The puzzle consists of a simple question – you need to pick a number between 0 and 100, and all 49,485 of the responses will be collated (assuming that every single one of the Times’ readership actually enters a number) and averaged. If your guess turns out to be the closest whole number to two-thirds of the average guess, you are clever and you win.

It’s a very well-known classic piece of game theory, sometimes called ‘Guess 2/3 of the Average‘ (mathematicians: they call it like they see it) and has been well analysed by game theorists. It originally appeared in Alain Ledoux’s magazine ‘Jeux Et Stratégie‘ as a tie-breaker question, and is a great example of a problem in which you have to make an educated guess, based on what you think other people will guess. And therein lies the problem (other people. ugh.)

This class of problem is sometimes called the Keynesian Beauty Contest, in reference to John Maynard Keynes’ example in his 1935 book *The General Theory of Employment, Interest and Money.* In the example, entrants must pick the six most beautiful faces from a collection of photos, and will win a prize if they pick the most popular faces – which means picking the ones you personally think look pretty is actually a bad strategy, if that’s not in line with popular opinion.

The NYT’s ‘2/3 of the average’ problem has a similar structure, and as in all game theory problems, the incentives and pressures on people playing the game will influence their behaviour. This means it’s possible to work out an optimal strategy, or at least have a stab at one. If you don’t want to hear about that, because you want to think about it and go and enter the competition first, go and do that, and don’t read the rest of this until you have. Here’s the competition entry page.

Assuming all the players in the contest behave rationally, and they’re all trying to guess a figure that’s 2/3 of the average, it makes no sense for anyone to pick a number bigger than 67, since there’s very little chance the average is going to be 100, and any number bigger than 67 will never be the correct answer. So, if everyone’s playing rationally, the numbers picked will all be between 0 and 67 – the option of picking numbers bigger than 67 is called a *weakly dominated strategy*, which means any other choice is as good or better.

This means the 2/3 of the average will never be more than 44. So nobody rational will pick anything bigger than 44. So the range of sensible guesses reduces again, and this thought process can be continued indefinitely until the only logical conclusion is to pick 0 (which works out pretty well, because if everyone is logical they’ll also do this, and the average will be 0, and 2/3 of the average will also be 0, so everyone wins). This is called a Nash Equilibrium, after game theory legend John Nash.

The above all applies to situations where any real number value is allowed – but in the NYT example, the game is restricted to integers. In this case, an interesting (integeresting) thing happens – instead of the only truly rational answer being 0, the options of 0 and 1 are both possible. If you suspect that a lot of people (more than a quarter of the players) will pick 0, then you should pick 0, but if you think more people will pick 1, it’s a more sensible choice. This is then a different game theory problem, where being in the majority means you win.

As the Wikipedia article on ‘Guess 2/3 of the Average’ so nicely puts it, “ordinary people” rarely turn out to be perfect game theorists, and so the NYT competition may find that many entries deviate from the optimal strategy, and instead rely on people’s instinct (and in fact, given this knowledge, the optimal strategy itself changes). It’s almost like they knew this would be a brilliant logical challenge, and presented it as a puzzle oh wait.

It’s also worth noting that all the deductions above rely on the additional assumption that everyone is trying to win – and if they’re not, then their strategy may not be the one most likely to result in them winning, but could be entirely random. You also don’t know what proportion of players have this attitude. Alfred was right when he said that some men just want to watch the world burn.

Past versions of this type of competition include one run by Danish magazine Politiken in 2005, using similar rules, in which 19,196 entrants turned out a winning value of 21.6 (the prize was shared by five people). As to whether NYT readers are more or less smug than Danish people, only time will tell.

Are You Smarter Than 49,485 Other New York Times Readers? at The Upshot

]]>Attempts to tile a plane with regular pentagons never really got off the ground, but until now there were 14 known ways to use irregular convex pentagons to cover a plane. Five were discovered in 1918 by German mathematician Karl Reinhardt, and it was assumed this was the complete list until more were discovered in 1968 and 1975, including four found by housewife Marjorie Rice upon reading about them in Scientific American (covered, of course, by Martin Gardner).

The Bothell researchers, Casey Mann, Jennifer McLoud and David Von Derau, employed an exhaustive computer search to find the new shape last month, and will shortly be publishing a paper on their work.

The new discovery, which involves a pentagon with angles $60^\circ$, $135^\circ$, $105^\circ$, $90^\circ$ and $150^\circ$, brings the total up to 15, all of which are illustrated below. Aren’t they beautiful? The new one is in the bottom right.

It’s still not known whether this is all the possible pentagonal tilings, or whether more are still out there. The team hope they can continue with their method to find other arrangements. Keep looking, pentagon pickers!

Attack on the pentagon results in discovery of new mathematical tile, by Alex Bellos at The Guardian

A new way to tile your floor (if you like pentagons), by Kevin Knudson at Forbes.com

At the end of an overnight flight from San Francisco to New York is hardly the ideal time to play “I Spy Mathematics” on a packed airplane. We were all grumpy and groggy from four scant hours of sleep. It seemed that nobody had watched any films en route and, like most of the other passengers, I didn’t have headphones or earplugs to hand.

Clearly there was no point in scanning the entertainment offerings with just 15 minutes to landing. But then I remembered spotting an ad on screen for **The Great Courses** as I’d settled into my seat in San Francisco. Was it possible that I’d blown a chance to watch Art Benjamin, David Bressoud, Judith Grabiner, David Kung, James Sellers or Mike Starbird in action? I decided to try to find out.

It wasn’t easy to locate The Great Courses; they weren’t listed under Drama, Comedy, or Documentaries, or anywhere else that made sense. When I finally did find them—under an audio menu—I was disappointed to discover only a small and decidedly non-mathematical selection (on Food, Nutrition and Stress) on offer.

But then, cruising a different menu out of sheer boredom, I spotted a short list of TED talks. I was down to 13 minutes, as shown below, and doubted that I could build a computer or learn a language in that amount of time, despite being a committed life long learner. But maybe I could finally learn the answer to the question “What Is Math For?”.

I clicked on this option excitedly. Was it going to explain why the giant plane we’d just flown 4000 km on had stayed up in the air the whole time? (Delta is a real stickler on this point.) The person sitting beside me needed to wake up and pay attention. I considered shaking her back to reality with reassuring comments like, “We’re not dead yet, and this short video on maths will explain why” but thought the better of it, and left her to her slumbers.

*“Mathematician Eduardo Sáenz de Cabezón answers the questions What is math good for? With humor and charm, he shows the beauty of math as the backbone of science”* promised the next screen.

It claimed that the video was 11 minutes long, and it seemed I now had just 12 minutes to landing time. Even better, the video was in Spanish, and had English subtitles. With nothing left to lose, I settled down to watch silently, having no particular expectations.

It started off like a standard standup comedy routine, immediately delving into that awkward social situation for mathematicians where we reveal our profession and the response is either “I was always terrible at maths” or “what is mathematics for?” Within two minutes, the speaker had the audience in stitches, having brushed aside the practical “bridges and computers” justifications for mathematics. But he’d also slipped in a mention of Hardy, which certainly got my attention.

Soon he’d pointed out that a piece of paper folded 50 times would basically stretch from the earth to the sun. Then he was on to sphere packing and space filling, highlighting recent breakthroughs by American mathematician Tom Hales and Irish physicists Dennis Weaire and Robert Phelan, contrasting the “obviously” true with what we have really proved to be 100% true. It was a masterful relaxed performance, and the large general audience went along with him every step of the way, laughing and learning in equal measure.

Sáenz de Cabezón, I later discovered, has a PhD on “Combinatorial Koszul Homology: Computations and Applications” and teaches and researches at Universidad de La Rioja, Logroño, in Spain. He’s on Twitter, where it turns out I’ve been following him without realising that he had a video with over a million online views.

The official (American English) title of his TEDx talk is “Math Is Forever” which ties in with the presenter’s central message that “theorems, not diamonds, are forever.”

It’s not just Delta who are making such high-brow videos available to their passengers, Jet Blue and Virgin are too. However, pending further investigation, I have no evidence that either of those airlines’ offerings include any mathematical content, so I’m sticking with my claim that given any Delta (flight), there exists Epsilon (mathematics to view), such that the overall effect is enjoyable and enlightening.

Does Epsilon depend on Delta? In other words, can Epsilon maths be found no matter how short the Delta flight is? Perhaps. Epsilon is certainly small, though not infinitesimal here. (In the great scheme of things, the observed case may have been an essential singularity.)

I do know my limits, and if all else fails and Epsilon mathematics can’t be found on a particular flight, there may well be Epsilons in the Erdős sense on board.

**Watch Eduardo’s talk below**

The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>I wrote previously on mathematician stereotypes, and suggested a few ideas for sources of biographies of historical and contemporary people working in maths who would be stereotype-breaking. I’m happy to report that since I wrote that post, This Is What A Scientist Looks Like has attracted more mathematicians.

Of course, in the background to all this is the issue of who counts as a mathematician. This issue has been well-discussed here, including by Katie Steckles, Liz Hind and on at least one previous occasion by me, plus the amusing satirical take by Christian Lawson-Perfect. My view is that if you’re using maths and are happy to think of yourself as a mathematician, that’s good enough for me.

Anyway, back to Kit and the #realfaceofmath, the hashtag on Twitter has attracted some interesting pictures — but there’s always room for more. I’m not keen on pictures of myself, and I don’t think I particularly break the stereotype, but I suppose if I am going to suggest you contribute a picture of yourself, I should play ball myself. So here we go.

]]>For @Kit_Yates_Maths, here I am doing something not mathematical. #realfaceofmath pic.twitter.com/YBr70ilCdg

— Peter Rowlett (@peterrowlett) August 9, 2015

Puzzlebomb is a monthly puzzle compendium. Issue 44 of Puzzlebomb, for August 2015, can be found here:

Puzzlebomb – Issue 44 – August 2015

The solutions to Issue 44 will be posted at the same time as Issue 45.

Previous issues of Puzzlebomb, and their solutions, can be found here.

]]>A group of applied mathematicians, including the University of Manchester’s Nick Higham, have been compiling a book on applied mathematics over the last few years, and they’ve announced it’s finally ready for publication. The book, which includes an introduction to applied mathematics, key concepts, and various examples of modelling problems, is aimed at undergraduate mathematicians and above (although some of the articles may be accessible to younger/lay readers) and comprises 186 articles by 165 authors from 23 countries. It’ll make a good companion (excuse the pun) to the *Princeton Companion to Mathematics*, edited by Tim Gowers and covering the pure end of the field. It will be published by Princeton University Press in September 2015.

- Nick Higham’s blog post about the book
- Princeton Companion to Applied Mathematics at Princeton University Press
- For some reason, there’s a Twitter feed

A recent court judgement ruled that the range “1 to 25″ can include the value 0.51, if you round to the nearest integer.

That’s a little bit interesting – it will certainly make people think twice before writing numbers in patents – but it’s been reported in the most fantastically mathematically illiterate fashion in The Independent, by someone who seems to have discovered what ’rounding’ is in the course of their research.

**Read: **What exactly does ‘one’ mean? Court of Appeal passes judgement on thorny mathematical issue, in The Independent

*(Via Tony Mann on Twitter)*

No, it’s not what happens when you try to do maths under pressure and forget everything you ever knew about calculus – Mathesia is a new crowdsourcing platform for mathematics, which companies can use to pitch mathematical problems to their collection of maths experts, who can then bid to be awarded the project. It also has a section for universities to advertise research posts.

One thing that does make me sad is that the site extensively uses the word ‘brainies’ to describe the mathematicians, and it looks like the pitchers are adopting this as standard terminology. Bit naff, right?

]]>Vi Hart, Andrea Hawksley, Henry Segerman and Marc ten Bosch each independently have long track records of doing crazy, innovative stuff with maths. Together, they’ve made Hypernom.

Hypernom is a game where you nom cells of four-dimensional objects by rotating, wiggling and twisting yourself around to move through a stereographic projection of a radial projection of a 4D polytope. It looks like this:

While from the outside it looks like this:

(That’s one of the creators of the game, Vi Hart, committing fully to the nom.)

I’ll make an attempt to explain the maths of what’s going on in the game. (Or you might just be better off reading the explanation in the paper, “Hypernom: Mapping VR Headset Orientation to $S^3$”) A polytope is a higher-dimensional analogue of a polyhedron: it sits in $n$-dimensional space, and its “sides” (also called cells) are flat $n-1$-dimensional objects. In Hypernom, you’re nomming up regular 4D polytopes, whose cells are regular 3D polyhedra.

In order to display the 4D shapes, you need to project them onto 3D space (like a shadow is the 2D projection of a 3D shape). *Hypernom* first radially projects the polytope onto a 4D hypersphere $S^3$ (that’s a 3 not a 4 because “sphere” means the edge, or shell, of a ball. The boundary of a 4D ball has three dimensions). Each cell of the polytope fills up a region of the sphere. That sphere is then stereographically projected into normal 3D space: the stereographic projection picks one origin point and a 3D hyperplane (imagine 3D space, but it also has a particular position in the fourth dimension), then projects each point on the hypersphere to the point where the line from the origin intersects the hyperplane.

That’s hard to imagine, so here’s a video Henry Segerman made to explain stereographic projection from 3D into 2D:

Segerman explains the 4D version of that in a paper titled “Sculptures in $S^3$”.

The next question is how you move through this space. Hart and co. have cleverly noticed that $SO(3)$, the set of orientations in three-dimensional space, is double covered by $S^3$, the set of points on the edge of a four-dimensional sphere – that is, for every orientation in 3D space, there are exactly two corresponding points on the 4D sphere.

Hypernom maps the orientation of your screen to a unit quaternion, which corresponds to a point of $S^3$. So, by wiggling around, you spin this four-dimensional ball, which looks to you like movement through three-dimensional space, and when you get close enough to the centre of a cell you “nom” it and it disappears. It’s designed for VR goggles, though it can also use your phone’s orientation sensor; I did a lot of twirling when playing on my phone, which I guess would be quite hard when the thing that needs to be twirled is your whole body. The aim of the game is to nom all the cells of your chosen polytope in as short a time as as possible.

Really, it’s all an excuse to mess about with odd geometry. It’s interesting to see which intuitions about how you move through space break down in this world, and how hard it is to develop new ones. The authors acknowledge this; the “motivation and artistic choices” section of their paper begins,

It may seem somewhat arbitrary to design a VR game that uses headset orientation data as a quaternion to map your position in $S^3$ to eat cells of regular polychora, but given the context of mathematical art and VR research the authors are immersed in, it seemed obvious, even necessary, to design exactly this game.

As for me, I’m curious to see if you can make a game where you only use a few cells, and maybe the aim is to travel from one to the other in a certain order. All the source code to Hypernom is available on GitHub, so maybe I’ll have a go at modifying it to do that.

Explanation, instructions and source code on the Hypernom GitHub page

Hypernom: Mapping VR Headset Orientation to $S^3$ – paper by Vi Hart, Andrea Hawksley, Henry Segerman and Marc ten Bosch.

Sculptures in $S^3$ – paper by Saul Schleimer and Henry Segerman

]]>This week, we’re investigating the Millennium Prize Problems – a set of mathematical equations that, if solved, will not only nab the lucky winner a million, but also revolutionise the world. Plus, the headlines from the world of science and technology, including why screams are so alarming, how fat fish help the human fight against flab, and what’s the future of money?

Better yet, the episode includes a contribution from our very own Katie Steckles talking topology, Poincaré and Perelman.

The episode is available to listen or download as a podcast or, less conveniently, at 5am tomorrow on Radio 5 Live (or later on iPlayer). Not a listener? Read a transcript.

]]>One of her favourite modes of attack is the “30 Second Challenge” from the Daily Mail. It looks like this:

You start with the number on the left, then follow the instructions reading right until you get to the answer at the end. It’s one of Grandma’s favourites because it’s very hard to do in your head when she’s just reading it out!

I decided it would be a fun Sunday morning mental excursion to make a random 30 second challenge generator.

Making a random challenge generator involves thinking about what the space of possible challenges is, and how to pick fairly from them corresponding to different difficulty levels. The strategy I came up with is for each difficulty level to have a pool of possible operations, and to pick at random from those for each step. An operation can involve more than one step, and each operation has a function which looks at the current state of the puzzle to decide if it can be applied.

As far as I know, each operation must leave you with a whole number. That means that a “divide by $N$” instruction can only appear when your number is divisible by $N$. Since some divisors are much more common than others, but I wanted to have a good distribution of numbers to divide by, I made the divide by” operation pick a number $N$, and then add a step to add or subtract the right amount to get to a multiple of $N$, before adding the step to divide.

Some pairs of operations shouldn’t appear next to each other – you shouldn’t get an “add” followed by a “subtract”, or a “halve it” followed by a “double it”.

With a rough system of making valid challenges in place, I needed to make three difficulty levels. My rough rule of thumb was that dividing is really hard, cubing is hard because it leads to big numbers, and adding and subtracting are quite easy. I could probably split the additions or multiplications into easier or harder versions – there’s some evidence that the 6 and 8 times tables are hardest – and add some more complicated operations like “square root of this”. At the moment, I don’t feel like the difficulty levels are consistent enough: sometimes you’ll get a really easy “hard” challenge, and sometimes you’ll get a pretty tricky “easy” one.

Finally, I decided to look at accessibility. Grandma continues to be unsatisfied with me because her son-in-law, who’s partially sighted, always solves the challenges much quicker than I can. She reads the steps out and he does the calculations in his head. With that in mind, I made sure the challenge is usable when you can’t see it. Thanks to modern web standards, that was easy – I set `tabindex=1`

on the step elements, so that you can navigate between them by pressing tab, and made sure all of the instructions make sense when read out by a screen reader: I had to add an `aria-label`

attribute to the fraction instruction with some alternate text to read out instead of the fancy formatting I use in the visual version. I tested it all with the ChromeVox extension, which works pretty well and is very easy to set up (arguably too easy – it went a bit mad reading every other tab I had open while I was testing).

In the end, this is how the finished game looks:

It was a fun excursion, and the game is pretty addictive. You can play it at christianp.github.io/30secondchallenge. Post your average times and record streaks in the comments!

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