Welcome to the 161th Carnival of Mathematics. This is a monthly digest of selected mathematical blogs, hosted each month on a different site. The 1st Carnival has been published in February 2007, so this tradition already continues for more than 11 years.

Following the Carnival’s tradition, first we collect some interesting facts about 161 = 7 x 23. First, it is an generalised octagonal number, and hence 3*161+1 is a perfect square. Next, it is also a greengrocer’s number. Finally, it is the number of isomorphism classes of groups of order 3080.

Which of these facts you would find most interesting? What is your favourite mathematical fact? Evelyn Lamb interviewed two mathematicians about their favourite theorems for her Roots of Unity blog. Holly Krieger chose the Brouwer fixed-point theorem, and Vidit Nanda – the Banach’s fixed-point theorem.

Other blog writers turned to mathematical facts that are not intuitively clear. For example, Elias Wirth used graph theory to describe the phenomenon called the Friendship Paradox: on average, most individual’s friends have more friends than the individual themselves. His blog, called Math Section, is dedicated to applications of mathematic in everyday life, and the friendship paradox is a perfect fit. In particular, the article refers to a study which applies this paradox to better predict outbreaks of contagious diseases.

While the friendship paradox’s explanation is accepted, that’s not yet the case for the proof of the abc conjecture, one of the central modern problems in number theory. You can read about the latest developments in the article by Erica Klarreich called “Titans of Mathematics Clash Over Epic Proof of ABC Conjecture” and in the post by Rachel Crowell called “Musings on a mathematicians duties“, placing the problem in the wider context of research ethics.

Chistina Heimken from the University of Münster interviewed her colleague Raimar Wulkenhaar who, together with Erik Panzer (Oxford) solved an equation from elementary particle physics, previously considered to be unsolvable. The interview reflects on 10 year long attempt to find a solution, and gives an interesting insight into the process of mathematical research.

But mathematical research is inseparable from learning and teaching mathematics. How we combine building theories and solving problems when we teach mathematics?Joshua Bowman looks at this from the philosophical viewpoint in “Dialectics of Mathematics“.

Continuing teaching theme, look also at “An Introduction to Newton’s Method” by Ari Rubinsztejn for geometric explanations and a visualisation of the algorithm.

John D. Cook‘s new post “Pi primes” was inspired by another blog post by Evelyn Lamb in which she had mentioned that 314159 is a prime number. Can we find another prime number formed by the initial digits of π? There must be an OEIS sequence for that! In his another recent post on statistics called “Six sigma events” John discusses the rarity of six-sigma events and concludes that they are much more common than the name implies.

Finally, Q is for QUESTION: especially for this issue of the Carnival, Peter Cameron wrote a blog post called “Q is for quantum?” about several standard uses of the letter q in mathematics.

Thank you to everyone who made contributions to the 161th Carnival! The next 162th Carnival of Mathematics is hosted by the team at the Chalkdust Magazine. Please see the main Carnival website for further details and the form to submit blog posts to the new Carnival.

Welcome to the 150th Carnival of Mathematics. This is a monthly digest of selected mathematical blogs, hosted each month on a different site. The 1st Carnival has been published in February 2007, so this tradition already continues for more than 10 years. Thanks to everyone who submitted their blog posts to the 150th Carnival!

Following the tradition, we first ask what do we know about 150. It is the number of groups of order 900, the number of integer solutions to x²+y²+z² = 15² (allowing zeros and distinguishing signs and order), and the number of squares on the 5x5x5 Rubik’s cube. It is a Niven number (aka Harshad number), i.e. it is divisible by the sum of its digits, and an abundant number, i.e. the sum of its divisors exceeds 2×150.

Now to the highlights of some recent mathematical blogs. Ben Orlin, the author of Math with Bad Drawings blog, published new post (with drawings!) called The State of Being Stuck. It is based on Andrew Wiles’ answer to the question by Ben asked at the Heidelberg Laureate Forum 2016. Being stuck is a part of the research process, and is not something to be afraid of – this is what Andrew Wiles would like to emphasize when talking about mathematics to a broader public. This year, Ben went to the Heidelberg Laureate Forum again, and you may find his account of the event here. Another blog post about the Heidelberg Laureate Forum 2017 has been published by Katie Steckles on the Aperiodical.

And if you’re stuck at something and need a break, then perhaps you may find inspiration through looking at nature’s beauty – for example, LThMath suggests to look at Symmetry and Butterflies, which may reflect concepts from various areas of mathematics, ranging from analysis to algebra and statistics.

Speaking of statistics, John D. Cook discusses an interesting application in Randomized response, privacy, and Bayes theorem. Suppose you have a database with sensitive information, and you would like deliberately corrupt it with random noise to anonymise records. How this can be done in a way to preserve privacy while still keeping the data statistically useful?

Rachel Traylor wrote several posts for The Math Citadel website: The Central Limit Theorem Isn’t A Statistical Silver Bullet, where she shows how The Central Limit Theorem, for all its power and popularity, is not a one-stop result to be used for all occasions, and Cauchy Sequences: The Importance Of Getting Close. The latter is a wonderfully accessible explanation of the Cauchy property of sequences, taking time to rigorously examine every piece of the definition.

Anthony Bonato has been interviewing prominent mathematicians in a series of blog posts, most recently Eugenia Cheng, researcher in category theory and the author of How to Bake Pi and Beyond Infinity, and Jennifer Chayes, one of the leading researchers in network science, working at the interface of mathematics, physics, computational science and biology.

Now to discrete mathematics. Reduce The Problem: Permutations And Modulo Arithmetic is another post by Rachel Traylor which in a very accessible manner explains permutation and introduces the concept of isomorphism.

In EKR, Steiner systems, association schemes, and all that, Peter Cameron discussed a class of graphs which give a context to two major results in combinatorial mathematics: the construction of Steiner systems, and the Erdős–Ko–Rado theorem (this story has a continuation in his later post here). Steiner systems also lead to a winning strategy in the card game described in the post MINIMOGs and Mathematical blackjack by David Joyner.

In the Quanta Magazine, Erica Klarreich writes about the recent result on O’Nan moonshine by Ken Ono, John Duncan and Michael Mertens (see their paper Pariah moonshine in Nature Communications) in her article Moonshine Link Discovered for Pariah Symmetries. Another new article in the Quanta Magazine is Mathematicians Measure Infinities and Find They’re Equal by Kevin Hartnett. It introduces recent results by Maryanthe Malliaris and Saharon Shelah, which lead to their Third Hausdorff Medal 2017 award.

Ian Gent published Why the world’s toughest maths problems are much harder than a chess puzzle, and well worth US$1m at The Conversation. There he explains the n-queens completion problem and gives some comments to the recent paper written by Chris Jefferson, Peter Nightingale and Ian Gent and published in the Journal of Artificial Intelligence Research, where they show that this problem is NP-complete.

Several posts in September reported news on mathematical software and its applications, in particular Parallel multivariate multiplication by Bill Hart, and Types of Gaussian Elimination and more technical High Performance Meataxe Interface redesign by Richard Parker. Katie Steckles reported about the new largest generalised Fermat prime, discovered by the PrimeGrid project.

Next Carnival

151th Carnival of Mathematics will be hosted by Frederick at White Group Mathematics. Please see the main Carnival website for further details and the form to submit blog posts to the new Carnival.