Simple City

Welcome to the 100th edition of this Carnival of Maths! Yes, that’s the 1³ + 2³ + 3³ + 4³th edition.

I gather a party is planned for the 101st issue (that being a prime number of course). All the same, in our bimanual pentadactyl species, one hundred is a number laden with cultural significance. Making it to a hundred can earn you a letter from the Queen or can even see your name on the honours board at Lords. Although I can’t offer a prize so grand, let me at least express hearty thanks and congratulations on behalf of all readers, contributors, and hosts to Alon Levy who got this thing started back in 2007, [added later: and to Mike Croucher at Walking Randomly who ran it 2009-2012], as well as to Katie Steckles, Peter Rowlett, and Christian Perfect at the Aperiodical who between them have done a stirling job of keeping the ball rolling. Now on with this month’s delights…

 

• Let’s start with the biggest mathematical news story at the moment. Back in May, Yitang Zhang announced a proof of the following great breakthrough: there are infinitely many pairs of primes which are at most 70,000,000 (= H) apart. Terry Tao reports on subsequent huge success:

This project is still ongoing, but we have made significant progress; currently, we have confirmed that [the same thing] holds for H as low as 12,006, and provisionally for H as low as 6,966 subject to certain lengthy arguments being checked.

 

• Sticking with the number theory, if you want to express 5 as a sum of positive integers there are 7 ways to do it: 1+1+1+1+1, 2+1+1+1, 2+2+1, 3+1+1, 3+2, 4+1, 5.

So we say that the partition number of 5 is 7, or P(5)=7. Partition numbers are a fascinating and deep branch of number theory. In this post, Adam Goucher delves into the world of partition numbers, unearthing some astonishing discoveries of Ramanujan, and what is surely this month’s most terrifying formula:

 

• And some more number theory! What is the probability that two randomly chosen whole numbers are coprime (i.e. the only number that divides them both is 1)? David Cushing leads us to the answer, introducing some classic number theoretic objects along the way.

 

• On the subject of the number 100, via Seek Echo (via Diogenes Laertius via Apollodorus), we hear that a certain classical mathematician may have celebrated his world-famous theorem in grizzly fashion:

When the great Samian sage his noble problem found,
A hundred oxen dyed with their life-blood the ground.

 

• Hat games are classic genre of mathematical puzzles. Usually the hats come in two colours, and a line of people have to deduce their own hat type. (E.g..) Here, Tanya Khovanova presents a new twist:

A sultan decides to give 100 of his sages a test. The sages will stand in line, one behind the other, so that the last person in the line sees everyone else. The sultan has 101 hats, each of a different color, and the sages know all the colors. The sultan puts all but one of the hats on the sages. The sages can only see the colors of the hats on people in front of them. Then, in any order they want, each sage guesses the color of the hat on his own head. Each hears all previously made guesses, but other than that, the sages cannot speak. They are not allowed to repeat a color that was already announced.

 

• Lê Nguyên Hoang solves a famous conundrum using a famous theorem:

In this article, we’ll present and solve the classical utilities problem, using one of the most beautiful formulas of mathematics, due to Leonhard Euler. This will have us walking a few steps in the stunning world of algebraic topology!

 

• Another well-known variety of puzzle is “What comes next…”. Tony Mann has some good examples, but also points to a problem with the whole formulation:

If, at the beginning of next season, I record the number of goals scored by my football team in their first 5 matches as 0, 0, 0, 0 and 0 (which is not unlikely), I might take comfort from the fact that the formula (n-5)(n-4)(n-3)(n-2)(n-1) has successfully predicted the goals in the first five matches and that therefore I can expect my team to score 120 in their sixth match.

 

• Sticking with the puzzles, here’s a nice geometrical challenge from Futility Closet:

There’s a long way to solve it and a very neat quick way. I did it the long way…

 

• How old is the oldest person you know? Mr Honner opines that this may not be the right question, and tells us about “the most viewed histogram of all time”. There’s a thoughtful discussion in the comments too. (Are the Pru perhaps capitalising on the availability heuristic?)

 

• In another blog-post inspired by an advert, Math Goes Pop challenges the man from AT&T to a game of What’s the biggest number you can think of?

 

• As part of the long-running debate about what mathematical education is, what it should be, what it’s for, and what it should be for, Alice Carey and Stephen Wilson discuss The Faulty Logic of the ‘Math Wars’ in the New York Times Opinionator blog.

 

• Continuing with the politics of mathematical education, here in the UK Michael Gove wants all children up to the age of nine to know their 12 times table. But in our decimal world, what purpose does this really serve? John McLoone at Wolfram Blog has done some investigations:

The “return on effort” drops very rapidly toward the 10 times table and then barely improves. It seems like a fairly compelling case for stopping our rote learning at 10.

 

• In Turkey we have a more dramatic combination of mathematics and politics, where Ali Nesin has been teaching mathematics amid the current turmoil and teargas in Istanbul. Alexandre Borovik has a photo of Ali speaking in Gezi Park (which adjoins Taksim Square, the centre of the protests).

There’s also news footage here. Ali was previously arrested for the crime of “Education Without Permission” – you may remember the extraordinary image of a blackboard with group-theoretic notes on it behind a police cordon. Once again, we wish Ali well.

 

• And even more on the theme of the number 100. What would the world look like if it was a village of 100 people? Jim Noble and students of the International School of Toulouse show us.

 

• Question: what do you get if you combine the ‘u’ species of a Gray Scott reaction diffusion model with the density of a simplified Navier Stokes fluid dynamics model?

I’ve no idea – I can’t even pretend to understand the question. But the answer appears to be solitons which are on fire. Whatever it may mean, this simulation by Simon Gladman is undoubtedly very cool, especially with the banging techno soundtrack.

 

• Math Tango has a glowing review of a compilation of mathematical articles from the New York Times.

 

• Of course book reviews are common things, as are film reviews, restaurant reviews… but every week^[1], the guys at the Aperiodical have been writing integer sequence reviews. And now they’ve expanded this project into a sort of integer sequence world cup, or as they put it Integer Sequence Review Mêlée Hyper-Battle DX 2000. Be that as it may, the winner of this week’s heat is an absolute corker – well worth taking the time to understand.

 

• It’s common to look at the average (mean) lifetimes of various things: people, pieces of technology, biological species… In a post at The Endeavour, John D. Cook introduces us to an important variant of this idea:

If something has survived this far, how much longer is it expected to survive? That’s the question answered by mean residual time.

It’s worth clicking through to John’s explanation of an interesting consequence, the Lindy effect: “The longer a technology has been around, the longer it’s likely to stay around.”

 

• Have you ever calculated the determinant of a 3×3 matrix or bigger? If so, you’ve probably wondered what it is you’re actually doing. Michelle at My Summation provides some insights.

 

And finally…

 

• Futility closet introduces us to some new units of measurement:

Since Helen’s face launched a thousand ships, Isaac Asimov proposed that one millihelen was the amount of beauty needed to launch a single ship.

 

• Which UK city googles for “maths” most often? Mr Gregg has investigated, but you won’t guess the answer.

 

• What colour uniform would mathematicians wear on the Starship Enterprise? Peter Rowlett has invested a surprising amount of thought into this question.

 

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