Welcome to the June 2024 issue of Mathematics Magazine!
We open the proceedings with combinatorics. Doug Ensley, Ji Young Choi, and Jesica Hoover consider the standard arithmetic triangle of binomial coefficients, but without the usual 1s down the sides. They make a number of surprising discoveries, with their main tools being recurrence relations and generating functions.
Continuing with the combinatorics theme, Lara Pudwell presents a novel discovery about the Catalan numbers. Everyone knows that the Catalan numbers count just about everything. But did you know that the squares of the Catalan numbers count things as well? In her article, MurphyKate Montee takes a combinatorial approach to some problems in hyperbolic geometry. Her investigations lead her to find the Fibonacci numbers in an unexpected place.
Geometry and trigonometry are well represented in this issue. Alexandru Tupan proves some fascinating theorems about Apollonian circle packings. Bahman Kalantari makes a notable contribution to the theory of equitable cake-cutting. André von Borries Lopes uses Heron’s formula for the area of a triangle to provide a clever proof of Apollonius’s theorem. Emmanuel Antonio José Garcia generalizes the law of tangents, which is a classic result in trigonometry. Roger Nelsen revisits a classic from Archimedes and uses cyclic quadrilaterals to find the incircle of an arbelos. Serhan Yarkan and Ali Boyaci show how to derive both the upper and lower bounds in Aristarchus’ inequality from a single, ingenious diagram. And Lǔdek Spíchal uses the golden section to approximate pi.
Our remaining articles defy simple characterization. Alan Roche continues a longstanding discussion in this magazine by providing a short proof that rational algebraic integers are necessarily integers in the familiar sense. David Ross proves the uncountability of the real numbers directly from the definition of Dedekind cuts. Chris Boucher studies a statistical problem related to digit sums.
Sylvie Corwin and Patrick Keef round out our selection of articles by investigating problems related to the enumeration of countably-infinite groups. Among other accomplishments, they provide a novel, accessible proof of a classic result due to Nash-Williams.
As always, we conclude the proceedings with proofs without words, original problems and solutions, and reviews. That should be enough to keep you busy until we do this all again with our October issue!