Letter From The Editor

Welcome to another mammoth double issue of Mathematics Magazine! Truly we have a bumper crop of mathematical munificence for your reading pleasure this month.

We open the proceedings with a fascinating discourse on a perennial favorite: Fibonacci and Lucas numbers. Greg Dresden and Yichen Wang study certain known convolution identities for these numbers. They generalize them to other, similarly-defined, recursive sequences, such as the Padovan and Perrin numbers. Their investigation leads them not just through number theory, but through generating functions and Newton’s identities as well.

From there we move to combinatorics. S. Adefiyiju, H. Baranek, A. Daly,X. Gonclaves, M. L. Karker, A. LaBarre, and S. Walker explore the “Game of Cycles.” This game, played on planar graphs, was introduced in Francis Su’s wonderful book, Mathematics for Human Flourishing. Players take turns placing arrows on unmarked edges, subject to the rule that they must never create a source or sink. The winner of the game is either the first to create a cycle around one of the graph’s planar regions, or the last to be able to make a legal move. As with so many combinatorial games, the rules are easy to understand, but the mathematics quickly becomes intense. The authors consider the game play on so-called “cactus graphs.” Graph theory articles are always fun for the diagrams alone, but in this case they are backed up with unusually engaging exposition.

Number theory is very well-represented in this issue. Mert Ünsal likes his number theory analytic. He uses Dirichlet’s famous theorem to prove an intriguing result: If a and b are positive integers with a deficient greatest common divisor, then the arithmetic progression $\mathit{\text{ak}}+b$ contains infinitely many primitive abundant numbers. (Recall that positive integers are classified as deficient, perfect, or abundant according to whether the sum of their proper divisors is, respectively, smaller than, equal to, or greater than the number itself). John Campbell and Paul Levrie likewise take their inspiration from a classic: the Basel problem. They provide a novel proof that $\zeta (2)={\pi }^2/6$, based on squaring Gregory’s series (which expresses $\pi /4$ as the alternating sum of unit fractions with odd denominators). Along the way they serve up some fascinating and little-known nuggets in the history of number theory.

The number theory continues with a contribution from Heng Huat Chan, Kuo-Jye Chen, and Warren Johnson. This time the inspiration comes from a third classic in this area: the fundamental theorem of arithmetic. Starting from the well-known formula for the number of divisors of a number expressed in terms of the exponents in its prime factorization, they derive a series of ever more elaborate polynomial identities. Their paper also serves as a nice introduction to “q-analysis.” After the exertions of the previous three articles, you can relax a little with the article by Crystal Brubaker and Verne Leininger. They consider some arithmetical pyrotechnics inspired by an old Martin Gardner essay.

We return to combinatorics with William Erickson’s intriguing discussion of the “Break Buddy Problem,” inspired by an actual situation that arose in the author’s career as a lifeguard. Who would have thought that group rings could arise in such a context? Piotr Pikul considers a problem in polyomino tilings. Specifically, if we build a planar set from polyominoes arising from the 11 distinct polyhedral nets of the unit cube, then what is the smallest perimeter that can be obtained? Pikul provides a complete answer to this question and provides food for thought for future investigations.

We also have several Euclidean geometry papers for your consideration. Shengping Zheng serves up a short and exceedingly clever proof of Heron’s formula for the area of a triangle. With short notes like this there is always the danger of anticipation, but none of the referees I consulted had ever seen this before. Jianke Chen and Kailiang Lin consider a higher-dimensional generalization of the classic Ptolemy-Euler theorem. This is the one that relates the lengths of the diagonals to the side lengths of cyclic quadrilaterals. Chen and Lin survey some of the known proofs of this result, and then provide an original one of their own.

Our Euclidean offerings continue with an article by Allan Berele and Stefan Catoiu. Their article is called, “You Can’t Cut Two Pancakes with Compass and Straightedge,” and if that title does not persuade you to read it with all possible speed, then I doubt there is anything I could say to persuade you. Meanwhile, Katiuscia Teixeira provides a transitional form between geometry and analysis. She asks a question you probably never thought to ask: How do we know that a straight line is the shortest distance between two points? She also provides a very elegant and readable answer to this question.

The remaining articles for this issue explore a variety of topics. Michael Jones and Jennifer Wilson draw some connections between certain voting methods and the problem of rating sports teams. Jeffrey Blanchard and Marc Chamberland show how to use integer programming to solve problems in college registration arising from the COVID-19 pandemic. (This article makes a nice companion piece to the article by J. K. Denny in our February 2024 issue.) And Gary Gordon wraps up our selection of articles with a clever synthesis of linear algebra and Galois theory.

We also have proofs without words, original problems, reviews, and a report on the 64th annual International Mathematical Olympiad. More than enough to keep you busy until we do this all again in our June issue.

Jason Rosenhouse, Editor