the final candidate positions can jump discontinuously as the voters’ loyalty toward their candidate wanes. The underlying mathematical mechanism is a blue sky bifurcation.

A note from May 1914 by a certain Agronomof in the Belgian journal Mathesis is frequently cited in the context of the mathematical history of the Tribonacci number sequence. We round out the historical perspective by providing details on Agronomof’s life and work, and highlight two late-nineteenth-century occurrences of this number sequence. One occurrence is connected to Charles Darwin’s On the Origin of Species. Furthermore, we show that the main identity in Agronomof’s note can be leveraged to derive several nontrivial identities with surprising ease and elegance. Some of these identities have only recently been proven by different and more laborious methods; others are new.

We consider how a mechanism of final-offer arbitration may be applied to a negotiation between N players attempting to split a unit of wealth. The game model is defined where the arbitrator chooses a fair split from a Dirichlet distribution. For the case of a uniform probability distribution the equilibrium strategy is found as a function of the Harmonic numbers.

The classical $\mathrm{AGM}$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields $\mathbb{F}_q,$ with $q\equiv 3\pmod 4,$ we introduce a finite field analogue $\mathrm{AGM}_{\mathbb{F}_q}$ that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a $\mathit{jellyfish~swarm},$ as the 3D renderings of the connected components resemble $\mathit{jellyfish}$ (i.e. tentacles connected to a bell head). These swarms turn out to be more than the stuff of child's play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of $\mathbb{F}_q$-points, which can be used to prove that each swarm has at least $(1/2-\varepsilon)\sqrt{q}$ jellyfish. Additionally, this interpretation gives a description of the $\mathit{class~numbers}$ of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish.