Happy New Year! And welcome to the first Carnival of Mathematics of 2014. The Carnival is a monthly roundup of blog posts on or related to mathematics, from all over the internet. Posts are submitted by authors and readers, and collated by the host, whose blog it’s posted on. This month, the Carnival has pulled in here at The Aperiodical, and we’re all ready with our party hats for the celebration of mathematical blogging that implies.

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### From the Mailbag: Dual Inversal Numbers

*Katie, one of our editors, has been contacted by Brendan with a question about some maths he’s been investigating. Read on to find out what he’s discovered, and read Katie’s response.*

Dear The Aperiodical,

I’ve noticed an interesting property of numbers, and I wondered if you could tell me if this is something which is already known to mathematicians? I’ve been calling them Dual Inversal Numbers, but I’d love to know if they have an existing name, and if there’s anything else you can tell me about them.

### Open Season: Prime Numbers (part 2)

*In this short series of articles, I’m writing about mathematical questions we don’t know the answer to – which haven’t yet been proven or disproven. This is the third article in the series, and across two parts will discuss various open conjectures relating to prime numbers. *This follows on from Open Season: Prime numbers (part 1).

So, we have a pretty good handle on how prime numbers are defined, how many of them there are, and how to check whether a number is prime. But what don’t we know? It turns out, quite a lot.

### Open Season: Prime Numbers (Part 1)

*In this short series of articles, I’m writing about mathematical questions we don’t know the answer to – which haven’t yet been proven or disproven. This is the third article in the series, and across two parts will discuss various open conjectures relating to prime numbers.*

I don’t think it’s too much of an overstatement to say that prime numbers are the building blocks of numbers. They’re the atoms of maths. They are the beginning of all number theory. I’ll stop there, before I turn into Marcus Du Sautoy, but I do think they’re pretty cool numbers. They crop up in a lot of places in maths, they’re used for all kinds of cool spy-type things including RSA encryption, and even cicadas have got in on the act (depending on who you believe).

### Maths at the Fringe

Starting next week, the historic city of Edinburgh will be taken over by entertainers of all types, performing comedy, dance, theatre and music, entertaining visitors to their massive world-famous festival fringe. Since discerning mathematicians sometimes also enjoy being entertained, I thought I’d write a roundup of the shows maths has non-empty intersection with.

First up, since we haven’t mentioned him in a while, it’s Alan Turing! No, his reanimated corpse isn’t performing edgy stand-up, but theatre company Idle Motion is performing a visual theatre piece entitled **That Is All You Need To Know**, celebrating the work of Bletchley Park codebreakers. Alan Turing Alan Turing Alan Turing.

### How to Win at Pointless

For the benefit of overseas readers, or British readers in full-time employment, I should briefly explain the concept of daytime TV quiz phenomenon Pointless. The pinnacle of British public service broadcasting, it’s shown at 5.15pm every weekday on BBC One and is hosted by Alexander Armstrong of comedy double-act Armstrong & Miller, and Richard Osman of comedy double-act Armstrong & Osman. We shall investigate how we can use maths to analyse the show, improve our chances of winning it, and ultimately perhaps improve the show itself.

The aim of the game is in each round to give the most obscure correct answer to a given question. Each question ($Q$) has a large set of valid answers $A_Q$, questions perhaps asking contestants to name “Films starring Bruce Willis” or “Countries without an O in their name”. All the questions have been asked to 100 members of the public prior to the quiz (call this set $P$), and they each have 100 seconds to name as many examples as they can (giving rise to a set $A_p\subseteq A_Q$ for each $p\in P$. The contestant gets a point for every one of the 100 people who named their answer $a$:

\[ \mbox{score}(a) = \begin{cases}

| \{p\in P : a\in A_p \} | & \mbox{if}\ a\in A_Q \\

100 & \mbox{if}\ a\not\in A_Q.

\end{cases} \]

So an obvious answer like Die Hard or France will score a lot of points, and an obscure answer like Striking Distance or Central African Republic will score fewer points. Points are bad (hence the title) so it’s better to dredge up an obscure answer than stick with something safe. However an incorrect answer like Avatar or Mexico scores the maximum 100 points. At the end of the round the contestant with the most points is eliminated.

### 2² number theory tricks

Here are some nice number facts and tricks you can try out on your friends. They will work without understanding how, but with a little investigation you should be able to figure out how each one works.