A novel cake-cutting puzzle reveals curiosities about numbers

Alan Frank introduced the “Muffins Problem” nine years ago. Erich Friedman and Veit Elser found some early general results. Now Bill Gasarch along with John Dickerson of the University of Maryland have led a team of undergraduates and high-schoolers (Guangqi Cui, Naveen Durvasula, Erik Metz, Jacob Prinz, Naveen Raman, Daniel Smolyak, and Sung Hyun Yoo) in writing two new papers that develop a full theory.

Today we discuss the problem and how it plays a new game with integers.

A pretty neat paper about a pretty neat theorem

 [ GLL edited ]

Mark Villarino, William Gasarch, and Kenneth Regan are terrific writers. Bill is a co-author of a famous blog located here. Ken is a co-author of another blog that we will not mention.

Today I would like to talk about one of their recent articles that just appeared in the Math Monthly.

Workshop happening this week—anyone can view it live

 IAS Weyl bio source

Hermann Weyl was one of the first members of the Institute for Advanced Study in Princeton. He made important contributions to many fields and even more contributions to groups. His work in group representation theory and polynomial invariant theory is being employed in a workshop being held this week at the Institute on “Optimization, Complexity, and Invariant Theory.” Weyl wrote a lovely book called Levels Of Infinity. One of the interesting chapters is on “Why is the world four-dimensional?” Indeed. Today we discuss some topics from the workshop. The talks are free and readers are invited to follow the remaining proceedings via the live link provided by IAS. Read more…

A possible source of interesting primes

 Study.com source

Pierre de Fermat was fluent in six languages. Yes I thought we would talk about Fermat today. Something I did not know about him is: he was fluent in French, Latin, Greek, Italian, Spanish, and Occitan. I am impressed since I never could handle another language, though Ken speaks several. Occitan is a relative of Catalan and some of its constituent dialects may be more-familiar names: Langue d’Oc, Provençal, Gascon, and Limousin.

Today Ken and I thought we would discuss something named for Fermat: Fermat primes. A Fermat prime is any prime of the form ${2^n + 1}$.

A diversion in mathematical consistency

 Cropped from source

Terrence Howard is an actor and singer who has been in a number of films and TV series. He was nominated for an Academy Award for his role in the movie Hustle & Flow. He currently stars in the TV series Empire.

Today Ken and I want to talk about his claim that ${1 \times 1 = 2}$.

Should we expect simplicity in a theory named for complexity?

 Amer. Phy. Soc. interview source

Sabine Hossenfelder is a physicist at the Frankfurt Institute for Advanced Studies who works on quantum gravity. She is also noted for her BackRe(Action) blog. She has a forthcoming book Lost in Math: How Beauty Leads Physics Astray. Its thesis is that the quest for beauty and simplicity in physics has led to untestable theories and diverted attention from concrete engagements with reality.

Today we wonder whether her ideas can be tested, at least by analogy, in computational complexity.

Triangulating proofs to seek a shorter path

 Cropped from 2016 Newsday source

Mehtaab Sawhney is an undergraduate student at MIT. His work caught my eye on finding his recent paper with David Stoner about permutations that map all three-term arithmetic progressions mod ${n}$ to non-progressions. Here a progression is an ordered triple ${(a,b,c)}$ where ${a -2b + c = 0 \pmod{n}}$. The paper addresses when such permutations can be found in certain small subgroups of ${S_n}$ while I am interested in senses by which they are succinct. This made me curious about Sawhney’s other work.

Today Ken and I wish to report on Sawhney’s simple new proof of the famous triangle inequality in ${\mathbb{R}^n}$.