Welcome to our second straight double-sized issue! Such is the volume of first-rate mathematical exposition the magazine receives that we have amassed a substantial backlog of accepted articles. As a result, some authors have had to wait entirely too long to see their papers in print. To alleviate this problem, Taylor and Francis has allowed us to dramatically increase our page count for a while to work through the backlog more quickly. In addition to thanking them for this largesse, let me also thank Amanda Gedney, Bonnie Ponce, and Annie Petitt for their invaluable assistance and endless patience in putting together these mammoth double issues.
So let’s see what have on tap!
Ezra Brown and Adrian Rice get us started with a marvelous exposition of Hurwitz’s classical result on sums of squares. At first blush, this theorem arises from a trivial algebraic observation: If each of two positive integers is the sum of two squares, then their product is also the sum of two squares. A similar claim could be made for sums of four and eight squares (or one square, for that matter, though that case seems a bit trivial). But why is such a claim false for other numbers, such as five or six squares? Answering that question quickly takes you into the deep waters of normed division algebras, and from there to the quaternions and octonions. It’s all very deep, but Brown and Rice’s flawless exposition makes it perfectly comprehensible.
Franklin Gould also takes his inspiration from the quaternions and octonions. Whereas Brown and Rice used a number-theoretic curiosity as their entry point, Gould takes a more geometrical and linear algebraic approach. It is fascinating to see similar questions explored from such different perspectives, and it is a pleasure to publish both of these wonderfully lucid articles.
The knot theorists in the audience will want to check out the article by Louis Kauffman, Devika Prasad, and Claudia Zhu. After introducing readers to the world of knots and their colorings, they provide an exceedingly clever proof of a result on Brunnian links (with the classic Borromean rings being the most famous example of such a link).
Julius Barbanel focuses on a classic geometrical construction task: doubling a cube. This is famously impossible using the standard Euclidean straightedge and compass. But if you relax the rules just a little, then this task becomes possible after all. Barbanel’s article is a fascinating mix of history and geometry. He provides an education on an unjustly forgotten chapter in ancient Greek geometry and ruminates on extensions of Greek methods to higher dimensions.
This issue will certainly provide the number theorists with plenty of food for thought. Ethan Berkove and Michael Brilleslyper find the Fibonacci numbers lurking in a problem of polynomial long division. The ensuing investigation leads them to the Tribonacci numbers, generating functions, and novel proofs of familiar summation formulas. Daeyeol Jeon and Heonkyu Lee investigate the endlessly fascinating figurate numbers. Specifically, they determine, among other things, which numbers are expressible as the difference of two polygonal numbers. Fang Chen rounds out our number theory offerings. She uses a classic brainteaser as a gateway for explicating the nature of mathematical research. In addition to the considerable mathematical interest of her article, she also has my personal favorite title from my time as Editor.
Geometry is well-represented in our table of contents. Atol Sasane generalizes a result due to Archimedes. Allan Berele and Stefan Catoiu study certain problems from convex geometry and the theory of equipartitions. And Greg Markowsky, Dylan Phung, and David Treeby generalize the familiar HM-GM-AM inequality by realizing each such mean as the centroid of a region in the Cartesian plane.
Mathematics Magazine publishes expository articles, and we include strong undergraduate students among our intended readers. Since analysis tends to be somewhat dense and technical, it does not always lend itself to the sort of conversational tone we like in the articles we publish. With that in mind, it is my great pleasure to be able to publish not one, but two such articles in this issue. Daniel Daners unifies three classic analysis results by showing how all can be proved using the same building block. In their article, Ehssan Khanmohammadi and Omid Khanmohamadi serve up an excursion into functional analysis. A consideration of basic questions regarding convergence and divergence of sequences leads them to a dual of the uniform boundedness principle.
Japanese pencil puzzles are a bottomless pit of interesting mathematical problems. Jacob Boswell, Jacob Clark, and Chip Curtis explore Nurikabe, which is not as well-known as, say, Sudoku or KenKen. For those of a mathematical cast of mind, which I assume includes all readers of this magazine, Nurikabe leads to some very natural combinatorial questions. Boswell, Clark, and Curtis offer some fascinating insights in this area.
Clifford Johnston addresses a very practical problem: how can we ensure that we receive truthful answers to sensitive survey questions? Johnston explores how some elementary probability theory provides an answer to this question, and he shows how these insights can be translated into classroom exercises. As the Editor of the magazine, I wish I had more opportunities to publish pieces of this sort. It shows that an article can be mathematically interesting even without a mountain of dense jargon and notation.
As always, we round out the issue with proofs without words, original problems, and short reviews of newsworthy articles. A fine way to finish off 2022. See you next year!