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Carnival of Mathematics 85

85 by brighterorange


Welcome to a new Carnival of Mathematics! Traditionally the Carnival opens with facts about the number, this time 85, but first I have an important point of admin to address.

From Carnival of Mathematics 59 in November 2009 until Carnival of Mathematics 84 in December 2011, the Carnival was coordinated by Mike Croucher. Mike said in a blog post (of course!) that he had “had a lot of fun doing so” but that: “Recently, however, I have struggled to find the time to give the CoM the attention it deserves and so it is time to hand over the baton.” We should all be very grateful to Mike for his effort over these two years. You can still find Mike blogging over at Walking Randomly.

Now, as a result, the Carnival has a new home (including an index of previous Carnivals) at The Aperiodical, a not-quite-yet-formally-announced blogging collaboration between Katie Steckles, Christian Perfect and me. We’ll do our best to look after the series for the time being. If you’ve been following the Carnival for a while you’ll know it only works because of a parade of volunteer hosts. The next few Carnivals are lined up but if you’d like to volunteer to host one on your blog later in the year please contact Katie. The other necessary element is submissions, and we have plenty of these this time, so let’s get to business.

Funky new Carnival logo by Katie Steckles


This is Carnival of Mathematics 85. The ever-faithful Number Gossip tells me that 85 has no unique or even rare properties, being merely composite, deficient, evil, odd, square-free and a Smith number. Beyond mathematics, Wikipedia tells me that 85 is the atomic number of astatine, the ISBN Group Identifier for books published in Brazil and the lower bound (due to incomplete research) found by Jorge Stolfi (2004) for The Hollywood Constant, the smallest non-negative integer that has never been used in the title of a movie.

Serious mathematics

(Not that the rest isn’t!)

Brent Yorgey at The Math Less Traveled wrote a series of four posts in response to Depressing Expressions by Patrick Vennebush over at Math Jokes 4 Mathy Folks. In these, he is working to prove why a certain iterative arithmetic algorithm always results in a factorial. Brent says:

In particular I’m proving it using a *combinatorial* proof, a lovely proof technique that (in my opinion) isn’t used or taught as widely as it ought. 

In part four Making Our Equation Count, he says,

I go through the different bits of the equation we’re trying to prove, and explain (with pictures) how to interpret each of them combinatorially.

Rebecka Peterson at Epsilon-Delta writes Extraneous Solutions of Log Equations–A Graphical Explanation. In submitting this post, Rebecka wrote:

Last semester a College Algebra student of mine asked why we sometimes get extraneous solutions when solving log equations. It was such a good question. I tried to do it justice in this post.

Drawing inspiration from the award of the Leroy P. Steele Prize for Mathematical Exposition to Aschbacher, Lyons, Smith, and Solomon for a work about the classification of finite simple groups, Gianluigi Filippelli at Doc Madhattan offers a post about finite simple groups and connections with physics in The classification of finite simple groups.

Frederick Koh from White Group Mathematics submitted Understanding MATTERS (4) saying:

Of late I noticed quite a few students (in online forums) experiencing difficulties in comprehending the concept of calculating distances between 3 dimensional vector planes in space, hence I am sharing an in depth explanation behind how things work.

Rohit Gupta at Kali & the Kaleidoscope investigates a new visualization of Möbius’ Mu, “a notorious function to classify all integers in three different boxes” in The 3 Pills of Möbius.


Thomas Egense from Thomas’ mathematical adventures writes about an attempt to create mathematics-inspired art algorithmically using something called “Fractal flames”. Thomas defines fractal flames, details the algorithmic work involved, and gives several examples of the generated artwork, including the image below (used with permission).

a Fractal flame

Thomas Egense also submitted Dimensions (series of nine videos embedded below) calling this an “impressive graphical visualization of hard to grasp mathematical concepts like higher dimensions and topology”.

Last week Gathering for Gardner 10, the meeting of mathematicians, magicians, puzzlers and others inspired by the life and work of Martin Gardner, took place. Edmund Harriss, from Maxwell’s Demon, previewed his G4G10 talk in a blog post, The 2×1 rectangle and Domes. Edmund begins with the humble 2×1 rectangle, “not one of mathematics most celebrated shapes”, and ends up with structures built as accommodation at the Burning Man event.

Pop culture

Matt over at Math Goes Pop! writes in The Probability Games about the process used to select participants to take part in a fight to the death in the book and film The Hunger Games. Matt says “the rules here practically beg for some mathematical analysis” and Katie Steckles, who submitted the post, called this “just the kind of unnecessary mathematical analysis of a situation I like to see”.

Video game mathematics

Drawing inspiration from the recent arXiv preprint Classic Nintendo Games are (NP-)Hard bringing video games “out of nerdy obscurity and into cutting edge computer science”, Sam Alexander from writes Toward the Mathematics of Video Game Glitches. Sam noticed a minor error in that article based on a glitch in Super Mario Brothers. Running with this theme, he defines video game glitches, game theoretically speaking, and focusing on “glitches which the player can exploit to win the game faster than intended”, defines a theorem (which he describes as “committing horrendous crimes against mathematics”!).

SNES vs. Xbox Triple60
SNES vs. Xbox Triple60 by avail

Performance and puzzles

Ben Nuttall writes about his experience as a maths busker.

“What is Maths Busking?” I hear you ask. Maths Busking is a street performance of mathematics whereby the buskers demonstrate mathematical ideas and engage the public in thinking like a mathematician

As well as a fuller explanation of maths busking, Ben shares some of the ‘busks’ he performed, his thoughts on the experience and some photos.

Birmingham City Centre by Maths Busking

Katie Steckles, writing at The Aperiodical, discusses a variant of a popular mathematics ‘mind reading’ trick. I don’t want to spoil the puzzle in case you want to play along at home, so go over and check out On Disreputable Numbers.

Paul Taylor, also at The Aperiodical, writes about a puzzle he designed for Katie Steckles’ Puzzlebomb. Puzzlebomb is a monthly puzzle sheet featuring all-new types of puzzles. Following the release of the April Puzzlebomb, Paul made the following claim on Twitter:

I guarantee there has never been, and will never be, another puzzle quite like Hilbert’s Space Filling Crossword

In Words to Fill Space he justifies this assertion and describes how he created the puzzle.

Paul also writes, again at The Aperiodical, about a class of puzzle in which a number of prisoners are all given hats and their fate depends on their ability to correctly determine the colour of their own hat. Paul offers “a nice variation on the theme that I heard about at a recent MathsJam” as Another black and white hats puzzle.

Maths Jam is a monthly meeting of maths enthusiasts in pubs worldwide to share stuff they like. “Puzzles, games, problems, or just anything they think is cool or interesting“. A recent development is blogging roundups of what happened at Maths Jam meetings. Recent outings, full of puzzles and mathematical goodies, include: Newcastle (February), Manchester (March), London (February) and Melbourne (January), and a set of photos from various February Maths Jam meetings.

Maths Jam London February 2012

Last month I attended Newcastle Maths Jam. While there we played with a puzzle that Out of the Norm states as:

By relabelling the faces of two dice, can you design a new, unusual pair of six-sided dice that achieves rolls with the same frequencies as a pair of normal dice? All the faces must have a positive number of spots.

Dice and Dissection: a puzzle discusses this puzzle and gives a nice diagrammatic way to view the solution.

History and society

John Cook of The Endeavour writes about a wedding invitation written under the collective pseudonym of that “semi-secret group of French mathematicians”, Nicolas Bourbaki. In Nicolas Bourbaki’s wedding invitation, John explains some of the mathematical references and reveals how the invitation “nearly cost Bourbaki member André Weil his life”.

A post on Pat’sBlog goes to primary sources to highlight an error in Wikipedia in relation to the origins of the four-fours problem. That is,

using four fours and whatever mathematical operations that were allowed to make a number, or a set of numbers. 

The origins of the problem apparently lie in earlier similar problems, including a problem of four threes. Read about it in Before There Were Four-Fours, There Were Four-Threes.

Guillermo Bautista at Mathematics and Multimedia writes A US President’s Proof of the Pythagorean Theorem about the proof of the Pythagorean theorem formulated by James Garfield, the 20th president of the United States of America.

Alexander Bogomolny of CTK Insights offers a little piece of mathematics in the history of chronology given in Florian Diacu’s The Lost Millennium in a post entitled Chinese Remainder Theorem: an Application to Chronology. Submitting this post, Alexander said:

Chinese Remainder Theorem is a staple of early puzzle books, both European and Eastern. This is very satisfying to learn that the theorem finds practical and important applications in the science of chronology. The book by Florian Diacu where I found this application is an exquisitely written compendium of history and mathematics of calendrical calculations. The book deserves every praise and gets my wholehearted recommendation.

In a blog post here on Travels in a Mathematical World, I was very taken with an answer given to a question about working outside traditional academia by Neil deGrasse Tyson in an interview with Samuel Hansen for the Strongly Connected Components Podcast episode 45. The whole interview (18 mins) is worth listening to. My reflections can be found as Culturally an academic.

Over at Second-Rate Minds, Samuel Hansen writes about The True Importance of Friends, explaining the origin of a result in social network theory and its implications in epidemiology. 

I really noticed the absence of the Carnival back in February when I thought I might submit a couple of blog posts which got a particularly warm reaction and found the submissions form deactivated. The posts were Apparently Gauss got in this bar fight with Hilbert… and the follow-up Why do we enjoy maths history misconceptions?


John Chase from Random Walks writes about the surprising features of Microsoft Office Equation Editor, in which he makes the bold claim: “LaTeX lovers will love it”. Find out why at Microsoft Office Equation Editor.

I’ve saved the last word for this revived Carnival to the previous coordinator, Mike Croucher. Mike’s month of math software for March 2012 offers the latest news in the world of mathematical software (a month of math software has been a monthly series since January 2011).

The End

Well, that’s that for this Carnival. You can help spread the word by blogging, tweeting, etc. about the revival of the Carnival and directing people to this edition. If you’re hungry for more mathematics blog posts then there’s the previous Carnival of Mathematics 84, the latest Math Teachers At Play Carnival 48 is over at Math Is Not A Four Letter Word and you can get a weekly selection of blog posts from the Weekly Picks.

Future outings for the Carnival of Mathematics are queued over at The Aperiodical. Carnival of Mathematics 86 will be posted in May by Brent at The Math Less Travelled. Submissions for this are now open, so keep your eye out for great posts and get writing!

Carnival of Mathematics in a world

For a while now (what, over a year?) the folks at have been choosing their weekly ‘picks’ of the blogs coming through their aggregator. The promise at the start of each post has amazed me:

We try to read every blog post that goes through For the Weekly Picks, we collect posts from last week that give you an impression of what the mathematical blogosphere has to offer.

This, to me, seems an incomprehensibly difficult task. There’s so much out there and a massive selection of blogs are now indexed at the site. In light of this resource, our current attempt to revive the Carnival of Mathematics did leave me wondering whether we were barking up the wrong tree in ‘a world’.

Now the incredible level of effort that must be needed for this ever-increasing task has proven unsustainable. In a post entitled “The Weekly Picks are dead, long live the Weekly Picks“, Peter Krautzberger outlines a plan to cope with this:

The goal of the Weekly Picks has always been to show off the wealth of the mathematical blogosphere and to offer an accessible introduction to the various types of mathematical blogs out there…
For the next few weeks, we will focus the Weekly Picks on a different category each week.  This will give us the room to present the full spectrum of the community.

This seems to be a reasonable plan to deal with this ever-growing task, and I’m delighted to hear the folks are human after all! However, there is a problem:

We realize this means we might be missing some great posts that just happen to appear on a week where we do not cover that category.

Aha, here is somewhere the Carnival can step in. If you feel a post you’ve written has been overlooked because wasn’t looking at your type of blog this week, you can submit it to the Carnival of Mathematics. (Of course, it’s then up to the current host to decide whether to include it.)

This approach has its problems too – much is missed, or determined by various biases or whether bloggers have heard of it or remember to submit. But while the ‘weekly picks’ represents a completist approach now focused on each category in turn, the Carnival attempts to catch the best posts across all with its anarchic, community-led method. Perhaps there is room in the world for both approaches, after all.

Reviving the Carnival of Mathematics

Those keeping score may have noticed there hasn’t been a new Carnival of Mathematics for a while. I’ve agreed to take a small part in running it from now on with Katie Steckles and Christian Perfect, as part of a secret new project we’re plotting. To get the ball rolling again, I’ve volunteered to host a new Carnival.

What is the Carnival of Mathematics? It’s a monthly (ish) mathematical blogging roundup. Here’s a description:

The Carnival of Mathematics accepts any mathematics-related blog posts: explanations of serious mathematics, puzzles, writing about mathematics education, mathematical anecdotes, refutations of bad mathematics, applications, reviews, etc. Sufficiently mathematized portions of other disciplines are also acceptable.

The previous Carnival of Mathematics was number 84, posted at Mathematics and Multimedia in December. So this is an announcement that the Carnival of Mathematics 85 will be hosted here on Travels in a Mathematical World in April. Please get your posts in by 2nd April. To submit articles, Katie has made a form which you should find embedded below or on the Carnival of Mathematics submission form.

You may recommend a post from your blog or a favourite you have read elsewhere. It’s helpful if you would put something in the comments box about the post and why you submitted it.

You can help by blogging, tweeting, etc. a link to this page or the submission form. Thank you!

Update (21/03/2012): Just to note that Mike Croucher, previous curator of the Carnival, has posted a blog post about the new arrangements: Carnival of Mathematics – The Next Generation.


Carnival of Mathematics 71 is published

Carnival Banner

proudly presents Carnival of Mathematics Number 71

Yes, the Carnival of Mathematics #71 has been published over at Robin Whitty’s Theorem of the Day.

Carnival of Mathematics #67

Carnival of Mathematics Issue 67


The Carnival of Mathematics #66 was hosted at Wild About Math! This is Carnival of Mathematics #67.

If you’re new to the Carnival of Mathematics, check out Mike Croucher’s introduction at Walking Randomly.

67Photo by woody1778a.

My first task is to choose an interesting fact about the number of the carnival. Well, the excellent Number Gossip tells me as well as being Odd, Prime, Square-free, Lucky, Odious, Deficient and a Lazy caterer, 67 is the largest prime which is not the sum of distinct squares, which seems interesting. Just to demonstrate the level of trivia in the world of numbers, here are some other contenders:

  • 67 is the smallest number which is palindromic in bases 5 and 6 (What’s Special About This Number?);
  • 67 is the only number such that the common alphabetical value of its Roman representation is equal to its reversal (LXVII – 12+24+22+9+9=76) (Number Gossip);
  • 67 is the sum of five consecutive primes (exercise for the reader to work out which) (Wikipedia);
  • 67 is the smallest prime which contains all ten digits when raised to the tenth power (Number Gossip).

Right, on with the Carnival…


Since I’ve started a ‘news’ theme, we’d better have some headlines.

Over at AMS Math in the Media, Allyn Jackson edits a collection of Summaries of Media Coverage of Math for June 2010.

It is worth taking a look at the MAA’s Math in the News archive for recent maths news.

The next Carnival of Mathematics host, Plus Magazine, have published their issue 55.

My own interest in mathematics news and maths in the media is fuelled by my role as the “maths” half of the new Math/Maths Podcast, a weekly conversation about mathematics between the UK and USA.


Mathematical BeadingAt the Make: Online blog, George Hart, for the Museum of Mathematics, writes a Math Monday article on Mathematical beading. This includes the image above – can you tell what it is? Go to the article to find out. George gives five examples by Bih-Yaw Jin, and asks What interesting shapes can you make with beads?

At General Musings, Daniel Colquitt considers the Sierpiński Triangle in Nothing inside infinity, giving an interesting roundup as part of a series of articles he has written on objects which are infinite in some dimensions but finite in others.

Edmund Harriss Penrose tiling based on Garamond
Edmund Harriss of Maxwell’s Demon has been playing with spreading text over tilings and gives some examples based on some of his favourite typefaces in Tiling Typography. The example above is a Penrose Tiling based on Garamond.

Alexander Bogomolny of CTK Insights writes A curious variant of the Pythagorean theorem, in which he gives a symmetric form of the Pythagorean theorem in which no one angle is being paid special attention.

Pythagorean TheoremPhoto by quinn.anya.


Joel Feinstein of Explaining Mathematics has been screencasting his lectures (read his case study), but recently has been struggling with the question of whether students really benefit from his doing this; so he asks: Should we make videos of our lectures available?

Tom DeRosa of I Want to Teach Forever writes a provocative post Why We Fail at Teaching the Language of Data, in which he gives his opinion on the emphasis placed on data analysis at school level and argues more time should be spent learning to look critically at data.


Mathematical notationPicture by wburris.

In The design of mathematical notation at The Number Warrior, Jason Dyer considers mathematical notation as a design issue and gives a series of examples of design that hinders, rather than aides, understanding.

On the subject of poorly designed notation, don’t get Murray Bourne of squareCircleZ started on the notation for natural logarithm! In Logarithms – a visual introduction, he motivates logarithms from a historical perspective, and uses an example to show how logarithms are useful and how they are used.


iPad birthday cakePhoto by Extra Ketchup.

Recently I hosted a seminar by Birgit Loch at the University of Nottingham and played with her new iPad, stuffed full with every free mathematics app in the Store. There was some interesting stuff there, so I am pleased to see Mike Croucher has started a new series of articles on Walking Randomly to explore the options for doing mathematics on this new platform. Start with Math on iPad #1. Meanwhile, David Warlick writes at 2¢ Worth with a roundup of tools for taking mindmap notes in Taking Notes on the iPad.

Computers and technology

Fëanor writes to say that at bit-player is Disentangling Gaussians, in which Brian Hayes writes about how ideas going back to the 1890s have been used recently to provide a computational (polynomial time) solution to a statistical question answered, to a mathematician’s satisfaction, in the 1950s and 60s.

At f(unctional)=>f(un), Samuel Jack asks: How do you guess what language a piece of text is written in? The answer is “using math!” Samuel shows a very simple, but surprisingly effective algorithm and provides an implementation in C#.

Computer codePhoto by Nat W.

John D. Cook at The Endeavour writes Math library functions that seem unnecessary, in which he gives examples of functions in the standard C math library that seem unnecessary at first glance, and the special cases that make them indispensable.

Fredrik Johansson posted Incomplete elliptic integrals complete, in which he describes his implementation of the arbitrary precision calculation of incomplete elliptic integrals in the free, open source mathematical software mpmath.

Katie O’Hare of NAG writes with a post by Mick Pont to The NAG Blog, which asks Why is writing good numerical software so hard? Mick discusses the reasons why software development is still needed.

At the Wolfram Blog, Ed Pegg Jr writes in The Circles of Descartes with a description of the Descartes Circle Theorem and an implementation in Mathematica.


Embracing the Wide Sky Embracing the Wide Sky“Shecky Riemann” of Math-Frolic! writes with The Savant Mind At Work, a book review of “Embracing the Wide Sky” by autistic savant & “math whiz” Daniel Tammet.


James Grime writes with a classic puzzle he recently featured on his YouTube channel: Two Trains One Fly. This YouTube video is below and the solution can be found in Two Trains One Fly Solution.

Meanwhile, over at Mind Your Decisions, Presh Talwalkar discusses solutions for another of James’ YouTube puzzles in Salem witches – a math puzzle.


XKCD cartoon

xkcd: Handy exam trick: when you know the answer but not the correct derivation, derive blindly forward from the givens and backward from the answer, and join the chains once the equations start looking similar. Sometimes the graders don’t notice the seam.


World cup and footballPhoto by CLF.

The Plus blog has a roundup of World Cup maths stories, for those who are that way inclined, while Tim Gowers is musing on a year of tennis in A mathematician watches tennis II.

And finally…

To end on a bit of fun, like all quality news reports, during a recording of the Math/Maths Podcast, Samuel Hansen showed me a couple of spoofs from David Simmons-Duffin: In arXiv vs. snarXiv, the game is to say which of two article titles is from the real arXiv, a “highly-automated electronic archive and distribution server for research articles”, and which is from the spoof snarXiv, a “ran­dom high-energy the­ory paper gen­er­a­tor incor­po­rat­ing all the lat­est trends, entropic rea­son­ing, and excit­ing mod­uli spaces”. Meanwhile, the Theorem of the Day generator is cooking up realistic looking ‘theorems’ and ‘proofs’ using a context free grammar.

Here ends Carnival #67. If you liked it, the sister carnival Math Teachers at Play #27 has been posted at Ramblings of a Math Mom. Carnival of Mathematics #68 will be hosted at Plus on 6th August. Please submit your articles via the carnival submission form.