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Welcome to the 112th edition of Carnival of Mathematics!
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I am proud to host Carnival for a number which is abundant, Belgian-0, biquadrateful, base-4 Colombian, congruent, early bird, equidigital, gozinta, Harshad, heptagonal, iban, icosagonal, idoneal, base-12 inconsumate, 4-Knödel, Loeschian, non-Størmer, non-unitary perfect, odious, practical, pseudoperfect, skinny and Zumkeller. And that's just a sample from over 900 pages returned by for sequences containing 112. There are 112 connected 6-vertex graphs, diplayed, mirabile dictu, by Mike Keith here. And 112 is the 17th largest prime gap \(p_{n+1}-p_n\), occuring when \(n = 31545\) and \(p_n=370261\) (prime gaps were THE big news of 2013, of course, as reported in Carnival no. 100).

The big news generally, over the last month, has been a certain quadrennial sporting fixture. A perennial favourite in recreational mathematics is predicting winners in such jamborees and Etienne Bernard pitched in with Predicting Who Will Win the World Cup with Wolfram Language. It's a technical piece and you have to scroll down a long way before learning that Brazil is the clear favourite. As it remains in his recent follow up. I leave it to aficionados to judge. The theme continues in a charmingly ghoulish submission from Matifutbol entitled The mathematical murderer: "A player has been killed in the locker room at halftime of a match of the World Cup Brazil 2014. A series of mathematical clues leads you to solve the mystery ... that will introduce the fascinating world of Combinatorics."

If you prefer a game where passions run less high, try A book that plays tic-tac-toe with you, spotted by's Mark Frauenfelder. It is a book which should not do what it says on the cover, where it claims you will lose! Unless you play randomly? In which case your odds are perhaps a nice puzzle from the 'fascinating world of'...

Much more rewarding, to my mind, are the books produced by Joel D. Hamkins as described in Math for seven-year-olds: graph coloring, chromatic numbers, and Eulerian paths and circuits.

But before leaving the theme of probabilities here's a wonderful little thing called Arbitrarily biasing a coin in 2 expected tosses which I picked up in a tweet from @icecolbeveridge. Possibly this doesn't qualify to appear here: Alexandr Makelov calls his blog Not really blogging. But who could resist the lure of "you can emulate extremely small, or irrational, probabilities with just two expected tosses".

And before leaving gaming, there is a game-theoretic flavour to Scott Aaronson's quite long (and made longer by a lot of serious reader feedback) essay on Eigenmorality. If moral people are defined as 'those who consort with moral people' how is this not circular? Read on: EigenMoses? EigenJesus?? — you will be hooked.

The people side of mathematics/computer science is Evelyn J. Lamb's preoccupation in The Human Side of Computer Science which offers a panorama of the justly famous Gödel’s Lost Letter and P=NP. This is a blog which is as much about researchers as research. And researchers "make mistakes, they go down dead-ends, they make progress. Also students may not know, but even experts sometimes forget, that ideas that we now see as easy were once unknown." I am reminded of the lovely aphorism of, I believe, the Danish scientist Piet Hein:
       Problems worthy of attack,
       Prove their worth by fighting back!
(Piet Hein, according to John D. Cook's post Swedish Superellipse, invented said superellipse. One short sweet post leading, as always at The Endeavour, to another, and another!)

Anyway, the topic of how mathematics and science's achievements are not just won but hard won is very rewarding stuff, not least for students, for whom the Arrivals Hall of science can mean much more if they've travelled, albeit vicariously, on the flight. David Bressoud has championed this teaching philosophy in his wonderful "Radical approach" books. He writes with great insight in Beyond the Limit I, the first of a series of blogs on how 'limit' need not be the first concept confronting a first year calculus student. We can step back even further with recent postgraduate Nathan, guest blogger at Mathbabe, in What is the goal of a college calculus course? while a first year physics student will probably find more to relate to in Jon Butterworth's Guardian post Rates of change (what is the goal of calculus in the first place?)

Baking and math has an even more basic message in Two things I tell calculus students (one is the squeeze theorem) (as well as a very nice explanation of squeeze). I'm pretty sure you will agree with it and it's entertainingly argued with a cooking analogy. And cooking is more than an analogy in Fibonacci Lemonade by the multi-talented Andrea Hawksley and in Salad dressing again starring Evelyn J. Lamb: "ingredients: lettuce, tomatos, power series expansions, ..."! We are back to limits and calculus.

As we are in A curious Geometry relation, and a Question at the wonderful Pat's Blog: why is it that, in a right triangle with sides \(a<b<c\), the angle opposite to \(a\) is so accurately approximated, in radians, by \(3a/(b+2c)\)? Tony Forbes, editor of M500 magazine, offered me a solution: compare series expansion \(\tan^{-1}x=x-x^3/3+x^5/5-\ldots\) with series expansion of \(3x/\left(1+2\sqrt{1+x^2}\,\right)\) (normalising by dividing through by \(b\)), which is \(x-x^3/3+7x^5/36-\ldots\). But have a look and see if you can see something else! There is usually more than one answer to a question — check out Uncountability of real numbers -- Topological proof by Shamil Asgarli: diagonalisation is not the only fruit!

And that any single mathematical truth manifests itself in so many ways is such a magical thing! This is precisely the ingenious SheckyR's point in Through the Looking Glass and (back to teaching) Keith Devlin's point in The Power of Dots. Mercurial Colignatus contributes to this theme on the subject of Euclid's Fifth Postulate; and I must mention Andrea Hawksley again, who always has a different angle on mathematics! Try her delightful Zip tie tangle.

Stephen Cavadino has a different take on this plurality though: "BIDMAS, BODMAS, BOMDAS, BIMDAS, PEMDAS, PEDMAS...": in Aaargh Ruddy BIDMAS he tells a scary tale of how mathematics gets lost in mnemonics. The cure, or at least a diagnosis, is at hand: Edmund Harriss in Twelve and the real life problems problem pinpoints the "converting one thing to another using ... symbols ... and mapping those moves onto the world" which is simultaneously BIDMAS-level arithmetic and category theory.

I should mention one other thing which belongs to last month: Tau Day! Aka 6.28. A combination of two perfect numbers, as Evelyn J. Lamb (moving sideways to Scientific American) points out in The Most Mathematically Perfect Day of the Year. Personally I happen to believe that the circumference of the unit circle is the correct circle constant, and have by now spent several years enjoying telling students that a quarter turn is \(\tau/4\) radians. It remains for now a personal thing, but there was certainly plenty of blog activity on the subject, which I will indulge myself with listing, in no particular order and without further comment: Pi vs. tau: Ultimate Smackdown; Two times Pi = Tau; Tau Day 2014 – A Quick Overview; Pi VERSUS Tau?!?; What are your thoughts on the pi v. tau debate?; and, last but not least, Mathematical Cat Fight (Happy Tau Day — June 28th) by no less a presence than Madeleine Begun Kane. Michael Hartl, who presides over the debate in a good natured and relatively impartial manner, contributes a quasi-blog in his State of the Tau.

But to end on a less equivocal note, here is something that everyone can celebrate: the launch of the Taking Maths Further podcasts in which Peter Rowlett and Katie Steckles talk to real maths-end-users about what they do. There was much excitment last month about the award of the first Breakthrough Prizes In Mathematics, overtly intended to make mathematics the coolest school subject ever. My money is on Rowlett and Steckles to do that, rather than Milner and Zuckerberg.

And now I will really end, with the last blog submission to arrive at Carnival, on 4th July; a little puzzle for you: Happy Independence Day by Gyora Benedek. Enjoy! Post a comment below!! Submit to Carnival 113 (followed by a big prime gap!), to be hosted by Mike at Walking Randomly!!!

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Posted by Robin01:11, 11 07
Thanks for all the wonderful submissions to Carnival 112 everyone!